Friday 22 September 2023

Imaginary Numbers, Imaginary Constant, ... History and Etymology of Math Terms

Imaginary Numbers The word imaginary was first applied to the square root of a negative number by Rene Descartes around 1635.  He wrote that although one can imagine every Nth degree equation had N roots, there were no real numbers for some of those imagined roots.  Around 1685 the English mathematician John Wallis wrote, "We have had occasion to make mention of Negative squares and Imaginary roots".  Some mathematicians have suggested the name be changed to avoid the stigma that it seems to create in young students, "Why do we have to learn them if they aren't even real."  (To be honest, in thirty years as an educator, I never heard the question.)  Perhaps the weight of history is too much to support the change.

Imaginary and real are found in English in 1668 in Philosophical Transactions. The words are found in a review of the book Geometriæ Universalis: "And for the like reason a Cubick Æquation, having three reals roots, can never be reduced to a pure Æquation, which hath but one onely root, for in these Æquations, Reduction shall no wise profit, for as much as 'tis impossible, by aid thereof to change an Imaginary root into a real one, and the Converse." [Google print search by James A. Landau]

As a way of removing the stigma of the name, the American mathematician Arnold Dresden (1882-1954) suggested that imaginary numbers be called normal numbers, because the term "normal" is synonymous with perpendicular, and the y-axis is perpendicular to the x-axis (Kramer, p. 73). The suggestion appears in 1936 in his An Invitation to Mathematics.

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors).

The first person ever to write about employing the square roots of a negative number was Jerome Cardin (1501-1576).  In his Ars Magna (great arts) he posed the problem of dividing ten into two parts whose product if forty.  After pointing out that there could be no solution, he proceeded to solve the two equations, x+y = 10, and xy=40 to get the two solutions,

$5±\sqrt{15}$  .  He then points out that if you add the two solutions, you get ten, and if you multiply them then the product is indeed forty, and concluded by saying that the process was "as subtle as it was useless."

Cardin also left a seed to inspire future work int he mystery of roots of negative numbers. Cardin had published a method of finding soltuons to certain types of cubic functions of the form  .  His solution required finding the roots of a derived equation.  For functions in which the value was negative, his method would not work, even if one of the three roots was a known real solution.
About thirty years later Rafael Bombelli found a way to use the approach to find a root to ${�}^{}$ with the known solution of four.  He went on to develop a set of operations for these roots of negative numbers.  By the eighteenth century the imaginary numbers were being used widely in applied mathematics and then in the nineteenth century mathematicians set about formalizing the imaginary number so that they were mathematically "real."
Some prefer the term "complex number" (a term first coined by Gauss) for combinations of a real and imaginary number like 2 + 3i, and reserve imaginary for numbers that have no real part.

Imaginary Unit The imaginary number with a magnitude of one used to represent the square root of $\sqrt{-1}$, has been the letter i since it was adopted by Euler in 1777 in a memoir to the St Petersburg Academy, but it was not published until 1794 after his death.  It seemed not to have gained much use until Gauss adopted it in 1801, and began to use it regularly.  The term, imaginary unit, was first created, it seems, by William Rowan Hamilton in writing about quaternions in 1843 to the Royal Irish Academy.  For his three-dimensional algebra of quaternions, Hamilton added two more imaginary constants, j, and k, which were both considered perpendicular to the i, and to each other.
The term IMAGINARY PART appears in 1748 in A Treatise of Algebra: In Three Parts by Colin Maclaurin. [Google print search by James A. Landau]