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42 votes

Why are quaternions more popular than tessarines despite being non-commutative?

Commutativity is over-rated: in fact, it holds back bicomplex numbers: It prevents your number system characterising non-commuting operations, e.g. rotations in $3$-dimensional space, Hamilton's ...
J.G.'s user avatar
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27 votes

Why are quaternions more popular than tessarines despite being non-commutative?

Your description "total uselessness of quaternions" in a comment above is poorly chosen, and reflects more on your interests than on the real state of knowledge of mathematics. The Hamilton ...
KCd's user avatar
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13 votes
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What was the motivation for Cauchy's Integral Theorem?

The original motivation, and an inkling of the integral formula, came from Cauchy's attempts to compute improper real integrals. Here is from A Brief History of Complex Analysis in the 19th Century: ...
Conifold's user avatar
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12 votes
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Euler's first proof of $e^{ix}=\cos(x)+i\sin(x)$

According to Boyer's A History of Mathematics, the identity first appeared in Euler's Introductio in analysin infinitorum of 1748. Euler started from the infinite series for $e^x$, $\sin x$, and $\...
nwr's user avatar
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11 votes

Why are quaternions more popular than tessarines despite being non-commutative?

Hamilton expected that the quaternions would be of physical interest. In this, he was right. But he was too early. He had discovered them in 1843, it was almost a century later, in 1928, when Dirac ...
Mozibur Ullah's user avatar
10 votes
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Who was the first person to notice logarithms of negatives numbers and for what reason?

As far as I known, logarithms of negative numbers first appeared (in the modern sense) in the 1751 paper of Euler De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres ...
user6530's user avatar
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10 votes
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Mathematics development can sometimes **exceed** the practical needs, right?

Yes, this happens many times. Mathematics developed purely for internal mathematical reasons, later turns out to have applications elsewhere. See the essay by Wigner, "The unreasonable effectiveness ...
Gerald Edgar's user avatar
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7 votes

History of complex analysis

I would say that there is no good book which satisfies your description. The Book of Bottazzini and Grey mentioned in the comments is OK, but it certainly does not cover the role of complex analysis ...
Alexandre Eremenko's user avatar
7 votes
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Were complex number first considered of limited usefulness?

Usefulness was never seriously questioned. Complex numbers are useful, for example, if you want to make Cardano's formula work. And this is how they were invented. They are useful for other ...
Alexandre Eremenko's user avatar
6 votes

Who pioneered the study of the sedenions?

There are (at least) two different types of numbers called "sedenions". The first ones were introduced by Muses in 1980, who called them $16$-ions, and renamed into "sedenions" by Carmody in 1988. ...
Conifold's user avatar
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6 votes
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History of a contour integral method for summing series

Whittaker and Watson credit this method to Mittag-Leffler, Acta Soc. Sci. Fenn., XI 1880 273-293, and Acta Math., IV 1884 1-79.
Alexandre Eremenko's user avatar
6 votes
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What is the history of using $i$/$\iota$ as the imaginary unit?

For an early occurrence of "iota", see : Alfred Cardew Dixon (English mathematician, 1865-1936), The elementary properties of the elliptic functions. With examples (London, 1894), p. 5 : §10. ...
Mauro ALLEGRANZA's user avatar
6 votes
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Original proof of the Schwarz lemma

The following paragraph in "Julius and Julia: mastering the art of the Schwarz lemma" by Professor Harold P. Boas (Amer. Math. Monthly, 117 (2010), no. 9, pp. 770-785) might be relevant: ...
José Hdz. Stgo.'s user avatar
6 votes

Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?

The problem with accepting complex numbers, when your only experience with "numbers" is the real number line, is lacking a visualization of them. No ordinary (i.e., real) number can square ...
KCd's user avatar
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6 votes

Who first considered signed area?

