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