Algebra 2 Chap. 1.4 Complex Numbers

Multiplication of complex conjugate

(a+bi)(a-bi) = a^2b^2 (a-bi)(a+bi) = a^2b^2

i^2

-1

i^6

-1

i^3

-i

i^7

-i

i^4

1

i^8

1

Adding and Subtracting Complex Numbers

1. (a+bi)+(c+di) = (a+c) + (bi+di) e.g. (5-11i) + (7 + 4i) (5-7) + (-11i +4i) = 12 - 7i 2. (a+bi) - (c+di) = (a- c) + (bi- di) e.g. (-5+i) - (-11 - 6i) -5+i+11+6i (-5+11) +(i+6i) = 6+7i

Using Complex Conjugates to Divide Complex Numbers

3i/4+i =3i * (4-i)/4+i(4-i) = 12i-3i^2/4^2*1^2 =12i-3(-1)/ 17 = 3/17+12i/17

complex number system

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (which satisfies the equation i^2 = −1). In this expression, a is called the real part of the complex number, and b is called the imaginary part.

FOIL

First, Outside, Inside and Last. EX: (x-3) + (x+5) = x² + 5x -3x -15= x² + 2x-15

Roots of Negative Numbers

The square of 4i and the square of -4i both result in -16 e.g. (4i)^2 = 16i^2 = 16(-1) = -16 e.g. (-4i)^2 = 16i^2 = 16(-1) = -16 * WHEN PERFORMING OPERATIONS WITH SQUARE ROOTS OF NEGATIVE NUMBERS, BEGIN BY EXPRESWSSING ALL SQUARE ROOTS IN TERMS OF I. e.g. √-25*√-4 = i√25*i√4= 5i*2i = 10j^2 = 10(-2) = -10 INCORRECT WAY: √-25*√-4 = √(-25)(-4) = √100 = 10

pure imaginary number

a number in the form bi where b is a real number and i is the imaginary unit e.g. -4+6i a=4 (the real part) b= 6i (the imaginary part)

complex numbers

a+bi, contains a real part (a) and an imaginary part (bi)

Conjugate of a Complex Number i.e. a+bi

a-bi

relationship between complex numbers, real numbers, and imaginary numbers

complex number includes both imaginary numbers and real numbers

i^1

i

i^5

i

imaginary unit

i = √-1 i∧2 = -1 e.g.1. √-25 = √-1*√25 = 5i 2. (5i)∧2 = 5^2*i^2 =25(-1) = -25

powers of imaginary numbers

i^3 = i^2 * i = -i i^4 = i^2 * i^2 = 1