# Imaginary Numbers

by Ron Kurtus (updated 18 January 2022)

The number √−1 is considered an imaginary number since there is no number that multiplied by itself will equal −1.

When combined with a real number, it is called a complex number. You can multiply and divide complex numbers.

Just as with real numbers, there is a conjugate of a complex binary number.

Questions you may have include:

• What is an imaginary number?
• What is a complex number?
• What is a complex conjugate number?

This lesson will answer those questions.

## Imaginary numbers

An interesting property of the square root concerns the square root of −1.

The number √−1 is considered an imaginary number since there is no number multiplied by itself that equals −1. Imaginary numbers are usually designated by the letter i.

Raising i to various powers results in:

i2 = −1

i3 = i2*i = −i

i4 = (i2)(i2) = (−1)(−1) = +1

and so on.

## Complex numbers

Real numbers combined with imaginary numbers are called complex numbers. Examples of complex numbers include:

7i

3 + 5i

a −bi

### Multiplying complex numbers

You can multiply complex number the same as you do with any polynomial.

In multiplying monomials:

(7i)(6i) = 42i2 = −42

In multiplying binomials, you can use the FOIL method:

(3 + 5i)(2 + 3i) =

3*2 + (3*3i + 5i*2) + 5i*3i =

6 + (9i + 10i) + 15I2 =

6 + 19i − 15 =

19i − 9 or − 9 + 19i

### Dividing complex numbers

You can also divide complex numbers as you would divide polynomials.

## Conjugate complex numbers

The conjugate of a binomial x + y is another binomial with one factor negated: x − y. A major feature of a conjugate is when you multiply the two expressions together, you get the difference of the squares of the terms:

(x + y)(x − y) = x2 − y2

The same holds for complex numbers:

The conjugate of 3 + 5i is 3 − 5i.

Thus (3 + 5i)(3 − 5i) = 9 − 25i2.

But since i2 = −1:

9 − 25i2 = 9 + 25 = 34

Another example is:

(a + bi)(a − bi) = a2 + b2

## Summary

The number √−1 is defined as an imaginary number since no number that multiplied by itself will equal −1. When combined with a real number, it is called a complex number. You can multiply and divide complex numbers. Just as with real numbers, there is a conjugate of a complex binary number.

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