# Why is it necessary to rationalize the denominator in complex division? (non-polar form)

I haven't been able to understand why the division algorithm requires this, (my high school textbook says that the division algorithm only works with natural numbers in the denominator), while rationalized answers in algebra after high school are not a requirement, I still don't get why I can't find a way to do the complex division without rationalizing the denominator and getting the imaginary part out of the denominator. Thanks for you help.

PD: I'm ignoring the polar form division because it's not in my textbook.

• You mean the denominator? I haven't seen people purposely make the numerator real in complex division – Triatticus Mar 18 '17 at 19:49
• @Triatticus Yes, sorry, the lack of sleep is playing with me. – Nick Cassol Mar 18 '17 at 19:50
• You don't know how to divide by $1+2$ if you only know how to divide by $1$ and $2$. Similarly you don't know how to divide by $1+i$, but you know how to multiply by $\frac{1-i}2$, which is the same thing. – Matt Samuel Mar 18 '17 at 19:53
• Because it makes it easier? – Travis Mar 18 '17 at 20:04
• The one thing to think is, how will you simplify a complex fraction into the form of $a+bi$ if you can't make the denominator a single number – Triatticus Mar 18 '17 at 20:21