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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    $\begingroup$ 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. $\endgroup$ – user13838 Oct 14 '11 at 2:15
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    $\begingroup$ Have you seen this? BTW, nice question. :) $\endgroup$ – J. M. is a poor mathematician Oct 14 '11 at 2:15
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    $\begingroup$ You may as well ask why $\sqrt2$ et al are called "irrational". $\endgroup$ – Gerry Myerson Oct 14 '11 at 3:11
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    $\begingroup$ 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 ..." $\endgroup$ – Arturo Magidin Oct 14 '11 at 3:17
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    $\begingroup$ @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. $\endgroup$ – Arturo Magidin Oct 14 '11 at 3:27

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.)


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