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I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition.

Why, then, are they called imaginary? I know there's some arbitrariness to why anything is called anything, but this name in particular probably leads a lot of people to be stuck on the question the poster of the linked questions asked. Almost as if the name itself it trying to encourage us not to trust this crazy "number."

Googling was able to get me that Descartes coined the term, but I couldn't find anything on why, or why the name stuck, even in translation.

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OctaviaQ
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    because they did not have math.SE back then and these answers you have read were not available to them. It was simply mind boggling without the tools that were developed over time only later. –  Oct 14 '11 at 02:15
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    Have you seen this? BTW, nice question. :) – J. M. ain't a mathematician Oct 14 '11 at 02:15
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    I'm still wishing there's a better name than "imaginary", but we seem to be stuck with this (as well as "complex", which I'm not too fond of either) until the end of time... – J. M. ain't a mathematician Oct 14 '11 at 02:19
  • @J.M. I think the tension is building up, I hear your complaint every now and then from others too. So I would say in 50 years we would get something better :) –  Oct 14 '11 at 02:23
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    You may as well ask why $\sqrt2$ et al are called "irrational". – Gerry Myerson Oct 14 '11 at 03:11
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    As I recall from reading a translation of The Geometry, Descartes actually talked about expressions involving "radicals of imaginary quantities", using "imaginary quantities" to refer to negative numbers, rather than to what we now call "imaginary numbers". So he would write a solution involving a square root, and say "if the quantity in the radical is imaginary, then ..." – Arturo Magidin Oct 14 '11 at 03:17
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    I wonder: were Real numbers called "Real" before imaginary numbers were called "imaginary"? Or did the name come out afterwards for emphasis? – OctaviaQ Oct 14 '11 at 03:20
  • Too bad Chuck Norris isn't on MSE. He's probably the only one who could give a good answer... – The Chaz 2.0 Oct 14 '11 at 03:22
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    @Gerry: I think this covers "rational/irrational"... – J. M. ain't a mathematician Oct 14 '11 at 03:22
  • @Arturo: I do remember that Descartes's concept of the plane we now have is that he preferred to be confined to the first quadrant... – J. M. ain't a mathematician Oct 14 '11 at 03:24
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    @J.M.: Exactly (also, he didn't restrict his axes to be perpendicular, but that's neither here nor there). The point is that, at the time, "imaginary" was an adjective used to describe negative numbers; saying Descartes coined the term "imaginary" in this context may be a bit misleading, I think. – Arturo Magidin Oct 14 '11 at 03:27
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    I have read that Gauss tried to change the name for this very reason. He succeeded in getting people to use the "complex number" used, but failed at excising the word "imaginary" completely; e.g. he failed to get us to use "lateral part" in favor of "imaginary part". –  May 09 '14 at 08:33

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A formula was found for expressing solutions of third-degree algebraic equations with real coefficients in terms of addition, subtraction, multiplication, division, and square and cube roots. But in some cases, the number whose square root was to be found was negative. But these paradoxical quantities canceled out, leaving a real number. And now the strange part: when such solutions were substituted into the equation, they checked! That was a reason to pay attention to them.

But they did not arise as quantities in geometry (lengths, areas, etc.) nor in accounting (which is where negative numbers came from), so they were not "real".

(I don't think the concept of "real number" existed until fairly late in the game---some time after all this was done.)

Michael Hardy
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