Complex Numbers
Square Root- For any real numbers a and b, if a 2 =b, then a is the square root of b. Imaginary Unit- I, or the principal square root of -1, i 2 = -1. Pure Imaginary Numbers- The square roots of negative real numbers, for any positive real number b, √-b 2 = √b 2 x √-1, or b x i. Square Root Property- For any real number n, if x 2 = n, then x = ±√n. Complex Number- Any number that can be written in the form a + bi, where a and b are real numbers and I is the imaginary unit. Complex Conjugates- Two complex numbers in the form a + bi and a – bi.
√ab = √a x √b Example o √20 o √12 o √45
√ a / b = √a / √b Examples o √ 9 / 25 o √ 4 / 49
i = √ -1 i 2 = -1 i 3 = -√ -1 i 4 = 1 i 5 = √ -1 i 6 = -1 i 7 = -√ -1 i 8 = 1 o Notice a pattern?
Factor out i first Then, break down all variables, if they are present, to the highest even power. With numbers present, look to factor into perfect squares. Simplify
√-20 √-150x 7 √-32 √-252y 6
When Given Square Roots o Factor out I, if necessary o Combine square roots o Factor into perfect squares o Simplify When Given i o Multiply i values o Factor out even number of i’s o Multiply non-i values o Simplify
-3i x 6i √-3 x √-15 4i x -5i √-6 x √-8
Solve for variable Don’t forget to use ± Example o 4x = 0 o 3x = 0 o x = 0
Find values for the variables that make the equation true Set corresponding parts equal to each other Solve for the variables Examples o i = 2m + 3ni o (r+1) +3si = 5 – 9i o (2x + 5) + (1 – n)i = i
Combine values that do not have i in them Combine values that do have i in them Examples (5-3i) + (2 + 4i) (2 - i) – (6 - 5i)
Multiply the top and the bottom by conjugates Simplify Examples
Worksheet 5-4