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Complex Numbers

Complex numbers are numbers of the form a=bi, where a and b are real numbers.

1 3-4i 3-4i

3+4i * 3-4i = 9+16

3 4

25 - 25

1+4i 5+12i 5+12i +20i + 48i²

5-12i *5+12i = 25 +144

-43 +32i

169

-43 +32

169 169

i1 = i i2 = –1

i3 = –i i4 = 1,

i5 = i i6 = –1

i7 = –i i8 = 1

i9 = i i10 = –1,

i11 = –i i12 = 1

i27 = i24*i3 = (i4)^6 *i3=1^6 *i3=-i

i101 = i100 *i1= (i4)^25*i = 1^25 *i=i

Solve each equation in the complex number system .

Example: x^2=4

x= + 4= +2

{-2,2}

the end !

Product of Complex Numbers

(a+bi)*(c+di) = (ac-bd) + (ad+bc)i

Example: (2i)(2i) = 4i² = -4

(2+i)(1-i) = 2-2i +i-i²

3-i

Multiplying a Complex Number by Its Conjugate

Conjugates

z = a+bi= a-bi

Example

z= 3+4i

z=3-4i

zz=(3+4i)(3-4i)

9-12i+12i-16i²

9+16

Example: 2+3i= 2-3i

-6-2i=-6+2i

25

Imaginary unit

i^2= -1

When a complex number is written in the form a+bi, where a and b are real numbers, we say it is in standard form.

Writing the Reciprocal of a Complex Number in Standard Form

A pure imaginary number is an imaginary number of the form a+bi where A is 0.

Example

1

3+4i

Add, Subtract, Multiply, and Divide Complex Numbers

Equality of Complex Numbers

a+bi=c+di if and only if a=c and b=d

i

Sum of Complex Numbers

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

Solving Equations

Difference of Complex Numbers

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

Writing the Quotient of Complex Numbers in Standard Form

Powers Of i

Solve Quadratic Equations with a Negative Discriminant

Product of Complex Numbers

Example

If N is a positive real number, we define the principal square root of -N, denoted by -N, as

Example

1+4i

5-12i

-N

Example: (3+5i) + (-2 +3i)

[3 +(-2)] + (5+3)i

=

Ni

where i is the imaginary unit is i^2= -1

Example:

Example :

-4 = 4i =2i

1+8i

Example: (5+3i)*(2+7i)

5*(2+7i) +3i(2+7i)

10+35i+6i+21i^2

10+41i+21(-1)

-1 = 1i =i

i

-11+41i

writing the Power of a Complex Number in Standard Form

Example

The Special product formula for (x+a)^3

(x+a)^3= x^3+3ax^2+3a^2x+a^3

Write (2+i)^3 in standard form.

(2+i)^3= 2^3 +3*i*2^2 +i^3

=8+12i+6(-1) + (-i)

=2+11i

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