Complex Variables: Definitions & Examples

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  • 0:03 Complex Numbers
  • 1:25 Comparing Complex Numbers
  • 1:38 Adding & Subtracting
  • 2:41 Multiplying
  • 4:22 Dividing
  • 5:52 Lesson Summary
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Lesson Transcript
Instructor: Russell Frith
This document presents fundamental details for the meaning of complex numbers. A definition is presented which shows the set of complex numbers is a superset of the set of real numbers. The arithmetic operations of complex addition, complex subtraction, complex multiplication, and complex division are presented.

Complex Numbers

Complex numbers are pretend numbers because they do not exist in reality. For example, the square root of -1. Complex numbers provide solutions to many math, science, and engineering problems that would otherwise have no solutions. For instance, consider finding the roots of the quadratic equation: y = x2 + 4x + 1. When graphed in the x-y plane, one readily sees that the graph never intersects the x-axis and thus has no real roots. With the use of complex numbers, however, this equation can be shown to have two complex roots.

A complex number is any number that can be written as a + bi, where a and b are real numbers and i is the square root of -1. In the complex number a + bi, a is called the real part and b is called the imaginary part. If b = 0, the complex number a + bi is simply a real number. If b is not zero and a = 0, the complex number 0 + bi (or just bi) is an imaginary number. An imaginary number is simply the square root of a negative number. Examples of imaginary numbers are:

Definition of i

Factoring an imaginary number as a product of a real number and i

Comparing Complex Numbers

Two complex numbers are said to be equal if both their real parts are equal and their imaginary parts are equal. Unlike real numbers, however, one complex number cannot be greater than or less than another.

Adding & Subtracting

To add two complex numbers, simply add the real parts of the complex numbers to get the real part of the sum and add the imaginary parts to get the imaginary part of the sum. Subtraction of two complex numbers is performed in the same manner, with the subtraction performed in place of addition. The distributive properties of addition and subtraction apply to complex numbers as well.

Let z1 = 4 + 7i and z2 = -2 + 6i. Then z1 + z2 = 2 + 13i.

Let z1 = 4 + 7i and z2 = -2 + 6i. Then z1 - z2 = 6 + i.

Let z1 = 4 + 7i, z2 = -2 + 6i and z3 = 5 - 8i. Then z1 - (z2 + z3) = z1 - (-7 - 3i) = z1 + 7 + 3i = 11 + 10i


Complex numbers may also be multiplied. Let a + bi be the first complex factor and let c + di be the second complex factor. When multiplying those two complex numbers, use the FOIL method for multiplying two binomials. The FOIL method stands for first, outer, inner, and last and is a mnemonic used to describe the order of operations in the multiplication of two binomials.

(a + bi)*(c + di) = ac + adi + bci + bdi2 = (ac - bd) + (ad + bc)i

Let z1 = 4 + 7i and z2 = -5 - 8i.

Then z1 * z2 = (-20 - 56) + (-32 - 35)i = -76 - 67i

Note the distributive property of multiplication with respect to addition or subtraction applies.

Let z1 = 4 + 7i, z2 = -5 - 8i, and z3 = 3 + 2i.

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