Key Concepts
- The square root of any negative number can be written as a multiple of [latex]i[/latex].
- To plot a complex number, we use two number lines, crossed to form the complex plane. The horizontal axis is the real axis, and the vertical axis is the imaginary axis.
- Complex numbers can be added and subtracted by combining the real parts and combining the imaginary parts.
- Complex numbers can be multiplied and divided.
- To multiply complex numbers, distribute just as with polynomials.
- To divide complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator to eliminate the complex number from the denominator.
- The powers of [latex]i[/latex] are cyclic, repeating every fourth one.
Glossary
- complex conjugate
- the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number
- complex number
- the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[/latex], where [latex]a[/latex] is the real part, and [latex]bi[/latex] is the imaginary part
- complex plane
- a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number
- imaginary number
- a number in the form [latex]bi[/latex] where [latex]i=\sqrt{-1}\\[/latex]
