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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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  • $\begingroup$ 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 '13 at 20:30
  • $\begingroup$ To add to the confusion, I've heard people call complex numbers in general "imaginary"... $\endgroup$ – vonbrand Feb 15 '13 at 1:46
  • $\begingroup$ The difference between a Complex Number and an Imaginary Number is a Real Number :D $\endgroup$ – Nick Oct 12 '14 at 15:47
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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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    $\begingroup$ 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 '13 at 19:10
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    $\begingroup$ Aren't $\pi + 3 i$, and probably $e + i \pi$, trancendent too? $\endgroup$ – vonbrand Feb 15 '13 at 1:44
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    $\begingroup$ isn't $0$ a pure imaginary number ? $\endgroup$ – mercio Oct 12 '14 at 15:47
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    $\begingroup$ @mercio, ok, my fault. I missed 'non zero $b$'. $\endgroup$ – Sigur Oct 15 '14 at 16:47
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    $\begingroup$ 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 '20 at 8:32
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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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  • $\begingroup$ ... except for $0=0i$ which is not an imaginary number. $\endgroup$ – hmakholm left over Monica Feb 14 '13 at 20:25
  • $\begingroup$ @HenningMakholm Mathematicians are professionals at finding these pathological examples... you're right :) $\endgroup$ – MyUserIsThis Feb 14 '13 at 21:01
  • $\begingroup$ @HenningMakholm why wouldn't 0 be considered an imaginary number? $\endgroup$ – Max May 2 '17 at 20:44

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