Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

Recently I was talking to my teacher about complex and imaginary numbers and he told me basically that $i$ is a complex number; its real part is just 0. However, this has made me wonder; if you can see $i$ as a complex number because you could argue its real part is 0, how can you differentiate between complex numbers and imaginary numbers?

share|cite|improve this question
It should be added that in modern mathematics there is almost never any reason to talk about imaginary numbers in general -- just about everything you can say about imaginary numbers is just as valid about all the complex numbers, so it is usually said in that more general form. – Henning Makholm Feb 14 '13 at 20:30
To add to the confusion, I've heard people call complex numbers in general "imaginary"... – vonbrand Feb 15 '13 at 1:46
The difference between a Complex Number and an Imaginary Number is a Real Number :D – Nick Oct 12 '14 at 15:47
up vote 10 down vote accepted

Every complex number can be written as $z=a+bi$, where $a,b\in \mathbb{R}$ (real numbers). The number $a$ is called real part of $z$ and the number $b$ is the imaginary part of $z$.

If the real part is zero then we call $z=bi$ as pure imaginary complex number.

Here is a diagram to show the inclusions:

enter image description here

share|cite|improve this answer
So all imaginary numbers are complex numbers too? But couldn't you then argue that all real numbers are complex numbers with imaginary part 0? – OmnipresentAbsence Feb 14 '13 at 19:09
Every real number is also a complex number since $\mathbb{R}\subset \mathbb{C}$, but they have a special property: their imaginary part is zero. – Sigur Feb 14 '13 at 19:10
Wow, the real numbers are a subset of the complex numbers? I didn't know that, it all makes sense now. – OmnipresentAbsence Feb 14 '13 at 19:16
@OmnipresentAbsence, see the image above. – Sigur Feb 14 '13 at 19:20
Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? – vonbrand Feb 15 '13 at 1:44

Imaginary numbers are numbers than can be written as a real number multiplied by the imaginary unit $i$, and complex numbers are imaginary numbers, plus numbers that has both real and imaginary parts. $i$ is both imaginary and complex. The imaginaries are a subset of the complex numbers, as the naturals are a subset of the integers.

share|cite|improve this answer
... except for $0=0i$ which is not an imaginary number. – Henning Makholm Feb 14 '13 at 20:25
@HenningMakholm Mathematicians are professionals at finding these pathological examples... you're right :) – MyUserIsThis Feb 14 '13 at 21:01

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.