Presentation on theme: "Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented."— Presentation transcript:
Imaginary & Complex Numbers
Once upon a time…
-In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. -These numbers were devised using an imaginary unit named i.
-The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i
Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i. Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i.
Powers of i 1.) Find i 23 2.) Find i ) Find i 37 4.) Find i 828
Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals (no fractions) pi, e Imaginary i, 2i, -3-7i, etc.
Simplify. 3.) 4.)5.) -Express these numbers in terms of i.
To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.
a + bi Complex Numbers real imaginary The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.
Add or Subtract
Multiplying & Dividing Complex Numbers Part of 7.9 in your book
REMEMBER: i² = -1 Multiply 1) 2)
You try… 3) 4)
You try… 6)
You try… 7)
Conjugate -The conjugate of a + bi is a – bi -The conjugate of a – bi is a + bi
Find the conjugate of each number… 8) 9) 10) 11)