Section 4.1 Intro to Complex Numbers

Section Objectives

  1. Identify and simplify complex numbers.
  2. Add, subtract, multiply, and divide complex numbers.
  3. Simplify powers of .



Imaginary Numbers

The imaginary unit is , where .


The square roots of negative numbers can be written in terms of .



Examples










Complex Numbers

A complex number is an expression of the form , where and are real numbers.



In the complex number , is called the real part and is called the imaginary part. The and terms cannot be combined---they are not like terms. But every complex number can be simplified to the standard form .




Examples
















Conjugates and Dividing

The complex conjugate of the complex number is the complex number . Notice that in forming the complex conjugate, we negate the imaginary part, but not the real part.



An important property of conjugates is the following:



Example




In order to divide by a complex number, we really just use the conjugate (and multiplication) to make the denominator real:



Examples