Article Sources and Contributors

Contents
Articles
Numerical cognition
1
Subitizing
5
Approximate number system
8
Estimation
11
Addition
13
Subtraction
32
Numerosity adaptation effect
42
Number sense
44
Ordinal numerical competence
45
References
Article Sources and Contributors
48
Image Sources, Licenses and Contributors
49
Article Licenses
License
51
Numerical cognition
1
Numerical cognition
Cognitive
psychology
Perception
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Visual perception
Object recognition
Face recognition
Pattern recognition
Attention
Memory
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Aging and memory
Emotional memory
Learning
Long-term memory
Metacognition
Language
Metalanguage
Thinking
Cognition
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Concept
Reasoning
Decision making
Problem solving
Numerical cognition
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Numerosity adaptation effect
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v
t
e [1]
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural
bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary
topic, and includes researchers in cognitive psychology, developmental psychology, neuroscience and cognitive
linguistics. This discipline, although it may interact with questions in the philosophy of mathematics is primarily
concerned with empirical questions.
Topics included in the domain of numerical cognition include:
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How do non-human animals process numerosity?
How do infants acquire an understanding of numbers (and how much is inborn)?
How do humans associate linguistic symbols with numerical quantities?
How do these capacities underlie our ability to perform complex calculations?
What are the neural bases of these abilities, both in humans and in non-humans?
• What metaphorical capacities and processes allow us to extend our numerical understanding into complex
domains such as the concept of infinity, the infinitesimal or the concept of the limit in calculus?
Numerical cognition
Comparative studies
A variety of research has demonstrated that non-human animals, including rats, lions and various species of primates
have an approximate sense of number (referred to as "numerosity") (for a review, see Dehaene 1997). For example,
when a rat is trained to press a bar 8 or 16 times to receive a food reward, the number of bar presses will approximate
a Gaussian or Normal distribution with peak around 8 or 16 bar presses. When rats are more hungry, their bar
pressing behavior is more rapid, so by showing that the peak number of bar presses is the same for either well-fed or
hungry rats, it is possible to disentangle time and number of bar presses.
Similarly, researchers have set up hidden speakers in the African savannah to test natural (untrained) behavior in
lions (McComb, Packer & Pusey 1994). These speakers can play a number of lion calls, from 1 to 5. If a single
lionness hears, for example, three calls from unknown lions, she will leave, while if she is with four of her sisters,
they will go and explore. This suggests that not only can lions tell when they are "outnumbered" but that they can do
this on the basis of signals from different sensory modalities, suggesting that numerosity is a multisensory concept.
Developmental studies
Developmental psychology studies have shown that human infants, like non-human animals, have an approximate
sense of number. For example, in one study, infants were repeatedly presented with arrays of (in one block) 16 dots.
Careful controls were in place to eliminate information from "non-numerical" parameters such as total surface area,
luminance, circumference, and so on. After the infants had been presented with many displays containing 16 items,
they habituated, or stopped looking as long at the display. Infants were then presented with a display containing 8
items, and they looked longer at the novel display.
Because of the numerous controls that were in place to rule out non-numerical factors, the experimenters infer that
six-month-old infants are sensitive to differences between 8 and 16. Subsequent experiments, using similar
methodologies showed that 6-month-old infants can discriminate numbers differing by a 2:1 ratio (8 vs. 16 or 16 vs.
32) but not by a 3:2 ratio (8 vs. 12 or 16 vs. 24). However, 10-month-old infants succeed both at the 2:1 and the 3:2
ratio, suggesting an increased sensitivity to numerosity differences with age (for a review of this literature see
Feigenson, Dehaene & Spelke 2004).
In another series of studies, Karen Wynn showed that infants as young as five months are able to do very simple
additions (e.g., 1 + 1 = 2) and subtractions (3 - 1 = 2). To demonstrate this, Wynn used a "violation of expectation"
paradigm, in which infants were shown (for example) one Mickey Mouse doll going behind a screen, followed by
another. If, when the screen was lowered, infants were presented with only one Mickey (the "impossible event") they
looked longer than if they were shown two Mickeys (the "possible" event). Further studies by Karen Wynn and
Koleen McCrink found that although infants' ability to compute exact outcomes only holds over small numbers,
infants can compute approximate outcomes of larger addition and subtraction events (e.g., "5+5" and "10=5" events).
There is debate about how much these infant systems actually contain in terms of number concepts, harkening to the
classic nature versus nurture debate. Gelman & Gallistel 1978 suggested that a child innately has the concept of
natural number, and only has to map this onto the words used in her language. Carey 2004, Carey 2009 disagreed,
saying that these systems can only encode large numbers in an approximate way, where language-based natural
numbers can be exact. One promising approach is to see if cultures that lack number words can deal with natural
numbers. The results so far are mixed (e.g., Pica et al. 2004); Butterworth & Reeve 2008, Butterworth, Reeve &
Lloyd 2008.
2
Numerical cognition
Neuroimaging and neurophysiological studies
Human neuroimaging studies have demonstrated that regions of the parietal lobe, including the intraparietal sulcus
(IPS) and the inferior parietal lobule (IPL) are activated when subjects are asked to perform calculation tasks. Based
on both human neuroimaging and neuropsychology, Stanislas Dehaene and colleagues have suggested that these two
parietal structures play complementary roles. The IPS is thought to house the circuitry that is fundamentally involved
in numerical estimation (Piazza et al. 2004), number comparison (Pinel et al. 2001; Pinel et al. 2004) and on-line
calculation (often tested with subtraction) while the IPL is thought to be involved in overlearned tasks, such as
multiplication (see Dehaene 1997). Thus, a patient with a lesion to the IPL may be able to subtract, but not multiply,
and vice versa for a patient with a lesion to the IPS. In addition to these parietal regions, regions of the frontal lobe
are also active in calculation tasks. These activations overlap with regions involved in language processing such as
Broca's area and regions involved in working memory and attention. Future research will be needed to disentangle
the complex influences of language, working memory and attention on numerical processes.
Single-unit neurophysiology in monkeys has also found neurons in the frontal cortex and in the intraparietal sulcus
that respond to numbers. Andreas Nieder (Nieder 2005; Nieder, Freedman & Miller 2002; Nieder & Miller 2004)
trained monkeys to perform a "delayed match-to-sample" task. For example, a monkey might be presented with a
field of four dots, and is required to keep that in memory after the display is taken away. Then, after a delay period
of several seconds, a second display is presented. If the number on the second display match that from the first, the
monkey has to release a lever. If it is different, the monkey has to hold the lever. Neural activity recorded during the
delay period showed that neurons in the intraparietal sulcus and the frontal cortex had a "preferred numerosity",
exactly as predicted by behavioral studies. That is, a certain number might fire strongly for four, but less strongly for
three or five, and even less for two or six. Thus, we say that these neurons were "tuned" for specific numerosities.
Note that these neuronal responses followed Weber's law, as has been demonstrated for other sensory dimensions,
and consistent with the ratio dependence observed for non-human animals' and infants' numerical behavior (Nieder
& Miller 2003).
Relations between number and other cognitive processes
There is evidence that numerical cognition is intimately related to other aspects of thought – particularly spatial
cognition. One line of evidence comes from studies performed on number-form synaesthetes. Such individuals report
that numbers are mentally represented with a particular spatial layout; others experience numbers as perceivable
objects that can be visually manipulated to facilitate calculation. Behavioral studies further reinforce the connection
between numerical and spatial cognition. For instance, participants respond quicker to larger numbers if they are
responding on the right side of space, and quicker to smaller numbers when on the left—the so-called
"Spatial-Numerical Association of Response Codes" or SNARC effect. This effect varies across culture and context,
however, and some research has even begun to question whether the SNARC reflects an inherent number-space
association, instead invoking strategic problem solving or a more general cognitive mechanism like conceptual
metaphor. Moreover, neuroimaging studies reveal that the association between number and space also shows up in
brain activity. Regions of the parietal cortex, for instance, show shared activation for both spatial and numerical
processing. These various lines of research suggest a strong, but flexible, connection between numerical and spatial
cognition.
Modification of the usual decimal representation was advocated by John Colson. The sense of complementation,
missing in the usual decimal system, is expressed by signed-digit representation.
3
Numerical cognition
Ethnolinguistic variance
The numeracy of indigenous peoples is studied to identify universal aspects of numerical cognition in humans.
Notable examples include the Pirahã people who have no words for specific numbers and the Munduruku people
who only have number words up to five. Pirahã adults are unable to mark an exact number of tallies for a pile of nuts
containing fewer than ten items. Anthropologist Napoleon Chagnon spent several decades studying the Yanomami in
the field. He concluded that they have no need for counting in their everyday lives. Their hunters keep track of
individual arrows with the same mental faculties that they use to recognize their family members. There are no
known hunter-gatherer cultures that have a counting system in their language. The mental and lingual capabilities for
numeracy are tied to the development of agriculture and with it large numbers of indistinguishable items.
Notes
[1] http:/ / en. wikipedia. org/ w/ index. php?title=Template:Cognitive& action=edit
References
• Butterworth, B.; Reeve, R. (2008), "Verbal Counting and Spatial Strategies in Numerical Tasks : Evidence From
Indigenous Australia", Philosophical Psychology 4 (21): 443–457
• Butterworth, Brian; Reeve, Robert; Reynolds, Fiona; Lloyd, Delyth (2008). "Numerical thought with and without
words: Evidence from indigenous Australian children" (http://www.pnas.org/content/105/35/13179).
Proceedings of the National Academy of Sciences 105 (35): 13179–13184. doi: 10.1073/pnas.0806045105 (http://
dx.doi.org/10.1073/pnas.0806045105).
• Carey, S. (2004), "Bootstrapping and the origins of Concepts", Daedalus: 59–68
• Carey, S. (2009), "Where our number concepts come from", Journal of Philosophy 106 (4): 220–254
• Dehaene, S. (1997), The number sense: How the mind creates mathematics (http://www.unicog.org), New
York: Oxford University Press, ISBN 0-19-513240-8
• Feigenson, L.; Dehaene, S.; Spelke, E. (2004), "Core systems of number", Trends in Cognitive Science 8 (7):
307–314
• Gelman, R.; Gallistel, G. (1978), The Child's Understanding of Number, Cambridge Mass: Harvard University
Press
• Lakoff, G.; Nuñez, R. E. (2000), Where mathematics comes from, New York: Basic Books., ISBN 0-465-03770-4
• McComb, K.; Packer, C.; Pusey, A. (1994), "Roaring and numerical assessment in contests between groups of
female lions, Panthera leo", Animal Behavior 47: 379–387
• Nieder, A. (2005), "Counting on neurons: The neurobiology of numerical competence", Nature Reviews
Neuroscience 6: 177–190
• Nieder, A.; Freedman, D. J.; Miller, E. K. (2002), "Representation of the quantity of visual items in the primate
prefrontal cortex", Science 297: 1708–1711
• Nieder, A.; Miller, E. K. (2003), "Coding of cognitive magnitude: Compressed scaling of numerical information
in the primate prefrontal cortex", Neuron 37: 149–157
• Nieder, A.; Miller, E. K. (2004), "A parieto-frontal network for visual numerical information in the monkey",
Proceedings of the National Academy of Sciences 101: 7457–7462
• Piazza, M.; Izard, V.; Pinel, P.; Le Bihan, D.; Dehaene, S. (2004), "Tuning curves for approximate numerosity in
the human intraparietal sulcus", Neuron 44: 547–555
• Pica, P.; Lemer, C.; Izard, V.; Dehaene, S. (2004), "Exact an Approximate Arithmetic in an Amazonian Indigene
Group", Science 306 (5695): 499–503
• Pinel, P.; Dehaene, S.; Riviere, D.; Le Bihan, D. (2001), "Modulation of parietal activation by semantic distance
in a number comparison task", Neuroimage 14 (5): 1013–1026
4
Numerical cognition
• Pinel, P.; Piazza, M.; Le Bihan, D.; Dehaene, S. (2004), "Distributed and overlapping cerebral representations of
number, size, and luminance during comparative judgments", Neuron 41 (6): 983–993
Subitizing
Subitizing, coined in 1949 by E.L. Kaufman et al. refers to
the rapid, accurate, and confident judgments of number
performed for small numbers of items. The term is derived
from the Latin adjective subitus (meaning "sudden") and
captures a feeling of immediately knowing how many items
lie within the visual scene, when the number of items present
falls within the subitizing range. Number judgments for
larger set-sizes were referred to either as counting or
estimating, depending on the number of elements present
within the display, and the time given to observers in which
to respond (i.e., estimation occurs if insufficient time is
available for observers to accurately count all the items
present).
