# Help with rearranging equation to get real and imaginary parts..

I know this is so simple but my algebra is totally failing me.. I have the equation 1/1+2i and I want to extract the real and imaginary parts so I have it in the form..

Re+Im could someone just show me the algebra steps for doing this please.. Thanks

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$\dfrac{1}{1+2i} = \dfrac{1-2i}{(1+2i)(1-2i)} = \ldots$. – njguliyev Oct 31 '13 at 11:00

## 1 Answer

Multiply by the conjugate: $$\frac{1}{1+2i}=\frac{1}{1+2i}\cdot \frac{1-2i}{1-2i}=\frac{1-2i}{5}.$$

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Being a non-native speaker: shouldn't it be called “expand with” instead of “multiply by”? – Michael Hoppe Oct 31 '13 at 11:05
@MichaelHoppe I use "multiply" because I am multiplying two complex numbers together. – Joe Johnson 126 Oct 31 '13 at 11:25
But you didn't multiply by the conjugate but with $1$ instead ... Just nitpicking – Michael Hoppe Oct 31 '13 at 11:51
@MichaelHoppe I multiplied by the conjugate twice, once on top and once on bottom. So, I did multiply by the conjugate. – Joe Johnson 126 Oct 31 '13 at 15:44
OK, but what is the proper verb for “to multiply the nominator and the denominator of a fraction by the same number“? It's called “erweitern” in german and I presumed that the english verb is “to expand by.” In case you multiply a fraction by a number you'll multiply the nominator of the fraction by this number and leave the denominator unchanged. – Michael Hoppe Oct 31 '13 at 15:56