Signed areas came up in connection with winding numbers even before Gauss, see Sunada, From Euclid to Riemann and Beyond: "Gauss briefly noticed in the letter to Bessel in 1811 , the winding ...
Conifold's user avatar
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6 votes

Translated articles of Fatou and Julia

I was pointed to this chain of messages and I would then like giving you my version of facts. The translations project was the very very early version of the contents that ended up in the book I ...
Sandro Rosa's user avatar
5 votes

The origins of complex differentiation/integration

There is no difference in differentiation. Derivative of a complex function of a complex variable is defined by the same formula $$f'(z)=\lim_{h\to 0} (f(z+h)-f(z))/h$$ as for the real variable. ...
Alexandre Eremenko's user avatar
5 votes

How much ground was prepared for Riemann so that he could conjecture Riemann hypothesis?

Riemann wrote his paper on the zeta-function in 1859. He was the first person to consider the zeta-function in the complex plane, first for ${\rm Re}(s) > 1$ and then he worked out an analytic ...
KCd's user avatar
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5 votes

How did Roger Cotes come up with logarithm form of Euler formula?

This answer is now mostly complete, and it's really long. I've made substantial changes since originally posting, but now it's pretty much in its final state. Finding the right interpretation of Cotes'...
Sam Gallagher's user avatar
5 votes
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History of the definition of complex derivative

In general, definitions in mathematics are not arbitrary. Useful definitions are those which lead to interesting and important objects. This particular definition is good because it leads to an ...
Alexandre Eremenko's user avatar
5 votes
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Reference request: What were the problems of accepting zero, negative numbers, and complex numbers? And how were they solved?

What you write suggests that you still have difficulty with complex numbers. I would suggest that you overcome such difficulties before you proceed to the next step such as delving deeply into real ...
Mikhail Katz's user avatar
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5 votes
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Why is the Mean Value Theorem (of holomorphic functions) called "Gauss's"?

The naming, apparently, derives from the corresponding theorem for harmonic functions, the mean value property. According to Netuka-Veselý's survey, Neumann originally called it der Gauss'sche Satz ...
Conifold's user avatar
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4 votes
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Who pioneered the study of the sedenions?

On Bibliography of Quaternions and Allied Mathematics by Alexander Macfarlane I found this: On page 72; James Byrnie Shaw 1896 Sedenions (title). American Assoc. Proc., 45, 26. I couldn't find this ...
Mr. J. Larios's user avatar
4 votes
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Why are complex numbers called 'complex'?

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 ...
Alexandre Eremenko's user avatar
4 votes
Accepted

Introduction of $\imath$ and $\jmath$ notations for the imaginary unit

In answer to your second part of the question regarding $j$ for $\sqrt{-1}$, this was introduced into text books describing Power System Analysis of AC power circuits in the early 1900s by Charles P. ...
K7PEH's user avatar
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4 votes

Mathematics development can sometimes **exceed** the practical needs, right?

It is well known when and why complex numbers were introduced. When you solve a cubic equation which has 3 real roots, using Cardano's formula, you obtain square roots of negative numbers in your ...
Alexandre Eremenko's user avatar
4 votes

Who introduced the stream function?

d'Alembert did originally introduce the stream function in "Remarques sur les lois du mouvement des fluides" in 1761. Page 149 says that $$p = \frac{\partial \omega}{\partial x} \,,$$ $$q = ...
Andrew Trettel's user avatar
3 votes

Did Gauss know the residues theorem in complex analysis in 1811?

For the completeness of the discussion in this post i must add a link to another relevant question - Meaning of passages by Gauss on the "convergence of expansions (in infinite series) of the (...
user2554's user avatar
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3 votes
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Who came up with the link between the spectrum of an operator and the poles of a meromorphic function?

A paper of Kneser (1904) strongly suggests that the idea does (indeed) go back to Cauchy, in connection with Sturm-Liouville problems (i.e. ordinary differential operators, as opposed to the Laplacian ...
Francois Ziegler's user avatar

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