The accuracy, speed, and confidence with which observers
make judgments of the number of items are critically
dependent on the number of elements to be enumerated.
Judgments made for displays composed of around one to
four items are rapid, accurate and confident. However, as the
number of items to be enumerated increases beyond this
amount, judgments are made with decreasing accuracy and
Counting or subitizing?
confidence. In addition, response times rise in a dramatic
fashion, with an extra 250–350 ms added for each additional item within the display beyond about four.
While the increase in response time for each additional element within a display is relatively large outside the
subitizing range (i.e., 250–350 ms per item), there is still a significant, albeit smaller, increase within the subitizing
range, for each additional element within the display (i.e., 40–100 ms per item). A similar pattern of reaction times is
found in young children, although with steeper slopes for both the subitizing range and the enumeration range. This
suggests there is no span of apprehension as such, if this is defined as the number of items which can be immediately
apprehended by cognitive processes, since there is an extra cost associated with each additional item enumerated.
However, the relative difference in costs associated with enumerating items within the subitizing range are small,
whether measured in terms of accuracy, confidence, or speed of response. Furthermore, the values of all measures
appear to differ markedly inside and outside the subitizing range. So, while there may be no span of apprehension,
there appear to be real differences in the ways in which a small number of elements is processed by the visual system
(i.e., approximately less than four items), compared with larger numbers of elements (i.e., approximately more than
four items). A 2006 study demonstrated that subitizing and counting are not restricted to visual perception, but also
extend to tactile perception, when observers had to name the number of stimulated fingertips.
5
Subitizing
Enumerating afterimages
As the derivation of the term "subitizing" suggests, the
feeling associated with making a number judgment within
the subitizing range is one of immediately being aware of the
displayed elements. When the number of objects presented
exceeds the subitizing range, this feeling is lost, and
observers commonly report an impression of shifting their
viewpoint around the display, until all the elements presented
have been counted. The ability of observers to count the
number of items within a display can be limited, either by the
rapid presentation and subsequent masking of items, or by
Numbers 1–10
requiring observers to respond quickly. Both procedures
have little, if any, effect on enumeration within the subitizing range. These techniques may restrict the ability of
observers to count items by limiting the degree to which observers can shift their "zone of attention" successively to
different elements within the display.
Atkinson, Campbell, and Francis demonstrated that visual afterimages could be employed in order to achieve similar
results. Using a flashgun to illuminate a line of white disks, they were able to generate intense afterimages in
dark-adapted observers. Observers were required to verbally report how many disks had been presented, both at 10 s
and at 60 s after the flashgun exposure. Observers reported being able to see all the disks presented for at least 10 s,
and being able to perceive at least some of the disks after 60 s. Despite a long period of time to enumerate the
number of disks presented when the number of disks presented fell outside the subitizing range (i.e., 5–12 disks),
observers made consistent enumeration errors in both the 10 s and 60 s conditions. In contrast, no errors occurred
within the subitizing range (i.e., 1–4 disks), in either the 10 s or 60 s conditions.
Brain structures involved in subitizing and counting
The work on the enumeration of afterimages supports the view that different cognitive processes operate for the
enumeration of elements inside and outside the subitizing range, and as such raises the possibility that subitizing and
counting involve different brain circuits. However, functional imaging research has been interpreted both to support
different and shared processes.
Balint's syndrome
Clinical evidence supporting the view that subitizing and counting may involve functionally and anatomically
distinct brain areas comes from patients with simultanagnosia, one of the key components of Balint's syndrome.
Patients with this disorder suffer from an inability to perceive visual scenes properly, being unable to localize objects
in space, either by looking at the objects, pointing to them, or by verbally reporting their position. Despite these
dramatic symptoms, such patients are able to correctly recognize individual objects. Crucially, people with
simultanagnosia are unable to enumerate objects outside the subitizing range, either failing to count certain objects,
or alternatively counting the same object several times.
However, people with simultanagnosia have no difficulty enumerating objects within the subitizing range. The
disorder is associated with bilateral damage to the parietal lobe, an area of the brain linked with spatial shifts of
attention. These neuropsychological results are consistent with the view that the process of counting, but not that of
subitizing, requires active shifts of attention. However, recent research has questioned this conclusion by finding that
attention also affects subitizing.
6
Subitizing
Imaging enumeration
A further source of research upon the neural processes of subitizing compared to counting comes from positron
emission tomography (PET) research upon normal observers. Such research compares the brain activity associated
with enumeration processes inside (i.e., 1–4 items) for subitizing, and outside (i.e., 5–8 items) for counting.
Such research finds that within the subitizing and counting range activation occurs bilaterally in the occipital
extrastriate cortex and superior parietal lobe/intraparietal sulcus. This has been interpreted as evidence that shared
processes are involved. However, the existence of further activations during counting in the right inferior frontal
regions, and the anterior cingulate have been interpreted as suggesting the existence of distinct processes during
counting related to the activation of regions involved in the shifting of attention.
Educational applications
Historically, many systems have attempted to use subitizing to identify full or partial quantities. In the twentieth
century, mathematics educators started to adopt some of these systems, as reviewed in examples below, but often
switched to more abstract color-coding to represent quantities up to ten.
Aleister Crowley advocated subitizing in 1913 in Liber Batrachophrenoboocosmomachia, published in The
Equinox. In the nineties, babies three weeks old were shown to differentiate between 1-3 objects, that is, to subitize.
A more recent meta-study summarizing five different studies concluded that infants are born with an innate ability to
differentiate quantities within a small range, which increases over time. By the age of seven that ability increases to
4-7 objects. Some practitioners claim that with training, children are capable of subitizing 15+ objects correctly [1]
Abacus
The hypothesized use of yupana, an Inca counting system, placed up to five counters in connected trays for
calculations.
In each place value, the Chinese abacus uses four or five beads to represent units, which are subitized, and one or
two separate beads, which symbolize fives. This allows multi-digit operations such as carrying and borrowing to
occur without subitizing beyond five.
European abacuses use ten beads in each register, but usually separate them into fives by color.
Dice and cards
Dice, playing cards and other gaming devices traditionally split quantities into subitizable groups with recognizable
patterns.
Twentieth century teaching tools
The idea of instant recognition of quantities has been adopted by several pedagogical systems, such as Montessori,
Cuisenaire and Dienes. However, these systems only partially use subitizing, attempting to make all quantities from
1 to 10 instantly recognizable. To achieve it, they code quantities by color and length of rods or bead strings
representing them. Recognizing such visual or tactile representations and associating quantities with them involves
different mental operations from subitizing.
7
Subitizing
References
[1] http:/ / www. brillkids. com/ teach-math/ what-does-a-lesson-look-like. php
Approximate number system
The approximate number system (ANS) is a cognitive system that supports the estimation of the magnitude of a
group without relying on language or symbols. The ANS is credited with the non-symbolic representation of all
numbers greater than four, with lesser values being carried out by the parallel individuation system, or object
tracking system. Beginning in early infancy, the ANS allows an individual to detect differences in magnitude
between groups. The precision of the ANS improves throughout childhood development and reaches a final adult
level of approximately 15% accuracy, meaning an adult could distinguish 100 items versus 115 items without
counting. The ANS plays a crucial role in development of other numerical abilities, such as the concept of exact
number and simple arithmetic. The precision level of a child's ANS has been shown to predict mathematical
achievement in school. The ANS has been linked to the intraparietal sulcus of the brain.
History
Piaget's theory
Jean Piaget was a Swiss developmental psychologist who devoted much of his life to studying how children learn. A
book summarizing his theories on number cognition, The Child’s Conception of Number, was published in 1952.
Piaget’s work supported the viewpoint that children do not have a stable representation of number until the age of six
or seven. His theories indicate that mathematical knowledge is slowly gained and during infancy any concept of sets,
objects, or calculation is absent.
Challenging the Piagetian viewpoint
Piaget’s ideas pertaining to the absence of mathematical cognition at birth have been steadily challenged. The work
of Rochel Gelman and C. Randy Gallistel among others in the 1970s suggested that preschoolers have intuitive
understanding of the quantity of a set and its conservation under non cardinality-related changes, expressing surprise
when objects disappear without an apparent cause.
Current theory
Beginning as infants, people have an innate sense of approximate number that depends on the ratio between sets of
objects. Throughout life the ANS becomes more developed, and people are able to distinguish between groups
having smaller differences in magnitude. The ratio of distinction is defined by Weber’s law, which relates the
different intensities of a sensory stimuli that is being evaluated. In the case of the ANS, as the ratio between the
magnitudes increases, the ability to discriminate between the two quantities increases.
Today, it is theorized that the ANS lays the foundation for higher-level arithmetical concepts. Research has shown
that the same areas of the brain are active during non-symbolic number tasks in infants and both non-symbolic and
more sophisticated symbolic number tasks in adults. These results suggest that the ANS contributes over time to the
development of higher-level numerical skills that activate the same part of the brain.
8
Approximate number system
Neurological basis
Brain imaging studies have identified the parietal lobe as being a key brain region for numerical cognition.
Specifically within this lobe is the intraparietal sulcus which is "active whenever we think about a number, whether
spoken or written, as a word or as an Arabic digit, or even when we inspect a set of objects and think about its
cardinality". When comparing groups of objects, activation of the intraparietal sulcus is greater when the difference
between groups is numerical rather than an alternative factor, such as differences in shape or size. This indicates that
the intraparietal sulcus plays an active role when the ANS is employed to approximate magnitude.
Parietal lobe brain activity seen in adults is also observed during infancy during non-verbal numerical tasks,
suggesting that the ANS is present very early in life. A neuroimaging technique, functional Near-Infrared
Spectroscopy, was performed on infants revealing that the parietal lobe is specialized for number representation
before the development of language. This indicates that numerical cognition may be initially reserved to the right
hemisphere of the brain and becomes bilateral through experience and the development of complex number
representation.
It has been shown that the intraparietal sulcus is activated independently of the type of task being performed with the
number. The intensity of activation is dependent on the difficulty of the task, with the intraparietal sulcus showing
more intense activation when the task is more difficult. In addition, studies in monkeys have shown that individual
neurons can fire preferentially to certain numbers over others. For example, a neuron could fire at maximum level
every time a group of four objects is seen, but will fire less to a group three or five objects.
