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
6
b) 3 = 0.272727… can be written as 0.27 or 0.27
11

Terminating decimal b) 3 = 0.75
Ex: a) 2 = 0.4 4
5

1

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

2

• 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

3

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
If the product of two surds is a
- When surds appear in the denominator of a fraction. rational number, for example
- Example : (2 + 3)(2 − 3) = 1
then (2 − 3) is called the
1 , 2 , 3 −1 conjugate surd of (2 + 3)
2 2− 7 3 +1 and vice versa.

- Method : multiply both the numerator and denominator Conjugate
with its conjugate.

Exercises:

Write down the conjugate for each of the following surds.

Surd Conjugate Surd

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 )

10

Examples :

Find the value of each of the following.

3

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

3

 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

15

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

16

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

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