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Does anyone know from a historical perspective what kinds of problems/situations resulted in the need and creation of complex/imaginary numbers?

Recently, my friend was asking about why we need complex numbers and what are there applications. I tried to tell him that there are certain functions that model phenomena in the natural world, and these functions sometimes have "complex roots". Thus, when working with these functions in engineering and physics domains (e.g. Laplace Transform, Fourier Transform), we need to be aware of these complex roots - but I could not think or find any better explanation.

Does anyone know if there was a particular moment in history where the need for complex/imaginary numbers became strikingly clear (e.g. https://www.youtube.com/watch?v=KufsL2VgELo @ 2:45 - the author does not explain why)? It would be interesting if I could show to my friend how some situation in history resulted in complex/imaginary numbers, which eventually lead to Fourier and Laplace Transforms, which eventually lead to the telephone in his pocket.

Thank you!

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    $\begingroup$ Look into the history of solving cubic equations. (BTW, this question might be better suited to History of Science and Mathematics .SE.) $\endgroup$
    – Blue
    Aug 9 at 19:45
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    $\begingroup$ I strongly recommend Mazur's book Imagining numbers (particularly the square root of minus fifteen). $\endgroup$ Aug 9 at 19:45
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    $\begingroup$ I'd have thought Cardano's solution of the general cubic brought the point home. The insight that imaginary numbers, introduced to provide formal solutions to quadratics, had more general life is quite important. $\endgroup$
    – lulu
    Aug 9 at 19:46
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    $\begingroup$ I think that the basic idea behind the introduction of complex numbers is to solve sqrt(x) when $x$ is a negative real number, but in order to fully understand this concept mathematicians were struggling to find its geometric interpretations. Moreover, somebody who found a way to solve a given set of polynomial equations could fail to interpret the method since they were basically subtracting some negative area when they translated the problem "geometrically". $\endgroup$
    – Marco
    Aug 9 at 19:48
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    $\begingroup$ Should be migrated to HSM SE for better answers $\endgroup$
    – qwr
    Aug 10 at 4:55

4 Answers 4

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In the 15th century Tartaglia presented a general solution to cubic polynomial equations that that required the square root of some combinations of the coefficients of the cubic. For some cubic equation, that could be proved by other methods to have (real number) solutions, Tartaglia's method required the square root of negative numbers.

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From "The beginnings & Evolution of Algebra" by Bashmakova and Smirnova published by the Mathematical Association of America, published 2000:

p. 71 (1) Cardano, like Tartaglia, was baffled by the "irreducible case."

In (1) the "irreducible case" is when solutions are imaginary/complex. This occurred when they were working on cubic and quartic equations, but, especially cubic.

pp71-73 (2) ... Rafael Bombelli (ca 1526-1573) ... When introducing complex numbers - which he regards as rather "sophistic" ...

So Bombelli worked with Diophantus' work and solutions of algebraic equations of the first four degrees. Cardano lived around 1501-1576 and Tartaglia lived around 1499-1557. But Tartaglia and Cardano got stuck on the imaginary or irreducible case. So complex numbers arose when looking at solutions to equations by Bombelli.

If you want a more detailed exposition then look at the referenced book pp 67-75 concerning Cardano and Tartaglia's "miss" and Bombelli's "find." I should add that we can conclude that complex numbers arose as the solutions to equations.

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Complex numbers originated in problems relating to expressing the solutions of cubic (not quadratic!) polynomials using radicals. That is, just as you have almost surely heard of the quadratic formula, there is also a "cubic formula" which expresses the value $x$ such that

$$ax^3 + bx^2 + cx + d = 0$$

for a given choice of coefficients $a$, $b$, $c$, and $d$. It's very complicated and definitely not easy to memorize. As can be expected, it involves cube roots - but also, involves square roots. And it has the interesting property that, even when all the coefficients are real, it is possible for the quantities under the square roots to be negative. In particular, if the cubic polynomial equation has 3 solutions, then the quantities under the square roots are always negative. Using the formula, thus, requires one to somehow be able to define a quantity that corresponds to taking the square root of a negative real number, and from this, the complex numbers were born.