Pathology
Damage to intraparietal sulcus
Damage done to parietal lobe, specifically in the left hemisphere, can produced difficulties in counting and other
simple arithmetic. Damage directly to the intraparietal sulcus has been shown to cause acalculia, a severe disorder in
mathematical cognition. Symptoms vary based the location of damage, but can include the inability to perform
simple calculations or to decide that one number is larger than another. Gerstmann syndrome, a disease resulting in
lesions in the left parietal and temporal lobes, results in acalculia symptoms and further confirms the importance of
the parietal region in the ANS.
Developmental delays
A syndrome known as dyscalculia is seen in individuals who have unexpected difficulty understanding numbers and
arithmetic despite adequate education and social environments. This syndrome can manifest in several different ways
from the inability to assign a quantity to Arabic numerals to difficulty with times tables. Dyscalculia can result in
children falling significantly behind in school, regardless of having normal intelligence levels.
In some instances, such as Turner syndrome, the onset of dyscalculia is genetic. Morphological studies have revealed
abnormal lengths and depths of the right intraparietal sulcus in individuals suffering from Turner syndrome. Brain
imaging in children exhibiting symptoms of dyscalculia show less gray matter or less activation in the intraparietal
regions stimulated normally during mathematical tasks.
9
Approximate number system
Further research and theories
Impact of the visual cortex
The intraparietal region relies on several other brain systems to accurately perceive numbers. When using the ANS
we must view the sets of objects in order to evaluate their magnitude. The primary visual cortex is responsible for
disregarding irrelevant information, such as the size or shape of the objects. Certain visual cues can sometimes affect
how the ANS functions.
Arranging the items differently can alter the effectiveness of the ANS. One arrangement proven to influence the
ANS is visual nesting, or placing the objects within one another. This configuration affects the ability to distinguish
each item and add them together at the same time. The difficulty results in underestimation of the magnitude present
in the set or a longer amount of time needed to perform an estimate.
Another visual representation that affects the ANS is the spatial-numerical association response code, or the SNARC
effect. The SNARC effect details the tendency of larger numbers to be responded to faster by the right hand and
lower numbers by the left hand, suggesting that the magnitude of a number is linked to a spatial representation.
Dehaene and other researchers believe this effect is caused by the presence of a “mental number line” in which small
numbers appear on the left and increase as you move right. The SNARC effect indicates that the ANS works more
effectively and accurately if the larger set of objects is on the right and the smaller on the left.
Development and mathematical performance
Although the ANS is present in infancy before any numerical education, research has shown a link between people’s
mathematical abilities and the accuracy in which they approximate the magnitude of a set. This correlation is
supported by several studies in which school-aged children’s ANS abilities are compared to their mathematical
achievements. At this point the children have received training in other mathematical concepts, such as exact number
and arithmetic. More surprisingly, ANS precision before any formal education accurately predicts better math
performance. A study involving 3-5 year old children revealed that ANS acuity corresponds to better mathematical
cognition while remaining independent of factors that may interfere, such as reading ability and the use of Arabic
numerals.
ANS in animals
Many species of non-human animals exhibit the ability to assess and compare magnitude. This skill is believed to be
a product of the ANS. Research has revealed this capability in both vertebrate and non-vertebrate animals including
birds, mammals, fish, and even insects. In primates, implications of the ANS have been steadily observed through
research. One study involving lemurs showed that they were able to distinguish groups of objects based only on
numerical differences, suggesting that humans and other primates utilize a similar numerical processing mechanism.
In a study comparing students to guppies, both the fish and students performed the numerical task almost identically.
The ability for the test groups to distinguish large numbers was dependent on the ratio between them, suggesting the
ANS was involved. Such results seen when testing guppies indicate that the ANS may have been evolutionarily
passed down through many species.
10
Approximate number system
Applications in society
Implications for the classroom
Understanding how the ANS affects students' learning could be beneficial for teachers and parents. The following
tactics have been suggested by neuroscientists to utilize the ANS in school:
•
•
•
•
Counting or abacus games
Simple board games
Computer-based number association games
Teacher sensitivity and different teaching methods for different learners
Such tools are most helpful in training the number system when the child is at an earlier age. Children coming from a
disadvantaged background with risk of arithmetic problems are especially impressionable by these tactics.
References
Estimation
Estimation is the process of finding an estimate, or approximation,
which is a value that is usable for some purpose even if input data may
be incomplete, uncertain, or unstable. The value is nonetheless usable
because it is derived from the best information available.[1] Typically,
estimation involves "using the value of a statistic derived from a
sample to estimate the value of a corresponding population
parameter".[2] The sample provides information that can be projected,
through various formal or informal processes, to determine a range
most likely to describe the missing information. An estimate that turns
out to be incorrect will be an overestimate if the estimate exceeded the
actual result,[3] and an underestimate if the estimate fell short of the
actual result.[4]
How estimation is done
Estimation is often done by sampling, which is counting a small
number of examples something, and projecting that number onto a
The exact number of candies in this jar can not be
larger population. An example of estimation would be determining
determined by looking at it, because most of the
how many candies of a given size are in a glass jar. Because the
candies are not visible. The amount can be
distribution of candies inside the jar may vary, the observer can count
estimated by presuming that the portion of the jar
that cannot be seen contains an amount
the number of candies visible through the glass, consider the size of the
equivalent to the amount contained in the same
jar, and presume that a similar distribution can be found in the parts
volume for the portion that can be seen.
that can not be seen, thereby making an estimate of the total number of
candies that could be in the jar if that presumption were true. Estimates
can similarly be generated by projecting results from polls or surveys onto the entire population.
In making an estimate, the goal is often most useful to generate a range of possible outcomes that is precise enough
to be useful, but not so precise that it is likely to be inaccurate. For example, in trying to guess the number of candies
in the jar, if fifty were visible, and the total volume of the jar seemed to be about twenty times as large as the volume
11
Estimation
containing the visible candies, then one might simply project that there were a thousand candies in the jar. Such a
projection, intended to pick the single value that is believed to be closest to the actual value, is called a point
estimate. However, a point estimation is likely to be incorrect, because the sample size - in this case, the number of
candies that are visible - is too small a number to be sure that it does not contain anomalies that differ from the
population as a whole. A corresponding concept is an interval estimate, which captures a much larger range of
possibilities, but is too broad to be useful. For example, if one were asked to estimate the percentage of people who
like candy, it would clearly be correct that the number falls between zero and one hundred percent. Such an estimate
would provide no guidance, however, to somebody who is trying to determine how many candies to buy for a party
to be attended by a hundred people.
Uses of estimation
In mathematics, approximation describes the process of finding estimates in the form of upper or lower bounds for a
quantity that cannot readily be evaluated precisely, and approximation theory deals with finding simpler functions
that are close to some complicated function and that can provide useful estimates. In statistics, an estimator is the
formal name for the rule by which an estimate is calculated from data, and estimation theory deals with finding
estimates with good properties. This process is used in signal processing, for approximating an unobserved signal on
the basis of an observed signal containing noise. For estimation of yet-to-be observed quantities, forecasting and
prediction are applied. A Fermi problem, in physics, is one concerning estimation in problems which typically
involve making justified guesses about quantities that seem impossible to compute given limited available
information.
Estimation is important in business and economics, because too many variables exist to determine how large-scale
activities will develop. Estimation in project planning can be particularly significant, because plans for the
distribution of labor and for purchases of raw materials must be made, despite the inability to know every possible
problem that may come up. Furthermore, such plans must not underestimate the needs of the project, which can
result in delays while unmet needs are fulfilled, nor must they greatly overestimate the needs of the project, or else
the unneeded resources may go to waste.
An informal estimate when little information is available is called a guesstimate, because the inquiry becomes closer
to purely guessing the answer. The "estimated" sign, ℮, is used to designate that package contents are close to the
nominal contents.
References
[1] C. Lon Enloe, Elizabeth Garnett, Jonathan Miles, Physical Science: What the Technology Professional Needs to Know (2000), p. 47.
[2] Raymond A. Kent, "Estimation", Data Construction and Data Analysis for Survey Research (2001), p. 157.
[3] James Tate, John Schoonbeck, Reviewing Mathematics (2003), page 27: "An overestimate is an estimate you know is greater than the exact
answer".
[4] James Tate, John Schoonbeck, Reviewing Mathematics (2003), page 27: "An underestimate is an estimate you know is less than the exact
answer".
External links
• Estimation chapter from "Applied Software Project Management" (PDF) (http://www.stellman-greene.com/
aspm/images/ch03.pdf)
12
Addition
13
Addition
Addition is a mathematical operation that represents the total amount
of objects together in a collection. It is signified by the plus sign (+).
For example, in the picture on the right, there are 3 + 2
apples—meaning three apples and two apples together, which is a total
of 5 apples. Therefore, 3 + 2 = 5. Besides counting fruits, addition can
also represent combining other physical and abstract quantities using
different kinds of objects: negative numbers, fractions, irrational
numbers, vectors, decimals, functions, matrices and more.
Addition follows several important patterns. It is commutative,
meaning that order does not matter, and it is associative, meaning that
when one adds more than two numbers, the order in which addition is
performed does not matter (see Summation). Repeated addition of 1 is
the same as counting; addition of 0 does not change a number.
Addition also obeys predictable rules concerning related operations
such as subtraction and multiplication. All of these rules can be proven,
starting with the addition of natural numbers and generalizing up
through the real numbers and beyond. General binary operations that
continue these patterns are studied in abstract algebra.
3 + 2 =5 with apples, a popular
[1]
choice in textbooks
Performing addition is one of the simplest numerical tasks. Addition of
very small numbers is accessible to toddlers; the most basic task, 1 + 1,
can be performed by infants as young as five months and even some
animals. In primary education, students are taught to add numbers in
the decimal system, starting with single digits and progressively
tackling more difficult problems. Mechanical aids range from the
ancient abacus to the modern computer, where research on the most
efficient implementations of addition continues to this day.
Addition of numbers 0-10. Line labels = addend.
X axis = addend. Y axis = sum.
Addition
14
Notation and terminology
Addition is written using the plus sign "+" between the terms; that is, in infix
notation. The result is expressed with an equals sign. For example,
(verbally, "one plus one equals two")
(verbally, "two plus two equals four")
(verbally, "three plus three equals six")
(see "associativity" below)
(see "multiplication" below)
There are also situations where addition is "understood" even though no symbol
appears:
The plus sign
• A column of numbers, with the last number in the column
underlined, usually indicates that the numbers in the column are to
be added, with the sum written below the underlined number.
• A whole number followed immediately by a fraction indicates the
sum of the two, called a mixed number.[2] For example,
3½ = 3 + ½ = 3.5.
This notation can cause confusion since in most other contexts
juxtaposition denotes multiplication instead.
The sum of a series of related numbers can be expressed through
capital sigma notation, which compactly denotes iteration. For
example,
The numbers or the objects to be added in general addition are called
the terms, the addends, or the summands; this terminology carries
over to the summation of multiple terms. This is to be distinguished
from factors, which are multiplied. Some authors call the first addend
the augend. In fact, during the Renaissance, many authors did not
consider the first addend an "addend" at all. Today, due to the
commutative property of addition, "augend" is rarely used, and both
terms are generally called addends.[3]
Columnar addition:
5 + 12 = 17
All of this terminology derives from Latin. "Addition" and "add" are English words derived from the Latin verb
addere, which is in turn a compound of ad "to" and dare "to give", from the Proto-Indo-European root *deh₃- "to
give"; thus to add is to give to. Using the gerundive suffix -nd results in "addend", "thing to be added".[4] Likewise
from augere "to increase", one gets "augend", "thing to be increased".
