Imaginary Numbers

i

Before we talk about imaginary numbers, let's talk about exponents. 

10⁰ =1

10¹=10

10²=100

10³=1,000

10⁴=10,000

10⁵=100,000

Notice something? The small number on top, called an exponent, tells us how many times we multiply a number. For example, 10⁴ or 10^4 is basically saying 10*10*10*10(which is 10,000).

The opposite of an exponent is a root. Instead of multiplying several times, you see what number times itself equals a certain number. For example, the square root of 4 is 2, since 2*2 is 4. The cube root of 27 is 3 because 3*3*3 is 27. The fourth root of 1 is 1 since 1*1*1*1 is 1.

For the answer to be correct, each number must be the same. That means you can't have 2*(-2) or 2*3. It has to be the same number.

When it comes to the square root of negative numbers there are no answers. Why? If you can multiply two numbers together to get a negative product, your equation will be negative times a positive. The problem is that these numbers are not the same. A positive times a positive is a positive and a negative times a negative is negative. That means that the square root of -1 does not exist. 

Mathematicians decided to make an imaginary number called i. i*i is -1 and the answer may vary depending on the value of i (e.g -i, 9i, -2i). Since i is expressed with a letter, we can write i^2 like this: ii

Although imaginary numbers aren't real, they are very useful in graphs that involve a number with an exponent of an even number and a certain y-intercept. For example, let's say we have a graph like this: x^2 + 1

The y-intercept is 1, but the x-intercept is i since it doesn't touch anywhere on the x-axis. i tells us when the shape touches the x-axis.

This is a very fuzzy topic, so if you don't understand, tell me in the comments below.