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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?

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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
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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

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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
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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
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Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? –  vonbrand Feb 15 '13 at 1:44
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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.

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... 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
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