14

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?

OmnipresentAbsence
  • 756
  • 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. – hmakholm left over Monica 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 01:46
  • 1
    The difference between a Complex Number and an Imaginary Number is a Real Number :D – Nick Oct 12 '14 at 15:47

2 Answers2

19

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

Sigur
  • 6,416
  • 3
  • 25
  • 45
  • 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
  • 3
    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
  • 2
    Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? – vonbrand Feb 15 '13 at 01:44
  • 1
    isn't $0$ a pure imaginary number ? – mercio Oct 12 '14 at 15:47
  • @mercio, zero is a real number. Imaginary pure should be non real. – Sigur Oct 13 '14 at 23:07
  • but the real part of $0$ is $0$, so as you say in your answer, $0$ is a purely imaginary number (and yes it is a real number too). In fact it's the only real purely imaginary number. – mercio Oct 14 '14 at 06:27
  • 1
    @mercio, ok, my fault. I missed 'non zero $b$'. – Sigur Oct 15 '14 at 16:47
  • Does anyone know of this source? Or a textbook with this sort of diagram? – PhysicalChemist Sep 11 '15 at 13:53
  • @vonbrand yes they are, the diagram shows this: Any number that is not algebraic is transcendental (the two are mutually exclusive) – Jon McClung Jan 29 '18 at 18:05
  • Thank you, the diagram is very informative. Is this diagram licensed by cc-by? – czlsws Aug 06 '19 at 12:02
  • 1
    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 09 '20 at 08:32
  • 1
    Oddly e^pi is an integer, -1, or in general, multiples of pi in this sort of exponent often gives integers. I suppose this is the root of why the so-called elliptical functions are so useful studying the integers and primes. (sorry, didn't know how to get a pi character here) – Gerry Gleason Jul 11 '20 at 09:16
  • In your answer you don't explain what an imaginary number is. Instead, you explain what is pure imaginary, and you also explain what an imaginary part of a complex number is. – michael_kuzmin Feb 16 '21 at 05:21
1

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.

MyUserIsThis
  • 3,702
  • 1
  • 24
  • 39