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LEAVING CERTIFICATE HIGHER LEVEL version 1.6
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DO NOT TURN ANY PAGE UNTIL YOU
COMPLETELY UNDERSTAND WHAT
YOU’VE READ SO FAR!
Read this from the front cover to the back cover. Care has been taken not to use any terminology
which wasn’t explained prior. If you come across a term that you don’t fully understand, then
read back over the previous pages and you should see what it is. You could also try looking up
the word in a dictionary or online search.
When you see the “mandatory exercises”, this means that you absolutely must not proceed to
the next page, before you have answered this question. The answer is on the next page, overleaf.
Some of these exercises are designed so that you can learn a great deal by completing them.
Others of these exercises help develop your problem solving skills - very important to develop
problem solving skills for exam success. It is crucial that you try your best to complete these
mandatory exercises. Take several attempts at each one, if you do not get it right, straight away.
Have patience. (It should be possible to figure out what the answer is from reading the material
that precedes the exercise, and by using your own brain-power.)
When you have finished reading each handout, do all the exercise questions. These questions, as
well as the ‘mandatory exercises’, are also designed to help develop your problem-solving ability,
specifically in relation to answering exam questions. Furthermore, if you do not do them, you
may be missing some parts of the syllabus. If you cannot figure out how to do any particular
exercises, keep coming back to them trying again and again. If all else fails you will need to
consult with your classmates or teachers as solutions are not directly provided.
You need to get a copy of the Formulae & Tables booklet, published by the State Examinations
Commission (SEC), if you don’t have one already. You’re also going to need good equipment such
as rulers, set-squares, and a good compass (with a sharp point.) For the ‘mandatory exercises’
and ‘practice exercises’ at the end of each handout, you’re going to want square-ruled paper to
write on. Indeed, you should only use square ruled paper for maths, from now on. You need
to get used to writing small and neatly on squared paper. This is because on the exam script,
the box in which you put your fully worked out answers is small and square ruled (not line
ruled.) The neater you write in an exam, the better, as it helps you see patterns easier, figure
out answers quicker and make less mistakes in your calculations, saving you time in an exam.
You can also fit more information in, increasing your chances of picking up more marks.
All honour to Allah - The Most Beneficent, The Most Merciful.
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Handout 1
Number Theory, Set Theory & More
1.1 Natural Numbers (N)
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−2
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0
1
2
3
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The natural numbers are basically all positive whole numbers, i.e. any number that you can
count with your fingers, as in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, etc., going on indefinitely. Decimals
are not included in this set (e.g. 0.25 ∈
/ N).1
As far as the Leaving Cert Project-Maths syllabus is concerned, ‘0’ is not a member of the set
of natural numbers. That’s why it is excluded from the number line above.2 If you want to
include 0 in your set of natural numbers, you would use the notation N0 . That ‘subscript zero’
lets everybody know that zero is included in your set, e.g. N0 = {0, 1, 2, 3, . . . }
1.2 Integers (Z)
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0
1
2
3
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The integers is the set of all whole numbers. Don’t incorrectly say that the integers are “the
set of positive and negative whole numbers” because guess what, zero is neither positive nor
negative, and zero is an element of the set of integers. (Zero is ‘unsigned’, i.e. no sign.)
The set of positive integers, Z+ , is {1, 2, 3, 4, . . . }. The set of negative integers, Z− , is {−1, −2, −3, −4, . . . }.
The set of positive integers including zero, Z+
0 , is {0, 1, 2, 3, . . . }.
The set of negative integers including zero, Z−
0 , is {0, −1, −2, −3, . . . }.
(The same notation applies for rational numbers Q, and real numbers R.)
1 ‘I.e.’ is short for the Latin id est and it basically means ‘that is’. ‘Etc.’ is a written abbreviation for the Latin
et cetera and means ‘and the rest’. ‘E.g.’ in Latin stands for exempli gratia and roughly means ‘for example’.
2 There’s a lack of agreement in maths and science communities with whether zero is included as an element
of the set of natural numbers, or not. Some people say it is, some people say it isn’t. For those of us who want
to be clear, we may notate the set of natural numbers excluding 0 as N+ , (with the ‘superscript plus’ symbol,)
and the set of natural numbers including 0 as N0 . In the Project-Maths syllabus, N = N+ .