Redrawn illustration from The Art of Nombryng,
one of the first English arithmetic texts, in the
[5]
15th century
"Sum" and "summand" derive from the Latin noun summa "the
highest, the top" and associated verb summare. This is appropriate not
only because the sum of two positive numbers is greater than either,
but because it was once common to add upward, contrary to the
modern practice of adding downward, so that a sum was literally
higher than the addends.[6] Addere and summare date back at least to
Addition
15
Boethius, if not to earlier Roman writers such as Vitruvius and Frontinus; Boethius also used several other terms for
the addition operation. The later Middle English terms "adden" and "adding" were popularized by Chaucer.[7]
Interpretations
Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there
are many possible interpretations and even more visual representations.
Combining sets
Possibly the most fundamental interpretation of addition lies in
combining sets:
• When two or more disjoint collections are combined into a
single collection, the number of objects in the single collection
is the sum of the number of objects in the original collections.
This interpretation is easy to visualize, with little danger of
ambiguity. It is also useful in higher mathematics; for the rigorous
definition it inspires, see Natural numbers below. However, it is
not obvious how one should extend this version of addition to
include fractional numbers or negative numbers.[8]
One possible fix is to consider collections of objects that can be
easily divided, such as pies or, still better, segmented rods.[9]
Rather than just combining collections of segments, rods can be
joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths of the rods.
Extending a length
A second interpretation of addition comes from extending an initial
length by a given length:
• When an original length is extended by a given amount, the final
length is the sum of the original length and the length of the
extension.
The sum a + b can be interpreted as a binary operation that combines a
and b, in an algebraic sense, or it can be interpreted as the addition of b
more units to a. Under the latter interpretation, the parts of a sum a + b
play asymmetric roles, and the operation a + b is viewed as applying
the unary operation +b to a. Instead of calling both a and b addends, it
is more appropriate to call a the augend in this case, since a plays a
passive role. The unary view is also useful when discussing
subtraction, because each unary addition operation has an inverse
unary subtraction operation, and vice versa.
A number-line visualization of the algebraic
addition 2 + 4 = 6. A translation by 2 followed by
a translation by 4 is the same as a translation by
6.
A number-line visualization of the unary addition
2 + 4 = 6. A translation by 4 is equivalent to four
translations by 1.
Addition
16
Properties
Commutativity
Addition is commutative, meaning that one can reverse the terms in a sum
left-to-right, and the result is the same as the last one. Symbolically, if a and b are any
two numbers, then
a + b = b + a.
The fact that addition is commutative is known as the "commutative law of addition".
This phrase suggests that there are other commutative laws: for example, there is a
commutative law of multiplication. However, many binary operations are not
commutative, such as subtraction and division, so it is misleading to speak of an
unqualified "commutative law".
4 + 2 = 2 + 4 with blocks
Associativity
A somewhat subtler property of addition is associativity, which comes up when one tries
to define repeated addition. Should the expression
"a + b + c"
be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the
choice of definition is irrelevant. For any three numbers a, b, and c, it is true that
(a + b) + c = a + (b + c).
2+(1+3) = (2+1)+3 with
segmented rods
For example, (1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3). Not all operations are
associative, so in expressions with other operations like subtraction, it is important to
specify the order of operations.
Identity element
When adding zero to any number, the quantity does not change; zero is the identity element for
addition, also known as the additive identity. In symbols, for any a,
a + 0 = 0 + a = a.
This law was first identified in Brahmagupta's Brahmasphutasiddhanta in 628 AD, although he
wrote it as three separate laws, depending on whether a is negative, positive, or zero itself, and
he used words rather than algebraic symbols. Later Indian mathematicians refined the concept;
around the year 830, Mahavira wrote, "zero becomes the same as what is added to it",
corresponding to the unary statement 0 + a = a. In the 12th century, Bhaskara wrote, "In the
addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same",
corresponding to the unary statement a + 0 = a.[10]
5 + 0 = 5 with bags
of dots
Addition
Successor
In the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least
integer greater than a, also known as the successor of a. Because of this succession, the value of some a + b can also
be seen as the
successor of a, making addition iterated succession.
Units
To numerically add physical quantities with units, they must first be expressed with common units. For example, if a
measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the
other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable;
this sort of consideration is fundamental in dimensional analysis.
Performing addition
Innate ability
Studies on mathematical development starting around the 1980s have exploited the phenomenon of habituation:
infants look longer at situations that are unexpected.[11] A seminal experiment by Karen Wynn in 1992 involving
Mickey Mouse dolls manipulated behind a screen demonstrated that five-month-old infants expect 1 + 1 to be 2, and
they are comparatively surprised when a physical situation seems to imply that 1 + 1 is either 1 or 3. This finding has
since been affirmed by a variety of laboratories using different methodologies.[12] Another 1992 experiment with
older toddlers, between 18 to 35 months, exploited their development of motor control by allowing them to retrieve
ping-pong balls from a box; the youngest responded well for small numbers, while older subjects were able to
compute sums up to 5.[13]
Even some nonhuman animals show a limited ability to add, particularly primates. In a 1995 experiment imitating
Wynn's 1992 result (but using eggplants instead of dolls), rhesus macaques and cottontop tamarins performed
similarly to human infants. More dramatically, after being taught the meanings of the Arabic numerals 0 through 4,
one chimpanzee was able to compute the sum of two numerals without further training.[14]
Discovering addition as children
Typically, children first master counting. When given a problem that requires that two items and three items be
combined, young children model the situation with physical objects, often fingers or a drawing, and then count the
total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three,
children count three past two, saying "three, four, five" (usually ticking off fingers), and arriving at five. This
strategy seems almost universal; children can easily pick it up from peers or teachers.[15] Most discover it
independently. With additional experience, children learn to add more quickly by exploiting the commutativity of
addition by counting up from the larger number, in this case starting with three and counting "four, five." Eventually
children begin to recall certain addition facts ("number bonds"), either through experience or rote memorization.
Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example,
a child asked to add six and seven may know that 6+6=12 and then reason that 6+7 is one more, or 13. Such derived
facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and
derived facts to add fluently.
17
Addition
18
Addition table
Children are often presented with the addition table of the 10 first numbers to memorize. Knowing this you can
perform any addition.
Addition table
Addition table of 1
1 +
0 =
1
1 +
1 =
2
1 +
2 =
3
1 +
3 =
4
1 +
4 =
5
1 +
5 =
6
1 +
6 =
7
1 +
7 =
8
1 +
8 =
9
1 +
9 = 10
1 + 10 = 11
Addition table of 2
2 +
0 =
2
2 +
1 =
3
2 +
2 =
4
2 +
3 =
5
2 +
4 =
6
2 +
5 =
7
2 +
6 =
8
2 +
7 =
9
2 +
8 = 10
2 +
9 = 11
2 + 10 = 12
Addition
19
Addition table of 3
3 +
0 =
3
3 +
1 =
4
3 +
2 =
5
3 +
3 =
6
3 +
4 =
7
3 +
5 =
8
3 +
6 =
9
3 +
7 = 10
3 +
8 = 11
3 +
9 = 12
3 + 10 = 13
Addition table of 4
4 +
0 =
4
4 +
1 =
5
4 +
2 =
6
4 +
3 =
7
4 +
4 =
8
4 +
5 =
9
4 +
6 = 10
4 +
7 = 11
4 +
8 = 12
4 +
9 = 13
4 + 10 = 14
Addition table of 5
5 +
0 =
5
5 +
1 =
6
5 +
2 =
7
5 +
3 =
8
5 +
4 =
9
5 +
5 = 10
5 +
6 = 11
5 +
7 = 12
5 +
8 = 13
5 +
9 = 14
5 + 10 = 15
Addition
20
Addition table of 6
6 +
0 =
6
6 +
1 =
7
6 +
2 =
8
6 +
3 =
9
6 +
4 = 10
6 +
5 = 11
6 +
6 = 12
6 +
7 = 13
6 +
8 = 14
6 +
9 = 15
6 + 10 = 16
Addition table of 7
7 +
0 =
7
7 +
1 =
8
7 +
2 =
9
7 +
3 = 10
7 +
4 = 11
7 +
5 = 12
7 +
6 = 13
7 +
7 = 14
7 +
8 = 15
7 +
9 = 16
7 + 10 = 17
Addition table of 8
8 +
0 =
8
8 +
1 =
9
8 +
2 = 10
8 +
3 = 11
8 +
4 = 12
8 +
5 = 13
8 +
6 = 14
8 +
7 = 15
8 +
8 = 16
8 +
9 = 17
8 + 10 = 18
Addition
21
Addition table of 9
9 +
0 =
9
9 +
1 = 10
9 +
2 = 11
9 +
3 = 12
9 +
4 = 13
9 +
5 = 14
9 +
6 = 15
9 +
7 = 16
9 +
8 = 17
9 +
9 = 18
9 + 10 = 19
Addition table of 10
10 +
0 = 10
10 +
1 = 11
10 +
2 = 12
10 +
3 = 13
10 +
4 = 14
10 +
5 = 15
10 +
6 = 16
10 +
7 = 17
10 +
8 = 18
10 +
9 = 19
10 + 10 = 20
Decimal system
The prerequisite to addition in the decimal system is the fluent recall or derivation of the 100 single-digit "addition
facts". One could memorize all the facts by rote, but pattern-based strategies are more enlightening and, for most
people, more efficient:[16]
• Commutative property: Mentioned above, using the pattern a + b = b + a reduces the number of "addition facts"
from 100 to 55.
• One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately,
intuition.
• Zero: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some
students are introduced to addition as a process that always increases the addends; word problems may help
rationalize the "exception" of zero.
• Doubles: Adding a number to itself is related to counting by two and to multiplication. Doubles facts form a
backbone for many related facts, and students find them relatively easy to grasp.
• Near-doubles: Sums such as 6+7=13 can be quickly derived from the doubles fact 6+6=12 by adding one more, or
from 7+7=14 but subtracting one.
Addition
22
• Five and ten: Sums of the form 5+x and 10+x are usually memorized early and can be used for deriving other
facts. For example, 6+7=13 can be derived from 5+7=12 by adding one more.
• Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 8 + 6 =
8 + 2 + 4 = 10 + 4 = 14.
As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently.
Many students never commit all the facts to memory, but can still find any basic fact quickly.
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting
from the ones column on the right. If a column exceeds ten, the extra digit is "carried" into the next column.[17] An
alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier,
but it is faster at getting a rough estimate of the sum. There are many other alternative methods.
• Fraction: Addition
• Scientific notation: Operations
Computers
Analog computers work directly with physical quantities, so their
addition mechanisms depend on the form of the addends. A mechanical
adder might represent two addends as the positions of sliding blocks, in
which case they can be added with an averaging lever. If the addends
are the rotation speeds of two shafts, they can be added with a
differential. A hydraulic adder can add the pressures in two chambers
Addition with an op-amp. See Summing
by exploiting Newton's second law to balance forces on an assembly of
amplifier for details.
pistons. The most common situation for a general-purpose analog
computer is to add two voltages (referenced to ground); this can be
accomplished roughly with a resistor network, but a better design exploits an operational amplifier.[18]
Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the
carry mechanism, is an important limitation to overall performance.
Part of Charles Babbage's Difference Engine
including the addition and carry mechanisms
automatically.
