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# 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:

i^{2}= −1

i^{3}= i^{2}*i = −i

i^{4}= (i^{2})(i^{2}) = (−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) = 42i^{2}= −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) + 15I^{2 }=

6 + 19i − 15 =

19i − 9or− 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)=x^{2}− y^{2}

The same holds for complex numbers:

The conjugate of **3 + 5i **is **3 − 5i**.

Thus

(3 + 5i)(3 − 5i) = 9 − 25i.^{2}But since

i:^{2}= −1

9 − 25i^{2}= 9 + 25 = 34

Another example is:

(a + bi)(a − bi)=a^{2}+ b^{2}

## 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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## Imaginary Numbers