Real Numbers and Complex Numbers

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Real Numbers and Complex Numbers
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Case Study
1.1 Real Number System
1.2 Surds
1.3 Complex Number System
Chapter Summary
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Case Study
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.
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1.1 Real Number System
We often encounter different numbers in our
calculations, For example,
These numbers can be classified into different
groups.
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1.1 Real Number System
A. 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.
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1.1 Real Number System
B. Rational Numbers
All of them are rational numbers.
Note that
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1.1 Real Number System
B. Rational Numbers
? 0.166 666 ............ (1)
10n ? 1.666 66 ............ (2)
(2) ? (1) 10n ? n ? 1.5
9n ? 1.5
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1.1 Real Number System
C. Irrational Numbers
Irrational numbers can only be written as
non-terminating and non-recurring decimals
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1.1 Real Number System
D. 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.
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1.1 Real Number System
D. Real Numbers
We can represent any real number on a straight
line called the real number line.
Real numbers have the following property
a2 ? 0 for all real numbers a.
For example
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1.2 Surds
In junior forms, we learnt the following
properties for surds
For any real numbers a and b, we have
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1.2 Surds
A. Simplification of Surds
For any surds, when we reduce the integer inside
the square root sign to the smallest possible
integer, such as
then the surd is said to be in its simplest form.
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1.2 Surds
B. Operations of Surds
When two surds are like surds, we can add them or
subtract them
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1.2 Surds
B. Operations of Surds
Example 1.1T
Solution
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1.2 Surds
B. Operations of Surds
Example 1.2T
Solution
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1.2 Surds
B. Operations of Surds
Example 1.3T
Solution
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1.2 Surds
C. Rationalization of the Denominator
Rationalization of the denominator is the process
of changing an irrational number in the
denominator into a rational number, such as
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1.2 Surds
C. Rationalization of the Denominator
Example 1.4T
Solution
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1.3 Complex Number System
A. Introduction to Complex Numbers
In Section 1.1, we learnt that
a2 ? 0 for all real numbers a.
For example
Therefore, in a real number system, equations
such as x2 ? ?1 and (x ? 1)2 ? ?4 have no real
solution
? ?i
? 1 ? 2i
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1.3 Complex Number System
A. Introduction to Complex Numbers
Properties of complex numbers
1. The complex number system contains an
imaginary unit, denoted by i, such that i2 ?
?1.
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.
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1.3 Complex Number System
A. Introduction to Complex Numbers
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.
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.
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1.3 Complex Number System
B. Operations of Complex Numbers
The addition, subtraction, multiplication and
division of complex numbers are similar to the
operations of algebraic expressions.
We classify the real part and the imaginary part
of the complex number as unlike terms in
algebraic expressions.
For complex numbers z1 ? a ? bi and z2 ? c ? di,
where a, b, c and d are real numbers, we have
(1) Addition
z1 ? z2 ? (a ? bi) ? (c ? di)
e.g. (3 ? 6i) ? (5 ? 8i)
? a ? bi ? c ? di
? (3 ? 5) ? 6 ? (?8)i
? (a ? c) ? (b ? d)i
? 8 ? 2i
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1.3 Complex Number System
B. Operations of Complex Numbers
(2) Subtraction
z1 ? z2 ? (a ? bi) ? (c ? di)
e.g. (9 ? 7i) ? (2 ? 3i)
? a ? bi ? c ? di
? (9 ? 2) ? ?7 ? (?3)i
? (a ? c) ? (b ? d)i
? 7 ? 4i
(3) Multiplication
This term belongs to the real part because i2 ?
?1.
z1z2 ? (a ? bi)(c ? di)
? ac ? adi ? bci ? bdi2
? (ac ? bd) ? (ad ? bc)i
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1.3 Complex Number System
B. Operations of Complex Numbers
Example 1.5T
Simplify (7 ? 2i)(5 ? 3i) ? 4i(3 ? i).
Solution
(7 ? 2i)(5 ? 3i) ? 4i(3 ? i) ? (35 ? 21i ? 10i ?
6i2) ? (12i ? 4i2)
? 35 ? 21i ? 10i ? 6 ? 12i ? 4
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1.3 Complex Number System
B. Operations of Complex Numbers
(4) Division
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1.3 Complex Number System
B. Operations of Complex Numbers
Example 1.6T
Solution
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Chapter Summary
1.1 Real Number System
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1.2 Surds
Chapter Summary
1. For any positive real numbers a and b
2. For any positive real numbers a and b
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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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Follow-up 1.1
1.2 Surds
B. Operations of Surds
Solution
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Follow-up 1.2
1.2 Surds
B. Operations of Surds
Solution
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Follow-up 1.3
1.2 Surds
B. Operations of Surds
Solution
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Follow-up 1.4
1.2 Surds
C. Rationalization of the Denominator
Solution
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Follow-up 1.5
1.3 Complex Number System
B. Operations of Complex Numbers
Simplify (5 ? 2i)(7 ? 3i) ? 3(2 ? 5i).
Solution
(5 ? 2i)(7 ? 3i) ? 3(2 ? 5i) ? (35 ? 15i ? 14i ?
6i2) ? (6 ? 15i)
? 35 ? 15i ? 14i ? 6 ? 6 ? 15i
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Follow-up 1.6
1.3 Complex Number System
B. Operations of Complex Numbers
Solution