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1.3 Rational Numbers (Q)
A rational number is any number which can be expressed in the form
b 6= 0. Basically it’s any fraction.
a
b,
where a, b ∈ Z, and
The reason why we don’t have b = 0 there, is because any non-zero number divided by zero
is ‘not defined’, and, 00 is ‘indeterminate’. ‘Not defined’, or ‘undefined’ means that the answer
doesn’t exist, and ‘indeterminate’ means that there can be an infinte number of answers.3
1.4 Why We Can’t Divide
1
0
Imagine we had a rectangle as below divided into 10 evenly sized segments.
0.1
Then imagine we had the same rectangle, except this time divided up into 100 segments:
0.01
If we added each of the segments together, we would eventually make up the full unit of the
rectangle.
We can say:
1
0.1
= 10.
We can also say:
1
0.01
= 100.
We cannot say that 01 equals anything, however, as if we try to add segments of size 0 together to
get the full unit, we would never be able to get this, as if you kept on adding segments of size zero,
you would just get them piling up on one side of the rectangle and not going anywhere:
0
In a way, it doesn’t make sense to ask the question “what is 1 divided by 0”.
0 does not divide into 1.
1.5 Classification Of Decimals
1
4
= 0.25
1
16
1
3
...
= 0.0625
= 0.3˙ = 0.33333 . . .
Terminating decimal. (i.e. it ends somewhere.)
...
Terminating decimal.
...
Non-terminating, repeating decimal. (i.e. it never ends.)4
1
13
˙
= 0.07692
3˙ = 0.076923076923076 . . .
1
26
˙
= 0.038461
5˙ = 0.038461538461538 . . .
...
...
3 More
4 It
Non-terminating, repeating decimal.
Non-terminating, repeating decimal.
on this later. See the ‘Limits’ sections of the ‘Lines’ handout, for a full explanation.
also has a repeating pattern. The 3 repeats itself. Hence being called a ‘repeating’ or a ‘recurring’ decimal.
2
The little dots above the numbers there let us know that the decimals are ‘recurring’ decimals.
E.g. 0.3˙ = 13 = 0.333333333333333333 . . . going on forever. (Also called ‘repeating decimals’.)
It’s worth reminding you at this stage that you can have two dots in the decimals as well, for
1
˙ 7˙ = 0.037037037037037 . . . going on forever. The first
= 0.03
example the rational number 27
dot marks the beginning of the recurring pattern, and the final dot marks the end where it then
repeats on and on again, indefinitely.
It should be noted that the location of where the repeating pattern begins can be delayed, like
1
˙
˙ which waits until the second decimal place before the repeating
= 0.038461
5,
for example in 26
1
˙
= 0.0192307
6˙ = 0.0192307692307692 . . . It waits until the
pattern starts. Another example is 52
third decimal place before it starts repeating.
1.6 Irrational Numbers
‘Irrational numbers’ are non-repeating, non-terminating decimals. (I.e. never ending and with
no repeating pattern.) An irrational number
√ √ is any number which cannot be expressed as a
fraction. Typical examples are ‘π’, ‘e’, 2, 3. These are decimals whose decimal places go
on forever (non terminating), e.g. π = 3.141592654 . . . and e = 2.718281828 . . . , and unlike
˙ which goes on forever, and can be represented as a fraction (0.083˙ = 1 ),
the decimal, 0.083,
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‘irrational numbers’, like ‘π’ and ‘e’, cannot be represented as fractions.
1.7 Real Numbers (R)
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0
1
2
3
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On the number line, the set of real numbers is represented as a continuous thick line.5 A real
number is effectively any number which exists on the number line. It can be a whole number, it
can be a decimal, it can be an irrational number. They are basically the set of ‘rational numbers’
plus ‘irrational numbers’.
1.8 What Is π?
π is the ratio of the circumference (c) of a circle to its’ diameter (d). It’s the same ratio for all
circles. And as written before, its’ value to nine decimal places’ accuracy is 3.141592654 . . .
c
MANDATORY ACTIVITY:
d
The formula for the circumference of a circle is c = 2πr.
Given that dc = π , can you derive the formula c = 2πr ?
(where ‘r’ stands for radius)
5 As you might remember from studying statistics, you can have ‘continuous data’ and ‘discrete data’. Examples
of ‘discrete’ objects are cans of cola, and examples of ‘continuous’ data sets is lists of heights. The sets of natural
numbers and integers are ‘discrete’ data sets, whereas the set of real numbers is a ‘continuous’ set.
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