Complex numbers have had a long and twisted history, largely because medieval mathematicians were consistently uncomfortable working with these concepts, which they considered “impossible” or “useless.” This sentiment is still popularly echoed today, even by those who are relatively comfortable with mathematics.
Initially developed to handle previously “unsolvable” quadratic and cubic equations, the idea of complex numbers has grown tremendously, to integrate with coordinate systems, vectors, matrices, and even quantum mechanics. As we discover more about advanced physics, complex numbers continue to grow ever more significant.
What is Complex Number?
A complex number is one of the form a + ib, where a and b are real numbers. a is called the real part of the complex number, and b is called the imaginary part ( a and b can also equal zero). The imaginary unit i (called iota) is defined by i= sqrt(-1).
Why do we need Complex Numbers?
The "problem" that leads to complex numbers concerns solution of equation x2 + 1 =0. This equation has no solutions because -1 does not have a square root. If this equation is to be given solutions, then we must create a sqrt(-1) i.e. i=Ö(-1).
The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid.
Many early mathematicians shared the opinion of the 9th-century Hindu Mahaviracarya, “The Square of a positive number, as that of a negative number, is positive. Hence the square root of a positive number is twofold, positive and negative. There is no square root of a negative number, for a negative number is not a square.”
During the sixteenth century, mathematicians attempted to solve various cubic equations. Scipione del Ferro found a formula for the cubic x3 + px = q sometime around 1600, but kept it a secret because the mathematical climate of those days was one of competition, contests, secrecy, and fame. When someone discovered a new technique, he would challenge another to a problem-solving contest and hopefully win glory and the favor of a rich patron. Still, del Ferro told almost no one except his student, Antonio Fior, who promptly challenged a far more talented mathematician, Tartaglia. Tartaglia, fearing that Fior would use del Ferro's secret, attempted to solve the cubic himself and finally did it before the contest, easily besting Fior. Tartaglia, too, valued his secrecy, and refused to reveal his techniques until Girolamo Cardano begged him for the answer. Once Cardano realized that Tartaglia was not the first to solve the cubic, he felt free to publish the solution in his book Ars Magna, giving del Ferro and Tartaglia credit.
It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in the 17th century. Descartes found that when these algebraic formulae popped up with imaginary numbers, the geometry failed.
The first person to get close to a graphical representation of complex numbers was the Englishman John Wallis in his De Algebra tractatus in 1685. He never achieved an actual system of representation, but his work on a geometric/algebraic problem hints toward the system we use today. Essentially, he worked with the SSA triangle, in which two side lengths are known and an angle which is not between them. If one of the sides is too short (i.e. if the algebra produces an imaginary number), then the triangle is not formed.
Wallis, however, realized that a triangle
is formed, but only if one shifts the base upward. The resulting triangle does not contain the original angle, but it does hint toward our understanding: imaginary numbers indicate a
vertical movement in the plane.
The existence of complex numbers was not completely accepted until the geometrical interpretation had been described by Caspar Wessel, a Norwegian surveyor,in 1799. Sadly, Wessel's paper outlining all these ideas was written only in Danish, and published only in a small journal read mostly inside Denmark. Indeed, his paper was virtually ignored until 1895, when it was rediscovered and recognized. Most of the credit for the geometric interpretation, therefore, commonly goes to the Swiss bookkeeper J.R. Argand. He published a small paper in 1806, essentially proposing the same ideas as Wessel, without even placing his own name on the title page. This, too, would have disappeared, if one of the men who received the paper had not mentioned it to a friend, who died and whose brother published the ideas in a well-known journal and invited Argand to claim the ideas. Ever since, the plane of real and imaginary axes has been known as the Argand diagram.
