User:MathMan64/CentigradeDegree

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[edit] Fractions

[edit] Repeating Decimals

A repeating decimal has digits in the decimal part that repeat forever, such as: 0.777... \

A shortcut way of writing this is 0.\overline 7

More examples:

13.515151... \ or 13.\overline{51}
8.6222... \ or 8.6\overline 2

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

[edit] Changing repeating decimals to fractions

[edit] If the entire decimal repeats

Change 0.\overline 7 to a fraction.

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

0.777... \times 10 = 7.777...

Then subtract the original number.

0.777... \times 10 = 7.777...
0.777... \times 1 \ = 0.777...

Subtract in two places: 10 - 1 = 9 \ and \ 7.777... - 0.777... = 7

[edit] Square root of a complex number

Each complex number has two square roots. Consider z=a+bi \ where r=\sqrt{a^2+b^2} and \phi = \arctan \left ( \frac b a \right ). The quadrant of \phi \ is determined by the signs of a and b.

The square roots are \pm\sqrt{\frac{r+a} 2 } \ \pm\ i \ \sqrt{\frac{r-a} 2} where the signs match if 0< \phi<\pi \ , but are different, if not.


This can be derived by expressing \sqrt {a+bi} = c+di

So a+bi=(c+di)^2 = c^2-d^2+2cdi \

Equating the real and imaginary parts: a=c^2-d^2 \ and b=2cd \

Solve the second equation for d: d=\frac b {2c}

Substitute into the equation for the real part above to get a=c^2-\frac {b^2} {4c^2}

Which simplifies to 4c^4 - 4ac^2 - b^2=0 \

Solving this for c^2 \ gives \frac{a+\sqrt{a^2+b^2}} 2

So c=\pm\sqrt{\frac{a+r} 2}

Solving b=2cd \ for c \ and doing similar work gives

d=\pm\sqrt{\frac{-a+r} 2}