NUMBER SYSTEMS

1: NUMBER SYSTEMS

1.1 REAL NUMBER, R

1.1.1 Sets of Real Numbers

 Natural numbers, N

N = { 1, 2, 3, 4, 5, ….}

Natural numbers are numbers which people used to count.
 Whole number, W

W = { 0, 1, 2, 3, 4, 5, ….}

Natural numbers together with 0.
 Integers, Z

Z = { …, -3, -2, -1, 0, 1, 2, 3, ….}

Z+ = { 1, 2, 3, ….}  positive integer

Z- = { ….., -3, -2, -1,}  negative integer

Consists of natural numbers, zero and the negative natural numbers.

 Rational numbers, Q

p { p, q  Z and q  0 }
- Numbers that can be written in the form (fraction)

q

- The decimal representations of rational numbers are repeating or terminating

Repeating decimal
Ex: a) 1 = 0.16666….. can be written as 0.16 or 0.16
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b) 3 = 0.272727… can be written as 0.27 or 0.27
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Terminating decimal b) 3 = 0.75
Ex: a) 2 = 0.4 4
5

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Converting decimal number (repeating number) to fraction form.

Example:

a) 0.16666….
= 0.1 + 0.06666…
 16
10 90
 90  60
900
 150  1
900 6

b) 1.272727…
 1 27
99
 1 3 or 14
11 11

Try this!

Convert the following to fraction form.

1. 2.13131313…. 2. 3.145

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 Irrational numbers, Q
- Cannot be written as fractions.
- The decimals of irrational numbers neither terminate nor repeat.
Ex: 5  2.236067978...
 = 3.141592654…

- Note that the square roots such as 9 is not irrational number since 9  3 .

All rational numbers and irrational numbers constitute the set of Real numbers, R.

REAL NUMBERS

Rational numbers, Q Irrational numbers, Q

Integers, Z Fractions
Whole numbers, W
Natural numbers, N

 3 1.4

-3 -2 -1 0 1 2 3

Real number line

Exercises :

Thick for the correct answers.

Natural number Integer Whole numbers Rational numbers Irrational
numbers

4
-3
6.5

- 1.090

2
e
9
13
-7

3
2

9

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1.1.2 Intervals

Display on a real Closed Interval notation Inequality
number line Open
Half-open [a, b] a≤x≤b
ab Half-open (a, b) a<x<b
ab Half-open [a, b) a≤x<b
ab Half-open (a, b] a<x≤b
ab [a, ∞)
a Half-open (a, ∞) x≥a
a Half-open x>a
(-∞, a]
a (-∞ a) x≤a
a x<a

Example:

Graph the following intervals and solve each of the following
a) (-1, 7)  [2 , 9)
b) ( 0, 6)  [-2, 4)  (3, 7]

1.1.3 Linear Inequalities

Linear inequality in one variable can be written in one of the following forms:

ax + b < 0 ax + b ≤ 0 ax + b > 0 ax + b ≥ 0

Example :

Solve the following inequalities. Give your answer in interval notation.

a) 4x – 9 ≥ 7 b) 9 - 4x > 15 b) -1 < 2x – 5 ≤ 9

Solution: Beware with the b) -1 < 2x – 5 ≤ 9
-1 + 5 < 2x ≤ 9 + 5
a) 4x – 9 ≥ 7 b) 9 - 4x > 15 sign & symbol! 4 < 2x ≤ 14
4x ≥ 7 + 9 -4x > 6 2<x≤7
4x ≥ 16 x < -1.5 27
x≥4 (2, 7]

4 [4, ∞) -1.5 4
(-∞, -1.5)

Exercises:

1. Given that A = {x : x2 – 4 = 0}, list all the elements in set A if
a) x  R
b) x  N

2. Express the following as a fraction in its lowest form.

a) 1.83 b) 2.45

3. Show each of the following intervals on the real number line.
a) (-4, 2)
b) [6, 9)
c)(-∞, 0)
d) [-2, ∞)
e) (-3, 2)  [1, 5]
f) (-∞, 1]  [-6, 3)

4. Solve the following inequalities and express your answer in interval notation.
a) 3x + 4 > 13
b) 3 – 5x ≥ 18
c)-10 ≤ 3x – 1 ≤ 5
d) -1 < 5 – 2x ≤ 9

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1.2 ABSOLUTE VALUES

- Modulus of a real number
- Written as x
- The absolute value of every real number is positive.

