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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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