Blaise Pascal invented the mechanical calculator in 1642,[19] it was the
first operational adding machine. It made use of an ingenious
gravity-assisted carry mechanism. It was the only operational
mechanical calculator in the 17th century[20] and the earliest automatic,
digital computers. Pascal's calculator was limited by its carry
mechanism which forced its wheels to only turn one way, so it could
add but, to subtract, the operator had to use of the method of
complements which required as many steps as an addition. Pascal was
followed by Giovanni Poleni who built the second functional
mechanical calculator in 1709, a calculating clock, which was made of
wood and which could, once setup, multiply two numbers
Addition
Adders execute integer addition in electronic digital computers, usually
using binary arithmetic. The simplest architecture is the ripple carry
adder, which follows the standard multi-digit algorithm. One slight
improvement is the carry skip design, again following human intuition;
one does not perform all the carries in computing 999 + 1, but one
bypasses the group of 9s and skips to the answer.[21]
Since they compute digits one at a time, the above methods are too
slow for most modern purposes. In modern digital computers, integer
"Full adder" logic circuit that adds two binary
digits, A and B, along with a carry input Cin,
addition is typically the fastest arithmetic instruction, yet it has the
producing the sum bit, S, and a carry output, Cout.
largest impact on performance, since it underlies all the floating-point
operations as well as such basic tasks as address generation during
memory access and fetching instructions during branching. To increase speed, modern designs calculate digits in
parallel; these schemes go by such names as carry select, carry lookahead, and the Ling pseudocarry. Almost all
modern implementations are, in fact, hybrids of these last three designs.[22]
Unlike addition on paper, addition on a computer often changes the addends. On the ancient abacus and adding
board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was
strong enough that early Latin texts often claimed that in the process of adding "a number to a number", both
numbers vanish.[23] In modern times, the ADD instruction of a microprocessor replaces the augend with the sum but
preserves the addend.[24] In a high-level programming language, evaluating a + b does not change either a or b; if the
goal is to replace a with the sum this must be explicitly requested, typically with the statement a = a + b. Some
languages such as C or C++ allow this to be abbreviated as a += b.
Addition of natural and real numbers
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first
defined on the natural numbers. In set theory, addition is then extended to progressively larger sets that include the
natural numbers: the integers, the rational numbers, and the real numbers.[25] (In mathematics education,[26] positive
fractions are added before negative numbers are even considered; this is also the historical route)[27]
Natural numbers
There are two popular ways to define the sum of two natural numbers a and b. If one defines natural numbers to be
the cardinalities of finite sets, (the cardinality of a set is the number of elements in the set), then it is appropriate to
define their sum as follows:
• Let N(S) be the cardinality of a set S. Take two disjoint sets A and B, with N(A) = a and N(B) = b. Then a + b is
defined as
.[28]
Here, A U B is the union of A and B. An alternate version of this definition allows A and B to possibly overlap and
then takes their disjoint union, a mechanism that allows common elements to be separated out and therefore counted
twice.
The other popular definition is recursive:
• Let n+ be the successor of n, that is the number following n in the natural numbers, so 0+=1, 1+=2. Define a + 0 =
a. Define the general sum recursively by a + (b+) = (a + b)+. Hence 1+1=1+0+=(1+0)+=1+=2.[29]
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an
application of the Recursion Theorem on the poset N2.[30] On the other hand, some sources prefer to use a restricted
Recursion Theorem that applies only to the set of natural numbers. One then considers a to be temporarily "fixed",
applies recursion on b to define a function "a + ", and pastes these unary operations for all a together to form the full
binary operation.[31]
23
Addition
24
This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in
the following decades.[32] He proved the associative and commutative properties, among others, through
mathematical induction; for examples of such inductive proofs, see Addition of natural numbers.
Integers
The simplest conception of an integer is that it consists of an absolute
value (which is a natural number) and a sign (generally either positive
or negative). The integer zero is a special third case, being neither
positive nor negative. The corresponding definition of addition must
proceed by cases:
• For an integer n, let |n| be its absolute value. Let a and b be integers.
If either a or b is zero, treat it as an identity. If a and b are both
positive, define a + b = |a| + |b|. If a and b are both negative, define
a + b = −(|a|+|b|). If a and b have different signs, define a + b to be
the difference between |a| and |b|, with the sign of the term whose
absolute value is larger.[33]
Although this definition can be useful for concrete problems, it is far
too complicated to produce elegant general proofs; there are too many
cases to consider.
A much more convenient conception of the integers is the
Grothendieck group construction. The essential observation is that
every integer can be expressed (not uniquely) as the difference of two
natural numbers, so we may as well define an integer as the difference
of two natural numbers. Addition is then defined to be compatible with
subtraction:
• Given two integers a − b and c − d, where a, b, c, and d are natural
numbers, define (a − b) + (c − d) = (a + c) − (b + d).[34]
Defining (−2) + 1 using only addition of positive
numbers: (2 − 4) + (3 − 2) = 5 − 6.
Rational numbers (fractions)
Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler
definition involves only integer addition and multiplication:
• Define The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic.[35]
For a more rigorous and general discussion, see field of fractions.
Addition
25
Real numbers
A common construction of the set of real numbers is
the Dedekind completion of the set of rational numbers.
A real number is defined to be a Dedekind cut of
rationals: a non-empty set of rationals that is closed
downward and has no greatest element. The sum of real
numbers a and b is defined element by element:
• Define
[36]
This definition was first published, in a slightly
modified form, by Richard Dedekind in 1872.[37] The
commutativity and associativity of real addition are
immediate; defining the real number 0 to be the set of
negative rationals, it is easily seen to be the additive
identity. Probably the trickiest part of this construction
pertaining to addition is the definition of additive inverses.[38]
Adding π2/6 and e using Dedekind cuts of rationals
Unfortunately, dealing with multiplication of Dedekind
cuts is a case-by-case nightmare similar to the addition
of signed integers. Another approach is the metric
completion of the rational numbers. A real number is
essentially defined to be the a limit of a Cauchy
sequence of rationals, lim an. Addition is defined term
by term:
• Define
[39]
This definition was first published by Georg Cantor,
also in 1872, although his formalism was slightly
different.[40] One must prove that this operation is
Adding π2/6 and e using Cauchy sequences of rationals
well-defined, dealing with co-Cauchy sequences. Once
that task is done, all the properties of real addition follow immediately from the properties of rational numbers.
Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous
definitions.[41]
Generalizations
There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions,
equations, strings, chains... —Alexander Bogomolny [42]
There are many binary operations that can be viewed as generalizations of the addition operation on the real
numbers. The field of abstract algebra is centrally concerned with such generalized operations, and they also appear
in set theory and category theory.
Addition
26
Addition in abstract algebra
In linear algebra, a vector space is an algebraic structure that allows for adding any two vectors and for scaling
vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (a,b) is interpreted as
a vector from the origin in the Euclidean plane to the point (a,b) in the plane. The sum of two vectors is obtained by
adding their individual coordinates:
(a,b) + (c,d) = (a+c,b+d).
This addition operation is central to classical mechanics, in which vectors are interpreted as forces.
In modular arithmetic, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the
integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition
operation it inherits is known in Boolean logic as the "exclusive or" function. In geometry, the sum of two angle
measures is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the
circle, which in turn generalizes to addition operations on many-dimensional tori.
The general theory of abstract algebra allows an "addition" operation to be any associative and commutative
operation on a set. Basic algebraic structures with such an addition operation include commutative monoids and
abelian groups.
Addition in set theory and category theory
A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers
in set theory. These give two different generalizations of addition of natural numbers to the transfinite. Unlike most
addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a
commutative operation closely related to the disjoint union operation.
In category theory, disjoint union is seen as a particular case of the coproduct operation, and general coproducts are
perhaps the most abstract of all the generalizations of addition. Some coproducts, such as Direct sum and Wedge
sum, are named to evoke their connection with addition.
Related operations
Arithmetic
Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself
a sort of inverse to addition, in that adding x and subtracting x are inverse functions.
Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set;
the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an
addition operation, an additive inverse operation, and an additive identity; for this reason, an additive group can be
described as a set that is closed under subtraction.[43]
Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the
product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1
yields the additive inverse of a number.
Addition
27
In the real and complex numbers, addition and multiplication can be
interchanged by the exponential function:
ea + b = ea eb.[44]
This identity allows multiplication to be carried out by consulting a
table of logarithms and computing addition by hand; it also enables
multiplication on a slide rule. The formula is still a good first-order
approximation in the broad context of Lie groups, where it relates
multiplication of infinitesimal group elements with addition of vectors
in the associated Lie algebra.[45]
There are even more generalizations of multiplication than addition.[46]
In general, multiplication operations always distribute over addition;
this requirement is formalized in the definition of a ring. In some
contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to
uniquely determine the multiplication operation. The distributive property also provides information about addition;
by expanding the product (1 + 1)(a + b) in both ways, one concludes that addition is forced to be commutative. For
this reason, ring addition is commutative in general.[47]
A circular slide rule
Division is an arithmetic operation remotely related to addition. Since a/b = a(b−1), division is right distributive over
addition: (a + b) / c = a / c + b / c.[48] However, division is not left distributive over addition; 1/ (2 + 2) is not the
same as 1/2 + 1/2.
Ordering
The maximum operation "max (a, b)" is a binary operation similar to
addition. In fact, if two nonnegative numbers a and b are of different
orders of magnitude, then their sum is approximately equal to their
maximum. This approximation is extremely useful in the applications
of mathematics, for example in truncating Taylor series. However, it
presents a perpetual difficulty in numerical analysis, essentially since
"max" is not invertible. If b is much greater than a, then a
straightforward calculation of (a + b) − b can accumulate an
unacceptable round-off error, perhaps even returning zero. See also
Loss of significance.
Log-log plot of x + 1 and max (x, 1) from x =
[49]
0.001 to 1000
The approximation becomes exact in a kind of infinite limit; if either a or b is an infinite cardinal number, their
cardinal sum is exactly equal to the greater of the two.[50] Accordingly, there is no subtraction operation for infinite
cardinals.[51]
Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of
real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:
a + max (b, c) = max (a + b, a + c).
For these reasons, in tropical geometry one replaces multiplication with addition and addition with maximization. In
this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical
"additive identity" is negative infinity.[52] Some authors prefer to replace addition with minimization; then the
additive identity is positive infinity.[53]
Tying these observations together, tropical addition is approximately related to regular addition through the
logarithm:
log (a + b) ≈ max (log a, log b),
Addition
which becomes more accurate as the base of the logarithm increases.[54] The approximation can be made exact by
extracting a constant h, named by analogy with Planck's constant from quantum mechanics,[55] and taking the
"classical limit" as h tends to zero:
In this sense, the maximum operation is a dequantized version of addition.[56]
Other ways to add
Incrementation, also known as the successor operation, is the addition of 1 to a number.
Summation describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of
the sum of a single number, which is itself, and the empty sum, which is zero.[57] An infinite summation is a delicate
procedure known as a series.[58]
Counting a finite set is equivalent to summing 1 over the set.
Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable
manifold. Integration over a zero-dimensional manifold reduces to summation.
Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier,
usually a real or complex number. Linear combinations are especially useful in contexts where straightforward
addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of
states in quantum mechanics.
Convolution is used to add two independent random variables defined by distribution functions. Its usual definition
combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side
addition; by contrast, vector addition is a kind of range-side addition.
In literature
• In chapter 9 of Lewis Carroll's Through the Looking-Glass, the White Queen asks Alice, "And you do
Addition? ... What's one and one and one and one and one and one and one and one and one and one?" Alice
admits that she lost count, and the Red Queen declares, "She can't do Addition".
