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Lesson 01
Numbers and Order Relation
1.1 Real number system1.2 Complex number system
1.3 Order relation of real numbers1.4 Linear inequalities
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1.1 Real number system
{ }| , , 0mn m n n Z
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1.1 Real number system
x2 = 2
To solve this, we require irrational numbers.
x = = 1.414 .....2
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1.1 Real number system
For any circle with
circumference Cand diameter d,
is irrational.Cd
=
Irrational Numbers =
Real Numbers, =
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1.1 Real number system
Ex 1. Determine which of the numbers
are
(a) integers, (b) rational numbers,
(c) irrational numbers, (d) real numbers.
Soln. (a) Integer:
(b) Rationals:
(c) Irrationals:
(d) Real numbers:
13, 4, 13,
5 2
4 2=
13, 4
5
13,2
13, 4, 13,
5 2
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1.2 Complex number system
What is the solution of or ?2 1 0x + = 2 1x =
To solve this we require complex numbers.
Construct an imaginary unit, i, which has theproperty i2 = 1 or
i = .
Then i is the solution of the equation .
We can use i to construct the complex number system.
A complex number is an expression of the form a + bi,where a, b
.
The set of complex numbers is denoted by .
12 1 0x + =
R
C
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1.2 Complex number system
The system of complex numbers iscomplete in the sense that
allowing
complex numbers enables everypolynomial equation to have
solutions.
Every complex number can be
represented as a point in the complexplane (Argand diagram).
C
Note that any real number is also acomplex number, for example,
5 = 5 + 0i.
Thus .R C.
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1.2 Complex number system
Ex 2. Solve x2= 9 over .
Soln.x =
C
9 (9)( 1) 3i = =
ab a b=Note: We may assume that the rule for radicals:in the
real number system carries over into the
complex number system.
Ex 3. Find the complex roots of (x 2)2 = 16.
Soln.x 2 = 16 (16)( 1) 4i = =
x = 2 4i
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1.2 Complex number system
Equality: a + bi = c + di a = c and b = d.
Addition and multiplication:The distributive law for real
numbers carries over intothe complex number system. Using the
distributive law,(a + bi)(c + di) = ac + bdi2 + adi + bci
= (acbd) + (ad+ bc)i, since i2 = 1. Thus addition and
multiplication are defined by:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (acbd) + (ad+ bc)i
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1.2 Complex number system
Ex 4. Write each expression in the form a + bi
(a) (2 + 3i) + (3 2i)
(b) (3 + 7i)(1 4i)
(c) (2 + i)2
Soln. (a) 5 + i
(b) 3 + 28 + (12 + 7)i = 31 5i
(c) (2 + i)(2 + i) = 4 1 + (2 + 2)i = 3 + 4i
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1.3 Order relation for real numbers
The order relation ( < or > ) can be defined for real
numbers(but not for complex numbers). It has the following
properties:
1. a < b a + c < b + c
e.g., 2 < 3 2 + 1 < 3 + 1; 2 + 1 < 3 + 1 2 < 3.
2. If c > 0, then a < b ac < bc
e.g., 2 < 3 2 5 < 3 5; 2 5 < 3 5 2 < 3.
3. If c < 0, then a < b ac > bc
e.g., 2 < 3 2 (1) > 3 (1); 2 (1) > 3 (1) 2 < 3.
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1.4 Linear inequalities
4 2x
Ex 5. SolveSketch the solution on a coordinate line.
Soln.
2 3 6x +
2 6 3x
32x
Compound inequalities: a
