Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am stuck on the problem:

Simplify : $\displaystyle \frac{3}{(2 + i)}$

Any explanations?


share|cite|improve this question
up vote 4 down vote accepted

They probably want you to get it in a form without $i$ in the denominator, which you do by multiplying top and bottom by the complex conjugate. For a complex number $c = a+bi$ (here $2+i$), the complex conjugate is $\bar c = a-bi$. This has the property that $c \bar c = a^2 + b^2$ is a real number.

So multiplying top and bottom of your fraction by the complex conjugate of $2+i$ will give you a fraction which has a real number in the denominator, which is often considered preferable.

share|cite|improve this answer
Thanks Wikipedia! Besides my previous lack of knowledge of the complex conjugate, you gave a very informative answer. Thanks! – Cole Weis Jan 23 '13 at 18:31

multiply both numerator and denominator by $(2-i)$. You will get $\frac{3(2-i)}{(2+i)(2-i)}=\frac{6-3i}{4-i^2}=\frac{6-3i}{4+1}=\frac{6-3i}{5}$

share|cite|improve this answer
Thanks for being straight-forward. – Cole Weis Jan 23 '13 at 18:28

In general, if you're dividing by a complex number $a+bi$, where $a,b$ are real and $b\neq 0$, then you need to multiply numerator and denominator by the complex conjugate $a-bi$. This will give you a real number on the bottom. At that point, simplify as far as you can.

share|cite|improve this answer
I am starting to see that... Thanks! – Cole Weis Jan 23 '13 at 18:38

You can multiply and divide by $2-i$ :) in general, to write a complex number in a $a+ib$ form you can multiply and divide by the conjugate of the denominator.

share|cite|improve this answer
Qn: how can i delete my answer if other n+1 people wrote the same thing in the meanwhile?? – pppqqq Jan 23 '13 at 18:24
Sorry, my question was too unexotic. – Cole Weis Jan 23 '13 at 18:37

Multiply by ${2-i \over 2-i}$ to get ${6-3i\over 5}$.

share|cite|improve this answer
Yepperdoodle sir. – Cole Weis Jan 23 '13 at 20:29

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.