# The Imaginary Number i

Posted by Professor Cram in Complex Numbers

## Complex Numbers: An Introduction

A complex number can be expressed in the form:

**a + b**

*i*where **a** and **b** are real numbers.

The number * i* is defined as the square root of -1. It is an

**imaginary number**, since the square root of any negative number is not a real number.

Complex numbers are not algebraic expressions. They are numbers, containing a real part (**a**) and an imaginary part (**b i**).

## Using i

In the world of real numbers, an expression like the square root of -9 is unsolvable. (Remember, because the square root of a negative number is not real.) However, by using the **imaginary number i** we can break this problem down as the square root of (9)(-1).

This can be expressed as the square root of 9 (which is 3) times the square root of -1 (which is *i*). Thus, the answer to this otherwise unsolvable problem is 3*i*.

## i to a Power

As with any real number, *i*^{0} = 1 and *i*^{1} = *i*. We know that *i*^{2} = -1. Also, *i*^{3} = (-1)(*i*) or -*i*. These four values (1,*i*, -1, -*i*) are the only possible results of raising the imaginary number *i* to any power.

To figure out any exponent higher than 4, just divide the exponent by four and use the remainder (0, 1, 2, or 3) as the new exponent. Match it up with the choices here and you have your answer. (Check the tutorial below to see this in action.)

## The Imaginary Number i

What is the imaginary number i, and how is it used? This tutorial will teach you all about it.

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## Get More Help!

Click one of these links to **get more help** from another Cramlet in this same chapter:

- Study Sheet of Complex Numbers
- Bottomless Worksheet of Subtracting Complex Numbers
- Bottomless Worksheet of Multiplying Complex Numbers
- Bottomless Worksheet of Dividing Complex Numbers
- Bottomless Worksheet of Adding Complex Numbers
- The Imaginary Number i
- Multiplying Complex Numbers
- Dividing Complex Numbers
- Adding and Subtracting Complex Numbers

How bout if imaginary number i is raise to i. .”i^i”??

Its a real no having magnitude e^(-pie/2) and other values as well. Use demoivre theorem

what is the demoivre theorem??

how to you get the remainder 0,1,2,3 when raisin i?

So if i^2 is equal to -1, and -1 is a real number, then two imaginary numbers can create a real number? So qualitativly, could this mean that if two imaginary particles were to collide, they could theoretically create a real particle? Does this not defy the first law of thermodynamics? Or does this simply mean that I’m misunderstanding the use of the word “imaginary”?