• In George Orwell's Nineteen Eighty-Four, the value of 2 + 2 is questioned; the State contends that if it declares 2
+ 2 = 5, then it is so. See Two plus two make five for the history of this idea.
Notes
[1] From Enderton (p.138): "...select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by
textbooks."
[2] Devine et al. p.263
[3] Schwartzman p.19
[4] "Addend" is not a Latin word; in Latin it must be further conjugated, as in numerus addendus "the number to be added".
[5] Karpinski pp.56–57, reproduced on p.104
[6] Schwartzman (p.212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the
other hand, Karpinski (p.103) writes that Leonard of Pisa "introduces the novelty of writing the sum above the addends"; it is unclear whether
Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe.
[7] Karpinski pp.150–153
[8] See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".
[9] Adding it up (p.73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard
to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their
nature."
[10] Kaplan pp.69–71
[11] Wynn p.5
[12] Wynn p.15
[13] Wynn p.17
28
Addition
[14] Wynn p.19
[15] F. Smith p.130
[16] Fosnot and Dolk p. 99
[17] The word "carry" may be inappropriate for education; Van de Walle (p.211) calls it "obsolete and conceptually misleading", preferring the
word "trade".
[18] Truitt and Rogers pp.1;44–49 and pp.2;77–78
[19] Jean Marguin, p. 48 (1994) ; Quoting René Taton (1963)
[20] See Competing designs in Pascal's calculator article
[21] Flynn and Overman pp.2, 8
[22] Flynn and Overman pp.1–9
[23] Karpinski pp.102–103
[24] The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz and Hill p.679; for ADD in 68k see p.767.
[25] Enderton chapters 4 and 5, for example, follow this development.
[26] California standards; see grades 2 (http:/ / www. cde. ca. gov/ be/ st/ ss/ mthgrade2. asp), 3 (http:/ / www. cde. ca. gov/ be/ st/ ss/ mthgrade3.
asp), and 4 (http:/ / www. cde. ca. gov/ be/ st/ ss/ mthgrade4. asp).
[27] Baez (p.37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to
understand than a negative apple!"
[28] Begle p.49, Johnson p.120, Devine et al. p.75
[29] Enderton p.79
[30] For a version that applies to any poset with the descending chain condition, see Bergman p.100.
[31] Enderton (p.79) observes, "But we want one binary operation +, not all these little one-place functions."
[32] Ferreirós p.223
[33] K. Smith p.234, Sparks and Rees p.66
[34] Enderton p.92
[35] The verifications are carried out in Enderton p.104 and sketched for a general field of fractions over a commutative ring in Dummit and
Foote p.263.
[36] Enderton p.114
[37] Ferreirós p.135; see section 6 of Stetigkeit und irrationale Zahlen (http:/ / www. ru. nl/ w-en-s/ gmfw/ bronnen/ dedekind2. html).
[38] The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton p.117
for details.
[39] Textbook constructions are usually not so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, drawn-out development of
addition with Cauchy sequences.
[40] Ferreirós p.128
[41] Burrill p.140
[42] http:/ / www. cut-the-knot. org/ do_you_know/ addition. shtml
[43] The set still must be nonempty. Dummit and Foote (p.48) discuss this criterion written multiplicatively.
[44] Rudin p.178
[45] Lee p.526, Proposition 20.9
[46] Linderholm (p.49) observes, "By multiplication, properly speaking, a mathematician may mean practically anything. By addition he may
mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."
[47] Dummit and Foote p.224. For this argument to work, one still must assume that addition is a group operation and that multiplication has an
identity.
[48] For an example of left and right distributivity, see Loday, especially p.15.
[49] Compare Viro Figure 1 (p.2)
[50] Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the
Axiom of Choice.
[51] Enderton p.164
[52] Mikhalkin p.1
[53] Akian et al. p.4
[54] Mikhalkin p.2
[55] Litvinov et al. p.3
[56] Viro p.4
[57] Martin p.49
[58] Stewart p.8
29
Addition
References
History
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ISBN 0-13-389015-5.
• Ferreirós, José (1999). Labyrinth of thought: A history of set theory and its role in modern mathematics.
Birkhäuser. ISBN 0-8176-5749-5.
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• Karpinski, Louis (1925). The history of arithmetic. Rand McNally. LCC QA21.K3 (http://catalog.loc.gov/
cgi-bin/Pwebrecon.cgi?Search_Arg=QA21.K3&Search_Code=CALL_&CNT=5).
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• Davison, Landau, McCracken, and Thompson (1999). Mathematics: Explorations & Applications (TE ed.).
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Education
• Begle, Edward (1975). The mathematics of the elementary school. McGraw-Hill. ISBN 0-07-004325-6.
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mthmain.asp) Adopted December 1997, accessed December 2005.
• D. Devine, J. Olson, and M. Olson (1991). Elementary mathematics for teachers (2e ed.). Wiley.
ISBN 0-471-85947-8.
• National Research Council (2001). Adding it up: Helping children learn mathematics (http://www.nap.edu/
books/0309069955/html/index.html). National Academy Press. ISBN 0-309-06995-5.
• Van de Walle, John (2004). Elementary and middle school mathematics: Teaching developmentally (5e ed.).
Pearson. ISBN 0-205-38689-X.
Cognitive science
• Baroody and Tiilikainen (2003). "Two perspectives on addition development". The development of arithmetic
concepts and skills. p. 75. ISBN 0-8058-3155-X.
• Fosnot and Dolk (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction.
Heinemann. ISBN 0-325-00353-X.
• Weaver, J. Fred (1982). "Interpretations of number operations and symbolic representations of addition and
subtraction". Addition and subtraction: A cognitive perspective. p. 60. ISBN 0-89859-171-6.
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ISBN 0-86377-816-X.
Mathematical exposition
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20060206145950/http://www.cut-the-knot.org/do_you_know/addition.shtml) from the original on 6
February 2006. Retrieved 3 February 2006.
• Dunham, William (1994). The mathematical universe. Wiley. ISBN 0-471-53656-3.
• Johnson, Paul (1975). From sticks and stones: Personal adventures in mathematics. Science Research Associates.
ISBN 0-574-19115-1.
• Linderholm, Carl (1971). Mathematics Made Difficult. Wolfe. ISBN 0-7234-0415-1.
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• Poonen, Bjorn (2010). "Addition" (http://www.girlsangle.org/page/bulletin.php). Girls' Angle Bulletin,
Volume 3, Numbers 3-5 (Girls' Angle). ISSN 2151-5743 (http://www.worldcat.org/issn/2151-5743).
• Smith, Frank (2002). The glass wall: Why mathematics can seem difficult. Teachers College Press.
ISBN 0-8077-4242-2.
• Smith, Karl (1980). The nature of modern mathematics (3e ed.). Wadsworth. ISBN 0-8185-0352-1.
Advanced mathematics
• Bergman, George (2005). An invitation to general algebra and universal constructions (http://math.berkeley.
edu/~gbergman/245/index.html) (2.3e ed.). General Printing. ISBN 0-9655211-4-1.
• Burrill, Claude (1967). Foundations of real numbers. McGraw-Hill. LCC QA248.B95 (http://catalog.loc.gov/
cgi-bin/Pwebrecon.cgi?Search_Arg=QA248.B95&Search_Code=CALL_&CNT=5).
• D. Dummit and R. Foote (1999). Abstract algebra (2e ed.). Wiley. ISBN 0-471-36857-1.
• Enderton, Herbert (1977). Elements of set theory. Academic Press. ISBN 0-12-238440-7.
• Lee, John (2003). Introduction to smooth manifolds. Springer. ISBN 0-387-95448-1.
• Martin, John (2003). Introduction to languages and the theory of computation (3e ed.). McGraw-Hill.
ISBN 0-07-232200-4.
• Rudin, Walter (1976). Principles of mathematical analysis (3e ed.). McGraw-Hill. ISBN 0-07-054235-X.
• Stewart, James (1999). Calculus: Early transcendentals (4e ed.). Brooks/Cole. ISBN 0-534-36298-2.
Mathematical research
• Akian, Bapat, and Gaubert (2005). "Min-plus methods in eigenvalue perturbation theory and generalised
Lidskii-Vishik-Ljusternik theorem". INRIA reports. arXiv: math.SP/0402090 (http://arxiv.org/abs/math.SP/
0402090).
• J. Baez and J. Dolan (2001). "From Finite Sets to Feynman Diagrams". Mathematics Unlimited— 2001 and
Beyond. p. 29. arXiv: math.QA/0004133 (http://arxiv.org/abs/math.QA/0004133). ISBN 3-540-66913-2.
• Litvinov, Maslov, and Sobolevskii (1999). Idempotent mathematics and interval analysis (http://arxiv.org/abs/
math.SC/9911126). Reliable Computing (http://www.springerlink.com/openurl.asp?genre=article&
eissn=1573-1340&volume=7&issue=5&spage=353), Kluwer.
• Loday, Jean-Louis (2002). "Arithmetree". J. Of Algebra 258: 275. arXiv: math/0112034 (http://arxiv.org/abs/
math/0112034). doi: 10.1016/S0021-8693(02)00510-0 (http://dx.doi.org/10.1016/S0021-8693(02)00510-0).
• Mikhalkin, Grigory (2006). "Tropical Geometry and its applications". To appear at the Madrid ICM. arXiv:
math.AG/0601041 (http://arxiv.org/abs/math.AG/0601041).
• Viro, Oleg (2001). "Dequantization of real algebraic geometry on logarithmic paper" (http://www.math.uu.se/
~oleg/dequant/dequantH1.html). In Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan;
Xambó-Descamps, Sebastià. European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I.
Progress in Mathematics 201. Basel: Birkhäuser. pp. 135–146. arXiv: math/0005163 (http://arxiv.org/abs/
math/0005163). ISBN 3-7643-6417-3.
Computing
• M. Flynn and S. Oberman (2001). Advanced computer arithmetic design. Wiley. ISBN 0-471-41209-0.
• P. Horowitz and W. Hill (2001). The art of electronics (2e ed.). Cambridge UP. ISBN 0-521-37095-7.
• Jackson, Albert (1960). Analog computation. McGraw-Hill. LCC QA76.4 J3 (http://catalog.loc.gov/cgi-bin/
Pwebrecon.cgi?Search_Arg=QA76.4+J3&Search_Code=CALL_&CNT=5).
• T. Truitt and A. Rogers (1960). Basics of analog computers. John F. Rider. LCC QA76.4 T7 (http://catalog.loc.
gov/cgi-bin/Pwebrecon.cgi?Search_Arg=QA76.4+T7&Search_Code=CALL_&CNT=5).
• Marguin, Jean (1994). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante
1642-1942 (in French). Hermann. ISBN 978-2-7056-6166-3.
• Taton, René (1963). Le calcul mécanique. Que sais-je ? n° 367 (in French). Presses universitaires de France.
pp. 20–28.
31
Addition
32
• Marguin, Jean (1994). Histoire des instruments et machines à calculer, trois siècles de mécanique pensante
1642-1942 (in French). Hermann. ISBN 978-2-7056-6166-3.
Subtraction
Subtraction is a mathematical operation that represents the operation
of removing objects from a collection. It is signified by the minus sign
(−). For example, in the picture on the right, there are 5 − 2
apples—meaning 5 apples with 2 taken away, which is a total of 3
apples. Therefore, 5 − 2 = 3. Besides counting fruits, subtraction can
also represent combining other physical and abstract quantities using
different kinds of objects: negative numbers, fractions, irrational
numbers, vectors, decimals, functions, matrices and more.
