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Chapter 1.5 Complex Numbers
Pindling
College Algebra  
by Example  Series

 The i-th Continuum
Acomplex number is any number than can be expressed in the form:

 a + bi

Where a and b are real numbers ( and i is called the imaginary unit: 

The standard form of complex numbers is a + bi where a is called the real part and b is called the imaginary part.

The conjugate of a + bi is a - bi and (a + bi)(a - bi) = a2+ b2 (a real number)

I. Convert the following to the standard form of the complex numbers a + bi
 
Problem Examples Standard form: a + bi
(1) 5 + 3i 5 + 3i (Already in standard form)
(2) 4 - 5i 4 + (-5)i
(3)  (note  is best written as 

)
(4) -6i 0 + (-6)i (when a=0, bi is called a pure imaginary number)
(5) 12 12 + 0i

II. Add the following complex number:

Principles: (a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

(a + bi) + (-a - di) = 0 (where -a - di is called the additive inverse)

Examples - Add and substract as indicated

(1) (-7 - 3i) + (-4 + 4i) = (-7 -4) + (-3 + 4)i = -11 + i

(2) (-2 - 3i) - (-1 - i) = (-2 -(-1)) + (-3 - (-1))i = -1 - 2i

(3) 

III. Mutipication and Division of Complex Numbers

Principles:

(1) 

(2) 

Use multiplication rules:

(1) (a + bi)(c + di) = a(c + di) + bi(c + di)

(2) (a + bi)2 = a2 + 2abi + (bi)2

(3) (a - bi)2 = a2 - 2abi + (bi)2

(4) (a + bi)(a - bi) = a2 - (bi)2 = a2 + b2 (conjugate used to simplify the quotient of 2 complex numbers)

Examples: Express each products in standard form:

(1) (4i)(3-2i) = 4i(3) + 4i(-2i) =12i - 8i2 = 12i - 8(-1) = 12i +8 = 8 + 12i

(2) (3 + 2i)(4 + 6i) = 3(4 + 6i) + 2i(4 + 6i) =12 + 18i + 8i + 12i2 = 12 + 26i + 12(-1) = 0 + 26i

                = 0 + 26i

(3) (4 + 5i)(2 - 9i) = 4(2 - 9i) + 5i(2 - 9i) = 8 - 36i + 10i - 45i2 = 8 - 26i - 45(-1) = 53 - 26i

(4) (3 + 4i)2 = (3 + 4i)(3 + 4i) = 32 + 2(3)(4)i + (4i)2 = 9 + 24i + 16i2 = 9 + 24i + 16(-1)

                = -7 + 24i

(5) (-1 - 2i)2 = (-1 - 2i)( -1 - 2i) = (-1)2 - 2(-1)(2)i + (2i)2 = 1 + 4i + 4i2 = 1 + 4i + 4(-1)

                = -3 + 4i

(6) Conjugate: (3 + 4i)(3 - 4i) = (-3)2 - (4i)2 = 9 - 16i2 = 9 - 16(-1) = 9 + 16 = 25

                = 25 + 0i

Examples: Express each quotients in standard form: (hint: use the conjugate)

(1)

(2) 

(3) 

(4) 

Absolute Value

Example

Example. Find

Example. Find

Example. Find

First write in a + bi form (use conjugate)

Remember that

So

Example. Find

Example. Find

Example. Find ,First write in a + bi form

So