Real Numbers and Complex Numbers

Real Numbers and Complex Numbers1

1.1 Real Number System

1.2 Surds

1.3 Complex Number System

Chapter Summary

Case Study

P. 2

As shown in the figure, after drawing the diagonal of the square, we can use a pair of compasses to draw an arc with radius OB and O as the centre.

Case Study

The point of intersection of the arc and the number line is the position of (i.e., point C).2

If you are given a pair of compasses and a ruler only, do you know how to represent the irrational numberon a number line?

2

I think I can do it by drawing a squareof side 1 first.

In junior forms, we learnt from Pythagoras’ theorem that the diagonal

of a square of side 1 is .211 22

P. 3

1.1 Real Number System1.1 Real Number System

We often encounter different numbers in our calculations,For example,

These numbers can be classified into different groups.

1, 2, 4, 7, 0, , 2.5, 0.16, , , …2

1 3.

0.16 means 0.166 666….

P. 4

A. IntegersA. Integers


1, 2, 4, 7 and 0 are all integers.

Positive integers (natural numbers) are integers that are greater than zero.

Negative integers are integers that are less than zero.

Integers

7, 4, 0, 1, 2

Negative Integers Positive Integers(Natural Numbers)

Zero is neither positive nor negative.

P. 5

B. Rational NumbersB. Rational Numbers

All of them are rational numbers.Recurring decimals are also called repeating decimals.

is a fraction, 2.5 is a terminating decimal and

0.16 is a recurring decimal.

2

1.

A rational number is a number that can be written in

the form , where p and q are integers and q 0.q

p

Note that

2

55.2

6

161.0 and .

Any integer n can be written as . Therefore, integers are also

rational numbers.1

n

Recurring decimals can be converted into fractions, as shown in the next page.


P. 6

B. Rational NumbersB. Rational Numbers

Express 0.16 as a fraction:.

0.166 666… ............ (1)Let n 0.16

.

10n 1.666 66… ............ (2)

(2) (1): 10n n 1.5 9n 1.5

n 9

5.1

6

1

In other recurring decimals, such as a 0.83 and b 0.803,. .

. .

consider 100a 83.83 and 1000b 803.803, then we obtain99a 83 and 999b 803.

.

.. .


P. 7

C. Irrational NumbersC. Irrational Numbers

Irrational numbers can only be written as non-terminating and non-recurring decimals:

Numbers that cannot be written in the form are irrational

numbers.q

p

Examples: , , and sin 453 11

...781 106 707.045sin

...79 624 316.311

...8 050 732.13

...65 592 141.3

is just an approximation

of .7

22


P. 8

D. Real NumbersD. Real Numbers

If we group all the rational numbers and irrational numbers together, we have the real number system.

That is, a real number is either a rational number or an irrational number.

Real numbers

1, 2, 4, 7, 0, , 3.5, 0.16, , 2

1 3

.

Rational numbers

1, 2, 4, 7, 0, , 3.5, 0.162

1

.

FractionsTerminating

decimalsRecurringdecimals

Integers

Irrational numbers

,

3

Negative integers Zero Positive integers


P. 9

D. Real NumbersD. Real Numbers

We can represent any real number on a straight line called the real number line.

.2

1 .3.1 .2.5 ..11

Real numbers have the following property:

For example:

is a real number since .17 017)17( 2 is not a real number since .1 01)1( 2

a2 0 for all real numbers a.


P. 10

1.2 Surds1.2 Surds

In junior forms, we learnt the following properties for surds:

baab 1.

b

a

b

a 2. In general,

baba

baba

For any real numbers a and b, we have

P. 11

1.2 Surds1.2 Surds

For any surds, when we reduce the integer inside the square root sign to the smallest possible integer, such as:

210210200 2

A. Simplification of SurdsA. Simplification of Surds

then the surd is said to be in its simplest form.

P. 12

1.2 Surds1.2 SurdsB. Operations of SurdsB. Operations of Surds

When two surds are like surds, we can add them or subtract them:

Like surds are surds with the same integer inside the square root sign, such as and .54 5

55554

P. 13

B. Operations of SurdsB. Operations of Surds

Example 1.1T

Solution:

Simplify .72532283

72532283 )26(5)24(2)22(3

2)3086( 216

2302826

1.2 Surds1.2 Surds

P. 14

Example 1.2T

Solution:

Simplify .152708

152708 15

30322

15

152322

232 12


1.2 Surds1.2 Surds

P. 15

Example 1.3T

Solution:

Simplify .)83)(23(

16

)83)(23( 166243 46623


1.2 Surds1.2 Surds

P. 16

1.2 Surds1.2 Surds

Rationalization of the denominator is the process of changing an irrational number in the denominator into a rational number, such as:

C. Rationalization of the DenominatorC. Rationalization of the Denominator

13

134

13

13

13

4

13

4

P. 17

C. Rationalization of the DenominatorC. Rationalization of the Denominator

Example 1.4T

Solution:

Simplify .3

6

3

3

3

6

3

3 3

3

3

6

3

3

3

36

3

3

3

35

1.2 Surds1.2 Surds

P. 18

1.3 Complex Number System1.3 Complex Number System

In Section 1.1, we learnt that

A. Introduction to Complex NumbersA. Introduction to Complex Numbers

For example:

is a real number since .17 017)17( 2 is not a real number since .1 01)1( 2

Therefore, in a real number system, equations such as x2 1 and (x 1)2 4 have no real solution:

112

xx

4141

4)1( 2

xx

x

i 1 2i

Complex numbers

Define .1i

i214)1(44

Then

a2 0 for all real numbers a.

P. 19

1. The complex number system contains an imaginary unit, denoted by i, such that

i2 1.


Properties of complex numbers:

2. The standard form of a complex number is

a bi,

where a and b are real numbers.

3. All real numbers belong to the complex number system.


P. 20

Notes:

1. For a complex number a bi, a is called the real part and b is called the imaginary part.

2. When a 0, a bi 0 bi bi, which is a purely imaginary number.

3. When b 0, a bi a 0i a, so any real number can be considered as a complex number.

4. When a b 0, a bi 0 0i 0.

Complex numbers do not have order. So we cannot compare which of the complex numbers 2 3i and 4 2i is greater.

Two complex numbers are said to be equal if and only if both of their real parts and imaginary parts are equal.

If a, b, c and d are real numbers, then

a bi c di

if and only if a c and b d.



P. 21

1.3 Complex Number System1.3 Complex Number SystemB. Operations of Complex NumbersB. Operations of Complex Numbers

The addition, subtraction, multiplication and division of complex numbers are similar to the operations of algebraic expressions.

In the operation of algebraic expressions, only like terms can be added or subtracted.

We classify the real part and the imaginary part of the complex number as unlike terms in algebraic expressions.

(1) Addition

z1 z2 (a bi) (c di)

a bi c di

(a c) (b d)i

For complex numbers z1 a bi and z2 c di, where a, b, c and d are real numbers, we have:

e.g. (3 6i) (5 8i)

(3 5) [6 (8)]i 8 2i

P. 22

This term belongs to the real part because i2 1.

B. Operations of Complex NumbersB. Operations of Complex Numbers

(2) Subtraction

z1 z2 (a bi) (c di)

a bi c di

(a c) (b d)i

(3) Multiplication

z1z2 (a bi)(c di)

ac adi bci bdi2

(ac bd) (ad bc)i

e.g. (9 7i) (2 3i)

(9 2) [7 (3)]i

7 4i


P. 23

Example 1.5T

Solution:

Simplify (7 2i)(5 3i) 4i(3 i).

(7 2i)(5 3i) 4i(3 i) (35 21i 10i 6i2) (12i 4i2)

35 21i 10i 6 12i 4

33 19i



P. 24

(4) Division

2

1

z

z

dic

bia

The process of division is similar to the rationalization of the denominator in surd.

dic

dic

dic

bia

222

2 )()(

idc

iadbcbdiac

(p q)(p q) p2 q2

22

)()(

dc

iadbcbdac

The denominator contains .1



P. 25

Example 1.6T

Solution:

Simplify and express the answer in standard form.i

i

56

27

i

i

i

i

56

56

56

27

i

i

56

27

222

2

56

10351242

i

iii

2536

)3512()1042(

i

61

4732 i

i61

47

61

32



P. 26

1.1 Real Number System

Chapter Summary

Real numbers

Rational numbers

FractionsTerminating

decimalsRecurringdecimals

Integers

Irrational numbers

Negative integers Zero Positive integers

P. 27

1.2 Surds

Chapter Summary

1. For any positive real numbers a and b:

2. For any positive real numbers a and b:

baab (a)

b

a

b

a (b)

b

ba

b

b

b

a

b

a

P. 28

1.3 Complex Number System

Chapter Summary

1. Every complex number can be written in the form a bi, where a and b are real numbers.

2. The operations of complex numbers obey the same rules as those of real numbers.

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Real Numbers and Complex Numbers