x = x if x > 0
-x if x < 0

Example :
a) If x = 6

6 =6

b) If x = -6
 6 = - (-6)
=6

1.2.1 Absolute Values Inequalities

Properties of Absolute Values

x < a  -a < x < a
x > a  x < -a or x > a

Example :

1. Solve the equation x  4  9

Solution: or x + 4 = - 9
x+4=9 x = - 13

x=5

2. Solve the following inequalities b) 5 2x  9
a) x  3  5

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Solution : b)
a)

Exercises:

1. Solve the following equations.
a) x  2  8
b) 1 2x  7

2. Solve each of the following inequalities and express your answer using interval notation.
a) 2x  3  9
b) 1  3x  7

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1.3 SURDS 4 , 9 ,2  4 are not surds

- Containing a root, because the solution are
- Irrational numbers rational numbers.
- Ex : 2 , 3 , 5 , 1  2 , 3  5
-

1.3.1 Properties of surds

a  b  ab

a a
bb

1.3.2 Simplifying surds Perfect square number:
4, 9, 16, 25, 36, 49, 64,
Example : 81, 100, 121, 144, 169

a) 50 = 252 d) 2 162  196  338
= 25 2
= 52

b) 2 8  200  4 18
= 2 42  1002  4 92
= 2(2 2) 10 2  4(3 2)
= 4 2 10 2 12 2
=2 2

Try this out!

c) 32  128  200

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1.3.3 Rationalising the denominator Conjugate surd

- When surds appear in the denominator of a fraction. If the product of two surds is a
- Example : rational number, for example
(2  3)(2  3)  1
1 , 2 , 3 1 then (2  3) is called the
2 2 7 3 1 conjugate surd of (2  3)
and vice versa.
- Method : multiply both the numerator and denominator
with its conjugate.

Exercises:

Write down the conjugate for each of the following surds.

Surd Conjugate Surd Conjugate

5 3 5

1 3 2 5

32 3 2 53

3 2 2 2 5

Example:
Rationalize the denominator for each of the following surds.

a) 6 c) 3  3 Try this out!
3 2 3
9
= 6 3
33

=63
3

=2 3

b) 2
5 2

= 2  5 2
5 2 5 2

= 2( 5  2)
( 5  2)( 5 2)

= 2 54
52 5 2 5 4

= 2 54
54

= 2 54

1.4 INDICES AND EXPONENTS

an = a x a x a x a x a x …..
n times

Example : 53 = 5 x 5 x 5

1.4.1 Law of exponents

am x an = am + n
am  an = am - n
(am )n = am n

1. zero exponent

a0 = 1

2. negative exponent

an  1
an

3. Fractional exponent

1

an  n a

m1

a n  (a n )m  (n a)m

m 1
n
an  (a m )  (n am )

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Examples :

Find the value of each of the following.

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2  16  4 3
 81 
(a) 273 (b) (c) 0.25 2

Solution: Ans: -

2 27
8
(a) 273  (3 27)2
Ans: -
 (3)2 8
9 11

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 16  4
 81 
(b) 

33

(c) 0.25 2  (0.52) 2

1.4.2 Exponential Equations

If 2x  25 then by equating the exponents, x  5

Example:
Solve the following exponential equations.

a) 9x = 27
b) 22x – 9. 2x + 8 = 0

Solution :

a) Choose the same base for both sides.
9x = 27

(32)x = 33
32x = 33
2x = 3
x3
2

b) 22x – 9. 2x + 8 = 0

(2x)2 – 9. 2x + 8 = 0  form a quadratic equation

Let 2x = y

y2 – 9y + 8 = 0
y = 1 or y = 8

2x = 1 2x = 8
2x = 20 2x = 23

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1.5 LOGARITHMS

In index form, index / exponent

23 = 8

base

In logarithms form,

log 8 = 32  logarithm of 8 to base 2 is 3

In general, loga N = x ; a, N  R+

ax = N

Exercises: Logarithm Exponent form Logarithm
log5 125 = 3 44= 256 log216 = 4
Complete the table. 3 = log3 27
Exponent form
32 = 5 Remember!
81 = 912
logaa = 1
1.5.1 Laws of Logarithms log33 = 1
loga xy = loga x + loga y log100100 = 1
lg a = log 10 a
loga x = loga x – loga y
y 13

loga xp = p loga x

1.5.2 Changing the Base of Logarithms

logb a = log c a
log c b

Example :

1. Express each of the following in terms of lg x, lg y and lg z.

a) log (xy2) c) log  x 
yz
= log x + log y2
= log x + 2 logy

b) log  xy  d) log  1 
 z  xyz

= log xy – log z

= log x + log y – log z

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1.5.3 Solving Logarithm Equations.