Subtraction follows several important patterns. It is anticommutative,
meaning that changing the order changes the sign of the answer. It is
not associative, meaning that when one subtracts more than two
numbers, the order in which subtraction is performed matters.
Subtraction of 0 does not change a number. Subtraction also obeys
predictable rules concerning related operations such as addition and
multiplication. All of these rules can be proven, starting with the
subtraction of integers and generalizing up through the real numbers
and beyond. General binary operations that continue these patterns are
studied in abstract algebra.
"5 − 2 = 3" (verbally, "five minus two equals
three")
Performing subtraction is one of the simplest numerical tasks.
Subtraction of very small numbers is accessible to young children. In
primary education, students are taught to subtract numbers in the
decimal system, starting with single digits and progressively tackling
more difficult problems. Mechanical aids range from the ancient
abacus to the modern computer.
An example problem
Subtraction
33
Subtraction of numbers 0–10. Line labels =
minuend. X axis = subtrahend. Y axis =
difference.
Basic subtraction: integers
Imagine a line segment of length b with the left end labeled a and the
right end labeled c. Starting from a, it takes b steps to the right to reach
c. This movement to the right is modeled mathematically by addition:
a + b = c.
From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction:
c − b = a.
Now, a line segment labeled with the numbers 1, 2, and 3. From
position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes
2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is
inadequate to describe what would happen after going 3 steps to the
left of position 3. To represent such an operation, the line must be
extended.
To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...).
From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The
natural numbers are not a useful context for subtraction.
The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). From 3, it takes 4 steps to the left to
get to −1:
3 − 4 = −1.
Subtraction as addition
Subtraction
34
Calculation results
•
•
•
v
t
[1]
e
Addition (+)
augend + addend =
sum
Subtraction (−)
minuend − subtrahend =
difference
Multiplication (×)
multiplicand × multiplier = product
Division (÷)
dividend ÷ divisor =
quotient
Exponentiation
power
baseexponent =
nth root (√)
degree
√radicand =
root
Logarithm
logbase(power) =
exponent
There are some cases where subtraction as a separate operation becomes problematic. For example, 3 − (−2) (i.e.
subtract −2 from 3) is not immediately obvious from either a natural number view or a number line view, because it
is not immediately clear what it means to move −2 steps to the left or to take away −2 apples. One solution is to view
subtraction as addition of signed numbers. Extra minus signs simply denote additive inversion. Then we have
3 − (−2) = 3 + 2 = 5. This also helps to keep the ring of integers "simple" by avoiding the introduction of "new"
operators such as subtraction. Ordinarily a ring only has two operations defined on it; in the case of the integers,
these are addition and multiplication. A ring already has the concept of additive inverses, but it does not have any
notion of a separate subtraction operation, so the use of signed addition as subtraction allows us to apply the ring
axioms to subtraction without needing to prove anything.
Algorithms for subtraction
There are various algorithms for subtraction, and they differ in their suitability for various applications. A number of
methods are adapted to hand calculation; for example, when making change, no actual subtraction is performed, but
rather the change-maker counts forward.
For machine calculation, the method of complements is preferred, whereby the subtraction is replaced by an addition
in a modular arithmetic.
Subtraction
The teaching of subtraction in schools
Methods used to teach subtraction to elementary school vary from country to country, and within a country, different
methods are in fashion at different times. In what is, in the U.S., called traditional mathematics, a specific process is
taught to students at the end of the 1st year or during the 2nd year for use with multi-digit whole numbers, and is
extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
Some American schools currently teach a method of subtraction using borrowing and a system of markings called
crutches[citation needed]. Although a method of borrowing had been known and published in textbooks prior,
apparently the crutches are the invention of William A. Brownell who used them in a study in November 1937[citation
needed]
. This system caught on rapidly, displacing the other methods of subtraction in use in America at that time.
Some European schools employ a method of subtraction called the Austrian method, also known as the additions
method. There is no borrowing in this method. There are also crutches (markings to aid memory), which vary by
country[citation needed].
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a
least significant digit, a subtraction of subtrahend:
sj sj−1 ... s1
from minuend
mk mk−1 ... m1,
where each si and mi is a digit, proceeds by writing down m1 − s1, m2 − s2, and so forth, as long as si does not exceed
mi. Otherwise, mi is increased by 10 and some other digit is modified to correct for this increase. The American
method corrects by attempting to decrease the minuend digit mi+1 by one (or continuing the borrow leftwards until
there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit
si+1 by one.
Example: 704 − 512. The minuend is 704, the subtrahend is 512. The minuend digits are m3 = 7, m2 = 0 and m1 = 4.
The subtrahend digits are s3 = 5, s2 = 1 and s1 = 2. Beginning at the one's place, 4 is not less than 2 so the difference
2 is written down in the result's one place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the
difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten
by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6.
The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in
the result's hundred's place. We are now done, the result is 192.
The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundred's digit by one. A small
mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what
number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's
hundred's place.
There is an additional subtlety in that the student always employs a mental subtraction table in the American method.
The Austrian method often encourages the student to mentally use the addition table in reverse. In the example
above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number,
when increased by 1, and 5 is added to it, makes 7.
35
Subtraction
36
Subtraction by hand
Austrian method
Example:
1+…=3
The difference is written under
the line.
9+…=5
The required sum (5) is too
small!
So, we add 10 to it and put a 1
under the next higher place in the
subtrahend.
9 + … = 15
Now we can find the difference
like before.
(4 + 1) + … = 7
The difference is written under
the line.
The total difference.
Subtraction
37
Subtraction from left to right
Example:
7−4=3
This result is only penciled in.
3−1=2
Because the next digit of the
minuend is smaller than the
subtrahend, we subtract one from
our penciled-in-number and
mentally add ten to the next.
15 − 9 = 6
Because the next digit in the
minuend is not smaller than the
subtrahend, We keep this
number.
Subtraction
38
American method
In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top
number is too small to subtract the bottom number from it, we add 10 to it; this 10 is 'borrowed' from the top digit to
the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until
every digit has been subtracted. Example:
3−1=…
We write the difference under the
line.
5−9=…
The minuend (5) is too small!
So, we add 10 to it. The 10 is
'borrowed' from the digit on the
left, which goes down by 1.
15 − 9 = …
Now the subtraction works, and
we write the difference under the
line.
6−4=…
We write the difference under the
line.
The total difference.
Subtraction
39
Trade first
A variant of the American method where all borrowing is done before all subtraction.[2]
Example:
1 − 3 = not possible.
We add a 10 to the 1. Because
the 10 is 'borrowed' from the
nearby 5, the 5 is lowered by 1.
6−4=2
4 − 9 = not possible.
So we proceed as in step 1.
Working from right to left:
11 − 3 = 8
14 − 9 = 5
Subtraction
40
Partial differences
The partial differences method is different from other vertical subtraction methods because no borrowing or carrying
takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller
than the subtrahend. The sum of the partial differences is the total difference.[3]
Example:
The smaller number is subtracted
from the greater:
700 − 400 = 300
Because the minuend is greater
than the subtrahend, this
difference has a plus sign.
The smaller number is subtracted
from the greater:
90 − 50 = 40
Because the minuend is smaller
than the subtrahend, this
difference has a minus sign.
The smaller number is subtracted
from the greater:
3−1=2
Because the minuend is greater
than the subtrahend, this
difference has a plus sign
+ 300 − 40 + 2 = 262
Nonvertical methods
Counting up
Instead of finding the difference digit by digit, one can count up the numbers between the subtahend and the
minuend. [4]
Example:
1234 − 567 = can be found by the following steps:
•
•
•
•
567 + 3 = 570
570 + 30 = 600
600 + 400 = 1000
1000 + 234 = 1234
Add up the value from each step to get the total difference: 3 + 30 + 400 + 234 = 667.
Subtraction
Breaking up the subtraction
Another method that is useful for mental arithmetic is to split up the subtraction into small steps.[5]
Example:
1234 − 567 = can be solved in the following way:
• 1234 − 500 = 734
• 734 − 60 = 674
• 674 − 7 = 667
Same change
The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend
does not change the answer. One adds the amount needed to get zeros in the subtrahend.[6]
Example:
„1234 − 567 =“ can be solved as follows:
• 1234 − 567 = 1237 − 570 = 1267 − 600 = 667
Units of measurement
When subtracting two numbers with units of measurement such as kilograms or pounds, they must have the same
unit. In most cases the difference will have the same unit as the original numbers.
One exception is when subtracting two numbers with percentage as unit. In this case, the difference will have
percentage points as unit; the difference is that percentages must be positive, while percentage points may be
negative.
Notes and references
[1] http:/ / en. wikipedia. org/ w/ index. php?title=Template:Calculation_results& action=edit
[2] The Many Ways of Arithmetic in UCSMP Everyday Mathematics (https:/ / sites. google. com/ a/ oswego308. org/ msimester/ home/ math/
algorithms/ subtraction) Subtraction: Trade First
[3] Partial-Differences Subtraction (http:/ / ouronlineschools. org/ Schools/ NC/ Demoschool/ 4thGrade/ Math/ PartialDifferences. htm); The
Many Ways of Arithmetic in UCSMP Everyday Mathematics (https:/ / sites. google. com/ a/ oswego308. org/ msimester/ home/ math/
algorithms/ subtraction) Subtraction: Partial Differences
[4] The Many Ways of Arithmetic in UCSMP Everyday Mathematics (https:/ / sites. google. com/ a/ oswego308. org/ msimester/ home/ math/
algorithms/ subtraction) Subtraction: Counting Up
[5] The Many Ways of Arithmetic in UCSMP Everyday Mathematics (https:/ / sites. google. com/ a/ oswego308. org/ msimester/ home/ math/
algorithms/ subtraction) Subtraction: Left to Right Subtraction
[6] The Many Ways of Arithmetic in UCSMP Everyday Mathematics (https:/ / sites. google. com/ a/ oswego308. org/ msimester/ home/ math/
algorithms/ subtraction) Subtraction: Same Change Rule
• Browell, W. A. (1939). Learning as reorganization: An experimental study in third-grade arithmetic, Duke
University Press.
• Subtraction in the United States: An Historical Perspective, Susan Ross, Mary Pratt-Cotter, The Mathematics
Educator, Vol. 8, No. 1 (original publication) and Vol. 10, No. 1 (reprint.) (http://math.coe.uga.edu/TME/
Issues/v10n2/5ross.pdf) PDF
41
Subtraction
42
External links
• Hazewinkel, Michiel, ed. (2001), "Subtraction" (http://www.encyclopediaofmath.org/index.php?title=p/
s091050), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Printable Worksheets: Subtraction Worksheets (http://www.math-drills.com/subtraction.shtml), One Digit
Subtraction (http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1214&CurriculumID=2&
Method=Worksheet&NQ=24&NQ4P=3), Two Digit Subtraction (http://www.kwiznet.com/p/takeQuiz.
php?ChapterID=1202&CurriculumID=2&Method=Worksheet&NQ=24&NQ4P=3), and Four Digit Subtraction
(http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1273&CurriculumID=3&Method=Worksheet&
NQ=24&NQ4P=3)
• Subtraction Game (http://www.cut-the-knot.org/Curriculum/Arithmetic/SubtractionGame.shtml) at
cut-the-knot
• Subtraction on a Japanese abacus (http://webhome.idirect.com/~totton/abacus/pages.htm#Subtraction1)
selected from Abacus: Mystery of the Bead (http://webhome.idirect.com/~totton/abacus/)
Numerosity adaptation effect
Cognitive
psychology
Perception
•
•
•
•
Visual perception
Object recognition
Face recognition
Pattern recognition
Attention
Memory
•
•
•
•
Aging and memory
Emotional memory
Learning
Long-term memory
Metacognition
Language
Metalanguage
Thinking
Cognition
•
•
•
•
Concept
Reasoning
Decision making
Problem solving
Numerical cognition
•
Numerosity adaptation effect
Numerosity adaptation effect
43
•
•
•
v
t
e [1]
The numerosity adaptation effect is a perceptual
phenomenon in numerical cognition which
demonstrates non-symbolic numerical intuition and
exemplifies how numerical percepts can impose
themselves upon the human brain automatically.
