Imaginary Numbers - Final Project - Algebra 2

6+3i

Simplify: (8+6i)-(2+3i)

-1

Simplify: i^38

6i

Simplify: √-36

multiplying complex numbers

Use foil and combine like terms Each part of the first complex number gets multiplied by each part of the second complex number (a+bi)(c+di) = ac + adi + bci + bdi^2 Example: (3 + 2i)(1 + 7i) (3 + 2i)(1 + 7i) = 3×1 + 3×7i + 2i×1+ 2i×7i = 3 + 21i + 2i + 14i^2 = 3 + 21i + 2i + 14(-1) = 3 + 21i + 2i - 14 = −11 + 23i

complex number (general form)

a+bi

adding complex numbers

add two complex numbers we add each part separately: (a+bi) + (c+di) = (a+c) + (b+d)i add the real numbers, and add the imaginary numbers: (3 + 2i) + (1 + 7i) = 3 + 1 + (2 + 7)i = (4 + 9i)

4-3i

(3+2i)+(1-5i)

To find i^350...

1) Divide the power by 4. 2) Determine the remainder. 3) If the remainder is 1, then your answer is i. If the remainder is 2, then your answer is -1. If the remainder is 3, then your answer is -i. If the remainder is 4, then your answer is 1.

To find the square root of -81...

1) Split it up. The square root of -1 times the square root of 81 is the same thing. 2) Split it up even further. The square root of -1 is i. The square root of 81 is 9. 3) Multiply. 9 times i is 9i. Write your answer like a variable with a constant.

-i

3i(1-2i)-2i(2-3i)

Imaginary Numbers

Numbers that don't exist; physically not able to be computed; represented as i

Real Numbers

Rational numbers (integers, whole numbers, and natural numbers) and irrational numbers (pi, square root 2)

42-2i

Simplify: (3-5i)(4+6i)

12

Simplify: (6i)(-2i)

When distributing an imaginary number...

Treat it like a variable until the very end. Then substitute i with square root of -1 and solve.

what does the "bi" represent in a+bi form?

the imaginary component

what does the "a" represent in a+bi form?

the real number

what does i represent?

the square root of -1