Imaginary Numbers and the Imaginary constant

 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. 

Leibniz wrote, "[...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^3−6x^2+13x−10=0 as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)

 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.  

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 \pm \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 \$ x^3 + ax + b \$ .  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 \$ x^3 - 15x -4 \$ 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 other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors). *MacTutor

Imaginary Unit The imaginary number with a magnitude of one used to represent \$ \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.  

In most fields of electronics the imaginary constant i is replaced by a j, perhaps to avoid confusion with the use of i for current in Ohm's law.