2. • Overview of Topic
• Teaching concepts
• Solving examples
• Homework
• Tests
• Solving past papers
• Any suggestion made by the student
GSCE Mathematics
Teaching methodology
3. • Numbers
• Algebra
• Graphs
• Shape and space
• Sets
• HANDLING DATA
• Probability
• Sequence and series
GSCE Mathematics
Chapters
4. • How to define the real numbers
• About factors, multiples and prime factors
• How to write a whole number as a product of prime factors
• The properties of integers
• The properties of rational numbers
• About rounding and significant figures
• How to geometrically construct root 2 and root 3
• About orders of magnitude and scientific notation
In this chapter you have learned to:
05
Chapter 1 – Real Numbers
5. Real numbers, R, are all numbers that lie along an infinitely long number line.
Natural numbers, N, are the ordinary counting numbers.
Integers, Z, are made up of zero and all the positive and negative whole numbers.
Rational numbers, Q, are numbers in the form of
𝑎
𝑏
where 𝑎, 𝑏 𝜖 𝑍, b ≠ 0
Irrational numbers, Q, are numbers that can’t be written
in the form of
𝑎
𝑏
where 𝑎, 𝑏 𝜖 𝑍, b ≠ 0
Real
Irrationals
Rationals
Naturals
Integers
05 Real Numbers
6. IS IT PRIME? IS IT PRIME?
Prime numbers are natural numbers that have two factors only.
The Primes from 1 to 20
05 Real Numbers
NUMBER FACTORS NUMBER FACTORS ISITPRIME?
1 11
2 12
3 13
4 14
5 15
6 16
7 17
8 18
9 19
10 20
1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
1, 11
1, 2, 3, 4, 6, 12
1, 2,7, 14
1, 13
1, 3, 5, 15
1, 2, 4, 8, 16
1, 17
1, 2, 3, 6, 9, 18
1, 19
1, 2, 4, 5, 10, 20
No
Yes
No
No
No
No
No
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
IS IT PRIME?
7. To prove: 3 is irrational. The proof of this result is
an example of proof by
contradiction.
Proof:
Assume that is rational and can therefore be written in the
form of
𝑎
𝑏
where 𝑎, 𝑏 𝜖 𝑍, b ≠ 0
Also, assume that the fraction
𝑎
𝑏
is written in simplest terms, i.e. HCF(a, b) = 1.
3 =
𝑎
𝑏
⟹ 3 =
𝑎2
𝑏2 (squaring both sides)
∴ 𝑎2 = 3𝑏2 (∗)
As 𝑏2
is an integer, 𝑎2
has to be a multiple of 3, which means that 3 divides 𝑎2
.
If 3 divides 𝑎2
, then 3 divides a.
∴ a = 3k, for some integer k. Substituting 3k for a in (*) gives,
∴ (3𝑘)2
= 3𝑏2
∴ 9𝑘2
= 3𝑏2
⟹ 𝑏2
= 3𝑘2
As 𝑘2 is an integer, 𝑏2 has to be a multiple of 3, which means that 3 divides 𝑏2.
Therefore, 3 divides b. If 3 divides 𝑎 and 3 divides b, then this contradicts the
assumption that HCF(𝑎, b) = 1.
Q.E.D
05 Real Numbers
Proof that 3 is Irrational