The discoverers of the formula - most notably Niccolò Fontana Tartaglia and Girolamo Cardano - were the first to attempt to manipulate these quantities as numbers, however their full fleshing out of them into a proper number system would come a bit later with Rafael Bombelli (1526-1572) - all three of whom, of course, were from Italy.

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The complex numbers simply have a long and complicated history. It began with a formula for the solution of cubic equations, but that was only the beginning of a long journey. Tyson Larsen in a 2015 semester project gave a condensed version of the History & Background of complex numbers:

  • 50 AD: ... 665 AD: ... 1484: ...
  • 1545: Around this time, imaginary numbers began to be recognized more heavily by society. ... Nicolo Fontana in 1535 ..., Girolamo Cardano ...
  • 1572: Using a common technique known today as conjugation, Rafael Bombelli expanded upon a previous idea that one can use the square roots of negative numbers to find real solutions. ...
  • 1637: Rene Descartes, a renowned mathematician, came up with the term "imaginary" to represent expressions that require the square root of a negative number. ...
  • 1673: John Wallis made improvements helping the idea of complex numbers to be accepted. ...
  • 1747: Beginning around this time, Leonhard Euler had some notable contributions that helped society begin to accept the use of imaginary numbers. ...
  • 1797: As mentioned previously, Caspar Wessel, who was a Danish Mathematician was ultimately credited for the geometrical interpretation of complex numbers. ...
  • 1831: Carl Friedrich Gauss made some monumental advancements in the realm of complex numbers. ...
  • 1835: William Hamilton is credited for the advancement of complex numbers being represented as pairs. ...

He often uses Biggus, J. (2010). Sketching the History of Hypercomplex Numbers as source, which is also condensed. Even A Short History of Complex Numbers is already substantially longer:

  1. The first to solve equation (1) (and maybe (2) and (3)) was Scipione del Ferro, professor of U. of Bologna until 1526, when he died. ... Cardano then proceeded to publish the formula for all three cases in his Ars Magna (1545). ...
  2. A difficulty in case (2) that was not present in the solution to (1) is the possibility of having the square root of a negative number appear in the numerical expression given by the formula. ...
  3. According to $[9]$, “Cardano was the first to introduce complex numbers $a + \sqrt{−b}$ into algebra, but had misgivings about it.” ...
  4. Rafael Bombelli authored l’Algebra (1572, and 1579), a set of three books.
  5. Rene Descartes (1596-1650) was a philosopher whose work, La Geometrie, includes his application of algebra to geometry from which we now have Cartesian geometry. ...
  6. John Wallis (1616-1703) notes in his Algebra that negative numbers, so long viewed with suspicion by mathematicians, had a perfectly good physical explanation, based on a line with a zero mark, and positive numbers being numbers at a distance from the zero point to the right, where negative numbers are a distance to the left of zero. ...
  7. Abraham de Moivre (1667-1754) left France to seek religious refuge in London at eighteen years of age. ...
  8. L. Euler (1707-1783) introduced the notation $i = \sqrt{-1}$ $[3]$, and visualized complex numbers as points with rectangular coordinates, but did not give a satisfactory foundation for complex numbers. ...
  9. Caspar Wessel (1745-1818), a Norwegian, was the first one to obtain and publish a suitable presentation of complex numbers. ...
  10. Jean-Robert Argand (1768-1822) was a Parisian bookkeeper.
  11. William Rowan Hamilton (1805-65) in an 1831 memoir defined ordered pairs of real numbers (a, b) to be a couple. ...
  12. Carl Friedrich Gauss (1777-1855). There are indications that Gauss had been in possession of the geometric representation of complex numbers since 1796, but it went unpublished until 1831, when he submitted his ideas to the Royal Society of Göttingen. ...
  13. Augustin-Louis Cauchy (1789-1857) initiated complex function theory in an 1814 memoir submitted to the French Académie des Sciences. ...
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