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# imaginary

An *imaginary number* is the product of a nonzero real number multiplied by an imaginary unit (such as $i$) but having having real part 0.

Any complex number $c\in\mathbb{C}$ may be written in the form $c=a+bi$ where $i$ is the imaginary unit $i=\sqrt{-1}$ and $a$ and $b$ are real numbers ($a,b\in\mathbb{R}$). So an imaginary number is a complex number $c$ such that $a=0$ in the above formulation, i.e. such that $c$ can be written as $c=bi$. Such a complex number is then sometimes called a purely imaginary number. For a purely imaginary number $c$, then it is the case that $\Re(c)=0$ and $\Im(c)\neq 0$ and $b\in\mathbb{R}*$. A few examples of purely imaginary numbers: $0+47i$, $0-\pi i$, $\frac{3}{4}i$.

Note that the imaginary numbers are closed under addition but not under multiplication, since $i\times i=-1$ is not imaginary.

Whether $0+0i$ is an imaginary number is a matter of debate. It is in the middle of the number line of the real numbers but it is also in the middle of the number line of purely imaginary numbers.

The term imaginary has had its definition expanded by analogy to several areas of mathematics. For example, if $V$ is vector space with a linear involution (denoted $v\rightarrow v^{*}$), then an imaginary element in $V$ is one such that $v^{*}=-v$.

Much in the same way that the Impressionist painters came to be known by a term initially intended to be derogatory, the term “imaginary number” comes from a sneer by Descartes as to the validity of the concept: “For any equation one can imagine as many roots [as its degree would suggest], but in many cases no quantity exists which corresponds to what one imagines.”

# References

- 1 Titu Andreescu & Dorin Andrica, Complex Numbers from A to… Z.
- 2 Bryan E. Blank, Book Review of An Imaginary Tale: The Story of $\sqrt{-1}$, Notices of the AMS 46 10 (1999): 1236
- 3 Paul Nahin, An Imaginary Tale: The Story of $\sqrt{-1}$. Princeton: Princton University Press (1998)

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## Comments

## Imaginary.html

The real part of 0 is 0; consequently, 0 is an imaginary number (?)

## Re: Imaginary.html

I'd say yes then

so? ;)

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## Re: Imaginary.html

by the way, 0 is ALSO a complex number ;)

f

G -----> H G

p \ /_ ----- ~ f(G)

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G/ker f

## Re: Imaginary.html

To drini

Naturally, 0 and 50 are complex numbers, but 50 is not imaginary. Do you mean that 0 is imaginary??

Jussi

## Re: Imaginary.html

In Finland, and I think also in continental Europe, the term "imaginary" means a complex number having the imaginary part distinct from zero. We have also the term "purely imaginary" which means that the real part is zero (so, e. g. 3i and 0 are purely imaginary -- although 0 is real, too) =o)

Jussi

## Re: Imaginary.html

Yes, I meant 0 is both imaginary and real

(just like it's als integer, complex, rational, etc).

The purely imaginary vs imaginary I don't think it's standard (it's more matter of style for agiven prof or inside a school), but if you asked me , I would have guessed imaingary is real part = 0 (thus 0 is imaginary) and purely imaginary the complex part different from zero (so 0 wouldn't be purely imaginary)

I guess conventions vary in differents parts of the world.

But yes, I meant 0 is imaginary

f

G -----> H G

p \ /_ ----- ~ f(G)

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G/ker f

## Re: Imaginary.html

So, the terminology is different in different parts of the world -- I have known it also before. But, in my opinion, the "Finnish terminology" is somewhat clearer, and I guess that in the history, when the _imaginary_ numbers were introduced (who did it?), nobody thought that the well-known zero were imaginary!

Jussi

## Re: Imaginary.html

hehehe,, well

when imaginary numbers wre introduced around 16 or 17 century, someitmes even negative numbers would not be considered numbers...

but by the time imaginary numbers were acknowledged as an algebraic structure C (call it field or whatever), 0 would already be considered an imaginary number (methinks) albeit a rather special one (kinda like empty set isa very special set, or 0! or some such particularities)

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f