7-7 Imaginary and Complex Numbers

Why Imaginary Numbers? n What is the square root of 9? n What is the square root of -9? no real number New type of number was defined for this purpose. It is called an Imaginary Number Imaginary numbers are NOT in the Real Set.

n The constant, i, is defined as the square root of negative 1: n Multiples of i are called Imaginary Numbers

n The square root of -9 is an imaginary number... n To simplify a square root with negative coefficient inside radical, write it as an imaginary number.

n Powers of i:

n This pattern repeats:

Multiples of i n We can find higher powers of i using this repeating pattern: i, -1, -i, 1 What is the highest number less than or equal to 85 that is divisible by 4? 84 So the answer is:

Powers of i - Practice n i 28 n i 75 n i 113 n i 86 n i i4-i4-i4-i 4i4i4i4i i4i4i4i

Negative Exponents Ex: Odd negative powers are opposite Even negative powers are the same!

Simplify: Ex 1: Ex 2 :

Multiply n Ex 3 n Ex 4 n Ex 5

Complex Numbers n Complex Number : a + bi, Where a and b are real #s and i is imaginary part n real and imaginary numbers are not like terms, n Examples: 3 - 7i, i, -4i, 5 + 2i

Complex #s Real #s Imaginary #s Irrational #sRational #s

Add and Subtract n Combine Like Terms (the real & imaginary parts). n Example: (3 + 4i) + (-5 - 2i) = i

Practice Add these Complex Numbers: n (4 + 7i) - (2 - 3i) n (3 - i) + (7i) n (-3 + 2i) - (-3 + i) = 2 +10i = 3 + 6i = i

Assignment n 7-7/323/1-41, 43-48, 56-64