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I'm a high school teacher, and I was just wondering why complex numbers are called 'complex'. I have read that Gauss coined the term. But I couldn't find any reference where it was explained.

I also know that 'komplex' in German comes from Latin 'complexus' which means 'complex' but also 'intertwined'. And I thought to myself, 'Well, complex multiplication does intertwine the real and imaginary parts'. But that's just a guess.

Does anyone have a reference on this?

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    $\begingroup$ The use of the word complex here is in the sense of having multiple related parts, in this case both a real and an imaginary part. Gauss introduced the term because he saw the need for having different names for $ai$ and $a + bi$, so he gave to the latter the Latin expression numeros integros complexos. The full quote from Gauss' paper "Theoria Residuorum Biquadraticorum, Commentatio secunda" is given on Jeff Miller's site. $\endgroup$
    – nwr
    Commented Jun 22, 2020 at 18:59
  • $\begingroup$ Just wait until you get to quaternions and octonians! $\endgroup$
    – Carl Witthoft
    Commented Jun 23, 2020 at 14:24

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Complex numbers were used long before Gauss. They appeared for the first time in 16th century when people found a formula for solving cubic equations. One problem with this formula is that even for simplest equations like $x^3-x=0$ which have 3 real solutions, square roots of negative numbers occur in the formula (they cancel in the end, when you do calculation correctly). So use of the formula requires calculations with complex numbers, and people started investigating the rules of such calculations. They were called various names, "imaginary" numbers, "impossible" numbers, all these terms reflecting peoples confusion with them lasting till the beginning of 19th century.

Gauss found how to represent them geometrically, but even here he was not the first. Predecessors were Jean-Robert Argand and Caspar Wessel. Various authors combined these three names with words such as plane, diagram etc. See Argand diagram in Wikipedia.

Eventually the modern terminology emerged in 19th century: "complex numbers", meaning that they consist of two parts, real and imaginary. "Imaginary number" is used sometimes to denote a complex number which is not real, or more frequently a number whose real part is zero (a.k.a "pure imaginary").

Gauss also investigated numbers of the form $m+ni$ where $m,n$ are integers. These are still called "Gauss integers", they have applications in questions about ordinary integers (number theory). The reason is that some primes can be factored using Gauss integers, like $5=(1+2i)(1-2i)$

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  • $\begingroup$ Great -- now you have me diving into the "Prime Race" algorithms :-) $\endgroup$
    – Carl Witthoft
    Commented Jun 23, 2020 at 19:11

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