14
$\begingroup$

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?

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$
    – hmakholm left over Monica
    Feb 14, 2013 at 20:30
  • $\begingroup$ To add to the confusion, I've heard people call complex numbers in general "imaginary"... $\endgroup$
    – vonbrand
    Feb 15, 2013 at 1:46
  • 1
    $\begingroup$ The difference between a Complex Number and an Imaginary Number is a Real Number :D $\endgroup$
    – Nick
    Oct 12, 2014 at 15:47

2 Answers 2

Reset to default
19
$\begingroup$

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

$\endgroup$
15
  • 2
    $\begingroup$ 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. $\endgroup$
    – Sigur
    Feb 14, 2013 at 19:10
  • 2
    $\begingroup$ Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? $\endgroup$
    – vonbrand
    Feb 15, 2013 at 1:44
  • 1
    $\begingroup$ isn't $0$ a pure imaginary number ? $\endgroup$
    – mercio
    Oct 12, 2014 at 15:47
  • 1
    $\begingroup$ @mercio, ok, my fault. I missed 'non zero $b$'. $\endgroup$
    – Sigur
    Oct 15, 2014 at 16:47
  • 1
    $\begingroup$ 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. $\endgroup$
    – john
    Apr 9, 2020 at 8:32
1
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ ... except for $0=0i$ which is not an imaginary number. $\endgroup$
    – hmakholm left over Monica
    Feb 14, 2013 at 20:25
  • $\begingroup$ @HenningMakholm Mathematicians are professionals at finding these pathological examples... you're right :) $\endgroup$
    – MyUserIsThis
    Feb 14, 2013 at 21:01
  • $\begingroup$ @HenningMakholm why wouldn't 0 be considered an imaginary number? $\endgroup$
    – Max
    May 2, 2017 at 20:44

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