1. Chapter 2: Number Systems
Investigate the representation of numbers and arithmetic operations so that you can perform the operations of addition, subtraction, multiplication, and division and understand the relationship between these operations and the properties: commutative, associative and distributive in N, Z and Q and in R\Q, including operating on surds.

2. Review: Revision of material from Active Maths 1
1. Natural numbers: The set of natural numbers is the set of counting numbers They are also called the positive whole numbers.
N = {1, 2, 3, 4, ...}
2. Integers: The set of integers is the set of natural numbers, their negatives and zero.
Z = {..., –3, –2, –1, 0, 1, 2, 3, ...}
3. Rational numbers: The rational numbers are numbers that can be written in the form 𝑎 𝑏 ,where 𝑎, 𝑏 ∈ Z and 𝑏 ≠ 0.
Q = {Any number of the form 𝑎 𝑏 ,where 𝑎, 𝑏 ∈ Z and 𝑏 ≠ 0.
N ⊂ Z ⊂ Q
A prime number is a natural number that has exactly two factors.

3. The Commutative, Associative and Distributive Properties
If a, b, c ∈ N, then:
(i) a + b = b + a (Addition is commutative)
3+4=4+3
(ii) a × b = b × a (Multiplication is commutative)
3×4=4×3
(iii) (a + b) + c = a + (b + c) (Addition is associative)
3+4 +2=3+ 4+2
(iv) (a × b) × c = a × (b × c) (Multiplication is associative)
3×4 ×2=3× 4×2
(v) a × (b + c) = a × b + a × c (Multiplication distributes over addition)
2× 3+4 =2×3+2×4

4. Irrational Numbers 𝐴 𝐵 𝑥 1 𝐷 𝐶
The length 𝑥 of the diagonal of the unit square can be found using the theorem of Pythagoras. 𝑥 1 𝐴 𝐵 𝐶 𝐷
𝑥 2 =
𝑥 2 =1+1
𝑥 2 =2
𝑥= 2 (as 𝑥 > 0)
2 is bigger than 1 =1 and smaller than 4 =2. Therefore, 2 lies on the number line between 1 and and 2 are both rational. What about ?
A real number is any number that can be represented on the number line.
It can be proved that cannot be written as a ratio of integers (i.e. in the form 𝑎 𝑏 ). We say that is irrational.
An irrational number is a real number that cannot be written in the form 𝑎 𝑏 , where 𝑎 is an integer and 𝑏 is a non-zero integer. In other words, an irrational number is a real number that cannot be written as a ratio of integers.

5. Real Numbers
The rational and irrational numbers together make up the real number system.
Irrationals
Reals
Insert each of the following numbers in its correct region on the diagram below.
(a) (b) (c) (d) (e) (f) –8 (g) (h) 𝜋 (i) 5𝜋 (j) 𝑹 𝟓𝝅 𝝅
𝟑 𝟒
𝟏𝟒𝟒
𝟏𝟒 𝟐
𝟓 𝟐
𝟑 𝟐 –8 𝟑 𝟓

6. 2 and 3 are examples of surds.
A surd is an irrational number that can be written as the root of a rational number.
2 and 3 are examples of surds.
Which of the numbers 9 , 𝜋, , 5 are surds?
9 is not a surd as 9 =3, which is rational.
𝜋 is not a surd as 𝜋 cannot be written as the root of a rational number.
3 8 is not a surd. Although is the cube root of a rational number, 3 8 =2 which is rational.
5 is a surd as it is both irrational and can be written as the root of a rational number.

7. Laws of Surds Law 1: 𝑎 𝑏 = 𝑎𝑏 , 𝑎≥0, 𝑏≥0 Law 2: 𝒂 𝒃 = 𝒂 𝒃 , 𝑎≥0, 𝑏≥0
(a) Show that 𝒂 𝒃 = 𝒂𝒃 , if 𝒂 =𝟗 and 𝒃 = 𝟒
(b) Show that 𝒄 𝒅 = 𝒄 𝒅 , if 𝑐 =100 and 𝑑 =25
(c) Show that 𝑝 𝑝 =𝑝,if 𝑝 =16
(b) = 10 5 =2
= 4 =2
(a) = 3 2 =6
(9)(4) = 36 =6
(c) = 4 4 =16

8. Adding, Subtracting and Reducing Surds
Reduce 𝟒𝟓 to its simplest surd form.
Step 1: Find the largest square number that is a factor of is the largest square factor of 45.
Step 2: = 9×5 =
=3 5
Simplify 𝟓𝟎 + 𝟖 − 𝟑𝟐 , without the use of a calculator.
− 32
= 25×2 + 4×2 − 16×2
= −
= −4 2
=3 2

9. Further Multiplication and Division
Use an area model to simplify the following:
𝐚 𝟑 𝟕 +𝟏
𝐛 𝟐+ 𝟑 𝟐 +𝟑
(a) Form a rectangle that is subdivided into two smaller rectangles, as shown in the diagram: 7 1 7 3 1 3 21 3 =
(b) A rectangle is subdivided into four smaller rectangles, as shown in the diagram. 2 3
Summing the area of the four smaller rectangles gives the solution: 2 3
2 2 6 = 6
3 3

10. Rationalising the Denominator
The method of eliminating a surd from the denominator in order to simplify an expression is known as rationalising the denominator.
Simplify 𝟏 𝟐
To rationalise the denominator, multiply both the numerator and denominator by
= × = =