Download Σ ProjectMathsNotes.ie 04/1437 Leaving Certificate Higher Level

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04/1437
Leaving Certificate Higher Level
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Handout 1
Number Theory, Set Theory & More
1.1 Natural Numbers (N)
−4
−3
−2
−1
0
1
2
3
4
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 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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−1
0
1
2
3
4
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+ .
1
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 (or ‘ratio’, hence the word ‘rational’.)
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 of size 0.1 or 0.01 together, we would eventually make up the
full unit of the rectangle, populating going from one side to the other.
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
Zero plus zero equals zero. And so on.
In a way, it doesn’t make sense to ask “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
= 0.3̇ = 0.33333 . . .
Terminating decimal. (i.e. it ends somewhere.)
...
Terminating decimal.
...
Non-terminating, repeating decimal. (i.e. it never ends.)4
1
13
= 0.0̇76923̇ = 0.076923076923076 . . .
1
26
= 0.03̇84615̇ = 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. 0.3̇ = 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
= 0.0̇37̇ = 0.037037037037037 . . . going on forever. The first
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
= 0.03̇84615̇, which waits until the second decimal place before the repeating
for example in 26
1
= 0.019̇23076̇ = 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
1
the decimal, 0.083̇, which goes on forever, and can be represented as a fraction (0.083̇ = 12
),
‘irrational numbers’, like ‘π’ and ‘e’, cannot be represented as fractions.
1.7 Real Numbers (R)
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−1
0
1
2
3
4
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.
3
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