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Fractions

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Repeating Decimals

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A repeating decimal has digits in the decimal part that repeat forever, such as:

A shortcut way of writing this is

More examples:

or
or

Notice that the line is only over the part of the decimal that repeats.

Changing repeating decimals to fractions

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If the entire decimal repeats

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Change to a fraction.

This one has only one digit that repeats. So multiply by ten.

Then subtract the original number.

Subtract in two places: and

Square root of a complex number

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Each complex number has two square roots. Consider where and . The quadrant of is determined by the signs of a and b.

The square roots are where the signs match if , but are different, if not.


This can be derived by expressing

So

Equating the real and imaginary parts: and

Solve the second equation for d:

Substitute into the equation for the real part above to get

Which simplifies to

Solving this for gives

So

Solving for and doing similar work gives