Post on 31-Mar-2015
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
1.1 Real Number System1.1 Real Number System
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.
1.1 Real Number System1.1 Real Number System
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.
.
.. .
1.1 Real Number System1.1 Real Number System
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
1.1 Real Number System1.1 Real Number System
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
1.1 Real Number System1.1 Real Number System
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.
1.1 Real Number System1.1 Real Number System
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
B. Operations of SurdsB. Operations of Surds
1.2 Surds1.2 Surds
P. 15
Example 1.3T
Solution:
Simplify .)83)(23(
16
)83)(23( 166243 46623
B. Operations of SurdsB. Operations of Surds
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.
A. Introduction to Complex NumbersA. Introduction to Complex Numbers
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.
1.3 Complex Number System1.3 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.
A. Introduction to Complex NumbersA. Introduction to Complex Numbers
1.3 Complex Number System1.3 Complex Number System
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
1.3 Complex Number System1.3 Complex Number System
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
B. Operations of Complex NumbersB. Operations of Complex Numbers
1.3 Complex Number System1.3 Complex Number System
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
B. Operations of Complex NumbersB. Operations of Complex Numbers
1.3 Complex Number System1.3 Complex Number System
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
B. Operations of Complex NumbersB. Operations of Complex Numbers
1.3 Complex Number System1.3 Complex Number System
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.