This effect was first described in 2008.
Presently, this effect is described only for
controlled experimental conditions. In the
illustration, a viewer should have a strong
impression that the left display (lower figure) is
more numerous than the right, after 30 seconds of
viewing the adaptation (upper figure), although
both have exactly the same number of dots. The
viewer might also underestimate the number of
dots presented in the display.
An example of the numerosity adaptation effect
Both effects are resistant to manipulation of the
non-numerical parameters of the display. Thus, this
effect cannot be simply explained in terms of size,
density, or contrast.
Perhaps the most astonishing aspect of these effects is that they happen immediately, and without conscious control
(i.e., knowing that the numbers are equal would not hamper their happening). This points to the operation of a
special and largely automatic processing system. As noted by Burr & Ross (2008):
Just as we have a direct visual sense of the reddishness of half a dozen ripe cherries, so we do of their sixishness.
“
”
Possible explanations
Few explanations were suggested to explain these phenomena. It was argued that they are heavily dependent on
density and less on numerosity. Also, it was suggested that numerosity may be correlated with kurtosis and that the
results may be better explained in terms of texture density such that only dots falling within the spatial region where
the test is displayed effectively adapt the region.
However, as the display in the original experiments was of spots uniformly either white or black, the kurtosis
account is inapplicable. The texture density explanation doesn't seem to disentangle the complexity of these
phenomena as in the display the left field adapts to many dots, the right field to few, and these adapters selectively
affect the relevant test stimuli. It is not the number of dots in the entire display that causes the adaptation but only
those within a particular area. At present, why adaptation have such profound effect on numerosity estimates remains
largely unexplained.
Numerosity adaptation effect
References
Number sense
In mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude,
relationships, and how they are affected by operations." Other definitions of number sense emphasize an ability to
work outside of the traditionally taught algorithms, e.g., "a well organised conceptual framework of number
information that enables a person to understand numbers and number relationships and to solve mathematical
problems that are not bound by traditional algorithms".
There are also some differences in how number sense is defined in math cognition. For example, Gersten and Chard
say number sense "refers to a child's fluidity and flexibility with numbers, the sense of what numbers mean and an
ability to perform mental mathematics and to look at the world and make comparisons."
In non-human animals, number sense is not the ability to count, but the ability to perceive changes in the number of
things in a collection.[1] All mammals and most birds will notice if there is a change in the number of their young
nearby. Many birds can distinguish two from three.[2] In humans, small children around fourteen months of age are
also able to notice something that is missing from a group that they are familiar with.[citation needed]
Researchers consider number sense to be of prime importance for children in early elementary education, and the
National Council of Teachers of Mathematics has made number sense a focus area of pre-K through 2nd grade
mathematics education. An active area of research is to create and test teaching strategies to develop children's
number sense.
Number Sense also refers to the contest hosted by the University Interscholastic League. This contest is a
ten-minute test where contestants solve math problems mentally—no calculators, scratch-work, or mark-outs are
allowed.
Concepts involved in number sense
The term "number sense" involves several concepts of magnitude, ranking, comparison, measurement, rounding,
percents, and estimation, including:[3]
•
•
•
•
•
•
•
estimating with large numbers to provide reasonable approximations;
judging the degree of precision appropriate to a situation;
rounding (understanding reasons for rounding large numbers and limitations in comparisons);
choosing measurement units to make sense for a given situation;
solving real-life problems involving percentages and decimal portions;
comparing physical measurements within and between the U.S. and metric systems; and
comparing degrees Fahrenheit and Celsius in real-life situations.
Those concepts are taught in elementary-level education.
44
Number sense
References
[1] http:/ / www. math. twsu. edu/ history/ topics/ num-sys. html#sense
[2] Dantzig, Tobias. Number: The Language of Science. New York: Macmillan Company, 1930.
[3] "Unit 1: Number and Number Sense" (20-day lesson), STPSB.org, St. Tammany Parish School Board, Covington, LA (USA), 2009,
overview webpage: ST-MathGrade7Unit-topics (http:/ / www. stpsb. org/ files/ MathGrade7Unit1090120034917. htm).
External links
• Number Worlds (http://clarku.edu/numberworlds/index.htm) - a site with number sense development
materials
Ordinal numerical competence
In human developmental psychology or non-human primate experiments, ordinal numerical competence or
ordinal numerical knowledge refers to the ability to 'count' objects in order and to understand the greater than and
less than relationships between numbers. It has been shown that children as young as 2 can make some ordinal
numerical decisions. There are studies indicating that some non-human primates, like chimpanzees and rhesus
monkeys have some ordinal numerical competence.
Ordinal Numerical Competence in Humans
Prenatal
There is no evidence to support prenatal ordinal numerical competence. Teratogens such as stress can alter prenatal
neural development, leading to diminished competence after birth. Physical effects of teratogens are common, but
endocrine effects are harder to measure. These are the factors that influence neural development and by extension the
development of ordinal numerical competence. Premature birth is also a risk factor for developmental problems
including reduced brain activity. Brain activity is measured from outside the body with electroencephalography.
Infants
There have been a vast number of studies done on infants and their knowledge of numbers. Most research confirms
that infants do in fact have a profound innate sense of number, both in abstract and finite ways. Infants as young as
49 hours can accurately match up images with a certain amount of objects, with sounds that contain the same number
("ra, ra, ra, ra") as the number of objects in the image. Because the sounds are abstract, or visibly there, we can see
that infants as young as 49 hours have some abstract numerical sense as well as concrete numerical sense shown by
their recognition of the image with the corresponding number of objects. Similarly, infants around the age of 7
months can also match up images of random objects.
Although children as young as 49 hours can match up the number of sounds with the number of objects, they can
only do so at certain ratios. When 1:3 ratios were used (4 sounds and 4 objects or 12 objects), around 90% of the
infants paid more attention to the corresponding image thus showing their recognition. However, when 1:2 ratios
were used, only 68% of infants showed recognition of the correct corresponding image. This tells us that although
infants can recognize corresponding numbers of sounds and objects, the two images of objects must be visibly
different - one must have a much larger number of objects, or a much smaller number of objects.
Although there has to be a stark difference in the choices for infants to recognize the correct matching set of numbers
(1:3 vs 1:2), this seems to prove that infants have an innate numerical sense, but it may not be the same numerical
sense as older children. Around the age of three and a half years children lose some of their numerical sense.
Whereas children younger than three can recognize that four pebbles spread out in a line is less than six pebbles
45
Ordinal numerical competence
scrunched together in a line, children around the age of three and a half mysteriously lose this ability. Researchers
believe that this is because children around this age begin to rely heavily on the physical properties of the world and
objects within it, such that longer equals more. Although the ability to recognize that six pebbles closely lined up
together is more than four pebbles spread out farther from one another goes away around that age, it comes back
around four years of age when children begin to count.
Adults
Both behavioral research and brain-imaging research show distinct differences in the way "exact" arithmetic and
"approximate" arithmetic are processed. Exact arithmetic is information that is precise and follows specific rules and
patterns such as multiplication tables or geometric formulas, and approximate arithmetic is a general comparison
between numbers such as the comparisons of greater than or less than. Research shows that exact arithmetic is
language-based and processed in the left inferior frontal lobe. Approximate arithmetic is processed much differently
in a different part of the brain. Approximate arithmetic is processed in the bilateral areas of the parietal lobes. This
part of the brain processes visual information to understand how objects are spatially related to each other, for
example, understanding that 10 of something is more than 2 of something. This difference in brain function can
create a difference in how we experience certain types of arithmetic. Approximate arithmetic can be experienced as
intuitive and exact arithmetic experienced as recalled knowledge.
The conclusions from behavioral research and brain-imaging research are supported by observations of patients with
injuries to certain parts of the brain. People with left parietal injuries can lose the ability to understand quantities of
things, but keep at least some ability to do exact arithmetic, such as multiplication. People with left-hemisphere brain
damage can lose the ability to do exact arithmetic, but keep a sense of quantity, including the ability to compare
larger and smaller numbers. This information confirms that distinct parts of the brain are used to know and use
approximate and exact arithmetic.
Various researchers suggest that the processing of approximate arithmetic could be related to the numerical abilities
that have been independently established in various animal species and in preverbal human infants. This may mean
that approximate arithmetic is an adaptive train that humans developed through evolution. The combination of this
potential evolutionary trait and language-based exact arithmetic may be the reason that humans are able to do
advanced mathematics like physics.
Non-Human Animals
Animals share a non-verbal system for representing number as analogue magnitudes.[1] Animals have been known to
base their rationality on Weber’s Law. This historically important psychological law quantifies the perception of
change in a given stimulus. The law states that the change in a stimulus that will be just noticeable is a constant ratio
of the original stimulus. Weber’s Law describes discriminability between values based on perceptual continua such
as line length, brightness, and weight.[2]
Rhesus Monkeys
Studies of rhesus monkeys' foraging decisions indicate that animals spontaneously, and without training, exhibit
rudimentary numerical abilities. Most animals can determine numbers in the values 1 through 9, but recent
experiments have discovered that rhesus monkeys can quantify values from 1 up to 30. Monkeys' numerical
discrimination capacity is imposed by the ratio of the values compared, rather than absolute set size.[3] This
computation process focuses around Weber’s Law and the expectation violation procedure. This suggests that rhesus
monkeys have access to a spontaneous system of representation, which encodes the numerical differences between
sets of one, two and three objects, and contrasts three objects from either four or five objects as well. These
representations indicate the semantics of an encoded natural language. These encoded natural languages are also seen
in experiments with many animals including pigeons and rats.
46
Ordinal numerical competence
Other Animals
Experiments have shown that rats are able to be trained to press one lever after hearing two bursts of white noise,
then press another lever after four bursts of white noise. The interburst interval is varied between trials so the
discrimination is based on number of bursts and not time duration of the sequence. Studies show that rats as well as
pigeons learned to make different responses to both short and long durations of signals. During testing, rats exhibited
a pattern called break-run-break; when it came to responding after a stint of little to no response, they would
suddenly respond in high frequency, then return to little or no response activity. Data suggests that rats and pigeons
are able to process time and number information at the same time. The Mode Control Model shows that these
animals can process number and time information by transmission pulses to accumulators controlled by switches that
operate different modes.
References
[1] Brannon, 2005; Brannon & Terrace, 1998-2000; Cantlon & Brannon, 2005; Feigenson, Dehaene, & Spelke, 2004; Gelman &Gallistel, 2004;
Nieder, Freedman, & Miller, 2002; Nieder & Miller, 2003
[2] http:/ / www. britannica. com/ EBchecked/ topic/ 638610/ Webers-law
[3] Brannon, E.M, and H.S Terrace.1998. Ordering of the numerosities 1 to 9 by monkeys. Science 282:746-749
47
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Rjwilmsi, YanShen, 10 anonymous edits
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