1. Express each of the following as a single logarithm.

a) 2 log2 5 + 1 log2 9  log2  1 
2  4 

b) 2 loga x + loga y - 1
2

2. Solve the following equations. ii. Add ‘log’ to both sides of equation
Example : 2x = 8 log 2x = log 8
i. Convert to logarithm form x log 2 = log 8
x = log2 8 x  log8
x  log8 log2
log2 x=3
x=3

a) 3x = 8

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b) 52x = 8
c) 3x+1 = 18

Example: Solve log4 x logx 4  2.5

Solution:

log4 x logx 4  2.5

log4 x log4 4  2.5  change the base to base 4
log4 x  multiply the equation by log4 x

log4 x  1 x  2.5
log4

log4 x2 1  2.5log4 x  let y = log4 x

y2 1  2.5y or y2 2.5y 1  0

y = 2 or y = 1
2

log4 x  2 log4 x  1
x  16 2

x 2

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d) 3 logx2 – log2 x = 2
e) log4 x + logx4 = 2.5

1.6 COMPLEX NUMBER

- Not a real number
- Cannot find the square root
- The square of complex number is negative

Numbers in the form a + bi are called complex numbers

 a and b are real number
 denoted by z
 a is the real part, Re(z) and bi is the imaginary part Im(z)
 ex: 2 – 3i, -1 + 2i, 3 1  5i

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What is i?

i - an imaginary number

- representing  1

1 = i

i= 1 ; i 2 = -1

Example :

 4   1 4   1  4 = 2i Imaginary numbers
 7   1 7   1  7 = 7i

Exercises:

Express each of the following complex number in the form of a + bi.

a) 5 +  4
b) 1   9
c) 6   2
d) x   16y2

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1.6.1 Operations on Complex Numbers

1. Addition and subtraction
When adding two complex numbers, the real parts are added separately from imaginary.

Example :
Given z1 = 3 – 7i, z2 = 1 + 2i, find z1 + z2.

2. Multiplication
The distributive law of multiplication is used to multiply complex numbers.

Example :
Find (3 + 4i)(1 – 2i)

If a complex number a + bi is multiplied by the complex number a – bi, then

(a + bi)(a – bi) = a2 – (bi)2 Real number
= a2 + b2

The multiplication of a + bi and a – bi gives a real number, therefore, a – bi is known as the
complex conjugate of a + bi. The conjugate of z is denoted by z*.

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Example :
Find the product of (2 – 3i) and its conjugate.

3. Division
- To divide a number by a complex number, the denominator has to be transformed into a real
number.
- This is done by multiplying the numerator and denominator by the conjugate of the
denominator.

Example :
Find 1  i
2  3i

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1.6.2 Equal of complex numbers

Example 1:
Given that (x + yi)(2 + i) = 6 – 2i, find the values of x and y.
Solution:

Example 2:
Find the square roots of the complex number 12 + 5i.
Solution:

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Exercises:

1. Find the value of a and b if a + ib = (3 – 2i)2.
2. Solve z2 + 2z + 3 = 0.
3. Find the square roots of 24 + 7i.

1.6.3 Argand Diagram

- to represent a complex number z = a + bi on coordinate plane.
- x-axis  representing the real number
- y-axis  representing the imaginary number

Example: y
z = 3 + 2i
z* = 3 - 2i

z x

o
z*

1.6.4 Modulus and Argument

If P represents the complex number z = x + yi in the Argand Diagram, the length of OP, denoted by r,
is called the modulus of z, z .

r = z  x2  y2 .

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y
P(x,y)

r y = r sin 

 x

O x = r cos 

The angle of xOP,  is called the argument of z. Written as arg z. It is measured in radians.

arg z  θ  tan1  y  (+) y y
x
x (-)
- <  < 
x
y x 
 y

(-) x


(+)

Modulus-argument form of the complex number, (r, )

z = x + yi
z = r cos  + (r sin )i

z = r (cos  + i sin )

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Example:
Express 3 –i in the form r (cos  + i sin ).

r  x2  y2 θ  tan 1  1 
r  32  (1)2 3
r  10
θ  tan 1  1 
3

θ  0.32

 3i  10[cos(0.32)i sin(0.32)]

Exercises:

Find the modulus and argument of each of the following complex numbers. Hence express in the
form z = r (cos  + i sin ).

a) z = 1 + i b) z = 2 – i c) z = -5 -12i d) 1 - i 3

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