Imaginary Numbers

Imaginary numbers are one of the two sets that make up complex numbers.

Imaginary numbers are written using the variable i. The
basic rule is:

**i = √-1**

Now using i, you can write the square roots of negative numbers. First, you must review a principle of square roots.

√a√b = √ab

This way, if you are trying to find √-7, you can write it was √-1√7, since -1*7=-7

You know now that √-1
= i, so √-7 = **i√7**.
The i is usually written before a radical it is multiplied by (unlike most
variables, which come after their coefficients) to avoid confusion between √7i
and √7i.

**√-64**
= √-1√64 = **8i**

Sometimes it takes longer to simplify one of these. For example, √-20.

√-20 = √-1√20 = i√20

However, √20 is not simplified completely.

√20 = √4√5 = 2√5 (Remember? You must factor it into its largest perfect square factor and the other factor with it, then simplify the square root of the perfect square.)

So √-20
= **2i√5**.

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Here's an interesting pattern with imaginary numbers:

i^{1} = **√-1**

i^{2} = √-1√-1 = **-1**

i^{3} = -1√-1 = **-i or
-√-1**

i^{4} = **1**

i^{5} = 1√-1 = **√-1**

...

As you can see, the pattern will repeat. Therefore, you can simplify any power of i by dividing the exponent by 4, then taking the remainder and using the value of i with that exponent.

i^{25} = **i** because
25/4 = 6 remainder 1, and i^{1} = 1

i^{31} = **-i**

i^{800} = **1
**i

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