# Difference between imaginary and complex numbers

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?

• 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. Feb 14, 2013 at 20:30
• To add to the confusion, I've heard people call complex numbers in general "imaginary"... Feb 15, 2013 at 1:46
• The difference between a Complex Number and an Imaginary Number is a Real Number :D
– Nick
Oct 12, 2014 at 15:47

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:

• 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. Feb 14, 2013 at 19:10
• Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? Feb 15, 2013 at 1:44
• isn't $0$ a pure imaginary number ? Oct 12, 2014 at 15:47
• @mercio, ok, my fault. I missed 'non zero $b$'. Oct 15, 2014 at 16:47
• Wikipedia does not make a distinction between 'imaginary' and 'pure imaginary' in its tldr article on complex numbers. An imaginary number is a number of the form $a+bi$ where $b\neq 0$. There is a nice diagram of this.
– john
Apr 9, 2020 at 8:32

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.

• ... except for $0=0i$ which is not an imaginary number. Feb 14, 2013 at 20:25
• @HenningMakholm Mathematicians are professionals at finding these pathological examples... you're right :) Feb 14, 2013 at 21:01
• @HenningMakholm why wouldn't 0 be considered an imaginary number?
– Max
May 2, 2017 at 20:44