15.1 - Introduction to Complex Numbers

Note: Mathematicians use the expression “… over the complex numbers” to mean that the number system under consideration is the complex numbers, and the expression “… over the real numbers” to mean that the number system under consideration is the real numbers. For example they will say that the equation x² = −9 cannot be solved for x over the real numbers but that it has a solution over the complex numbers.

Imaginary Numbers

First we introduce the imaginary numbers.

An imaginary number is the square root of a negative number. The imaginary unit i is defined as:

Note: Electrical engineering is one application that makes extensive use of complex numbers. Unfortunately the letter i is already used to represent electric current. Therefore electrical engineers usually use the letter j to represent the imaginary unit. The Algebra Coach has an option that allows you to set either i or j to represent the imaginary unit.

Here are some examples of arithmetic with imaginary numbers. The first example shows how the square root of any negative number can be expressed as a multiple of the imaginary unit i. The second example shows that squaring an imaginary number gives a negative number. The last example shows how any power of i can be simplified.

Complex Numbers

A complex number z is the sum of a real number plus an imaginary number. It can be written in the form:

z = a + b i

where a and b are both real numbers. a is called the real part of z and b is called the imaginary part of z. We write this as a = Re(z) and b = Im(z).

Example: If z = − 5 − 7 i then Re(z) = −5 and Im(z) = −7.

The complex conjugate of any complex number a + b i is defined to be a − b i. (The imaginary part just has its sign reversed.) The complex conjugate of z is denoted z*. For example if z = − 5 − 7 i then z* = − 5 + 7 i.

Arithmetic with Complex Numbers

We demand that imaginary numbers and complex numbers have all the properties that real numbers have, plus a few new ones that we will discover along the way. Here are some examples of arithmetic with complex numbers:

To add two complex numbers just combine like terms.

To subtract two complex numbers just combine like terms.

To multiply a real times a complex just distribute.

To multiply an imaginary times a complex just distribute and then use i² = − 1.

To multiply two complex numbers just distribute and then use i² = − 1.

Multiplying a complex number by its complex conjugate (defined above) always results in a positive real number because the cross-terms cancel.

To divide a complex number by a real just break the fraction into two parts.

The reciprocal of i is − i. You can verify this by cross-multiplying.

To divide a complex number by an imaginary number just break the fraction into two parts and then use the fact that the reciprocal of i is − i.

To simplify the reciprocal of a complex number multiply numerator and denominator by the complex conjugate of the denominator. This is designed to produce a positive real number in the denominator which can then be divided into each term of the numerator.

A shortcut for the previous example is to replace a reciprocal by the complex conjugate in the numerator over a Pythagoras type sum of squares in the denominator.

To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. This produces a Pythagoras type sum of squares in the denominator and two complex numbers in the numerator that can be multiplied as described previously.

Graphing Complex Numbers

Complex numbers can be displayed as points or arrows on the complex plane. The real part of the complex number is plotted along the real (horizontal) axis and the imaginary part is plotted along the imaginary (vertical) axis. Real numbers lie on the real axis and imaginary numbers lie on the imaginary axis.

Complex numbers are similar to 2-dimensional vectors in the way they are added: you add horizontal components and you add vertical components. But complex numbers can also be multiplied whereas vectors can't.

Algebra Coach Exercises

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	To add two complex numbers just combine like terms.
	To subtract two complex numbers just combine like terms.
	To multiply a real times a complex just distribute.
	To multiply an imaginary times a complex just distribute and then use i² = − 1.
	To multiply two complex numbers just distribute and then use i² = − 1.
	Multiplying a complex number by its complex conjugate (defined above) always results in a positive real number because the cross-terms cancel.
	To divide a complex number by a real just break the fraction into two parts.
	The reciprocal of i is − i. You can verify this by cross-multiplying.
	To divide a complex number by an imaginary number just break the fraction into two parts and then use the fact that the reciprocal of i is − i.
	To simplify the reciprocal of a complex number multiply numerator and denominator by the complex conjugate of the denominator. This is designed to produce a positive real number in the denominator which can then be divided into each term of the numerator.
	A shortcut for the previous example is to replace a reciprocal by the complex conjugate in the numerator over a Pythagoras type sum of squares in the denominator.
	To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. This produces a Pythagoras type sum of squares in the denominator and two complex numbers in the numerator that can be multiplied as described previously.