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:

i¹ = √-1
i² = √-1√-1 = -1
i³ = -1√-1 = -i or -√-1
i⁴ = 1
i⁵ = 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.

i²⁵ = i because 25/4 = 6 remainder 1, and i¹ = 1
i³¹ = -i
i⁸⁰⁰ = 1
i⁸⁰² = -1

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