Remarks on the History of Complex Numbers

The study of numbers comes usually in succession. Children start with the counting numbers. Move to the negative integers and fractions. Dig into the decimal fractions and sometimes continue to the real numbers. The complex numbers come last, if at all. Every expansion of the notion of numbers has a valid practical explanation.

Negative number were needed to solve a + x = b, even when a > b. The fractions helped solve ax = b, when b was not divisible by a. The realization of the existence of reals was a response to the need to solve x² = 2. And finally, complex numbers came around when evolution of mathematics led to the unthinkable equation x² = -1. All in due course.

The historical reality was much too different. Strange and illogical as it may sound, the development and acceptance of the complex numbers proceeded in parallel with the development and acceptance of negative numbers.

Square roots of negative numbers appeared in Ars Magna (1545) by Girolamo Cardano, who would consider several forms of quadratic equations (e.g., x² + px = q, px - x²= q, x² = px + q) just in order to avoid using negative numbers. Which is hardly surprising in view of the fact that the tools Cardano used are usually described as geometric algebra. This is yet in the tradition of, say, Euclid II.5 and II.6, Al-Khowarizmi [Smith, pp. 446-447], and many others. Algebraic symbolism was still evolving and cumbersome and the proofs have been geometric.

Cardano's internal conflict is tangible in his writing. He handles the problem [Source Book, pp. 201-202] that nowadays would be described as solving the quadratic equation x² - 10x + 40 = 0:

A second type of the false position makes use of roots of negative numbers. I will give an example: If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion.

Driven by either genius or curiosity, Cardano goes on to solve an impossible question! When algebraic manipulations lead to a square root of a negative number, Cardano writes:

... This, however, is closest to the quantity which is truly imaginary since operations may not be performed with it as with a pure negative number, nor as in other numbers. ... This subtlety results from arithmetic of which this final point is as I have said as subtle as it is useless.

The next step in adopting complex numbers has been made by Rafael Bombelli in his Algebra (1572). He was by far more comfortable around negative numbers and annunciated the rules of handling the signed quantities:

Plus times plus makes plus
Minus times minus makes plus
Plus times minus makes minus
Minus times plus makes minus.

In relation to the complex numbers he wrote [Swetz, pp. 264-265]

... This kind of square root has different arithmetical operations from the others and a different denomination, ... But I shall call it 'plus of minus' when it is to be added, and when it is to be subtracted I shall call it 'minus of minus', and this operation is most necessary. ... This will seem to many more artificial than real, and I held the same opinion myself, until I found the geometrical demonstration ...

He then provides the rules of multiplication:

Plus of minus times plus of minus makes minus
Plus of minus times minus of minus makes plus
Minus of minus times plus of minus makes plus
Minus of minus times minus of minus makes minus.

However, when solving cubic equations with three real roots, he would omit negative roots [La Nave, p. 92], still not considering the negative ones as solution.

John Wallis (1616-1703), who gave the very first geometric interpretation of complex numbers, held a strange belief that negative numbers were larger than infinity but not less than 0 [Kline, p. 253]. This belief was shared by L. Euler. Euler, who used complex numbers extensively, who introduced i as the symbol for √-1 and linked the exponential and trigonometric functions in the famous formula

e^it = cos(t) + i·sin(t),

wrote in his Introduction to Algebra [Kline, p. 594]

Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negative numbers cannot be included among the possible numbers [real numbers]. Consequently we must say that these are impossible numbers. And this circumstance leads us to the concept of such number, which by their nature are impossible, and ordinarily are called imaginary or fancied numbers, because they exist only in imagination.

(By the way, the unfortunate term imaginary with exactly such a connotation has been coined by Descartes. He also called the negative roots of an equation false which, fortunately, did not stuck.) Jean Le Rond d'Alembert, in his Encyclopédie (1751 - 1772), passed entirely over complex and wrote ambiguously about negative numbers [Kline, p. 597]

... the algebraic rules of operation with negative numbers are generally admitted by everyone and acknowledged as exact, whatever idea we may have about this quantities.

The modern geometric interpretation of complex numbers was given by Caspar Wessel (1745-1818), a Norwegian surveyor, in 1797. His work remained virtually unknown until the French translation appeared in 1897. He correctly observed that to accommodate complex numbers one has to abandon the two directional line [Smith, pp. 55-66]:

... direction is not a subject for algebra except in so far as it can be changed by algebraic operations. But since these cannot change direction (at least, as commonly explained) except to its opposite, that is, from positive to negative, or vice versa, these are the only directions it should be possible to designate ...

It is not an unreasonable demand that operations used in geometry be taken in a wider meaning than that given to them in arithmetic.

Wessel treats complex numbers as vectors (without using the term) and derives most of their properties, including, say, multiplication in the trigonometric form, without designating the latter as algebraic.

Gauss, who gave a proof of the Fundamental Theorem of Algebra in 1799, thought (1825) that "the true metaphysics of √-1 is illusive." He overcame his doubts by 1831 with application of complex numbers to Number Theory, which gave a tremendous boost to the acceptance of complex numbers in the mathematical community. Still, the acceptance was not universal. Augustus De Morgan (1806-1871), a famous mathematician and logician wrote in 1831 [Kline, p. 593]:

The imaginary expression √-a and the negative expression -b have this resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absurdity. As far as real meaning is concerned, both are equally imaginary, since 0 - a is as inconceivable as √-a.

Sir William Hamilton, 9th Baronet, (1805-1865) is responsible for the abstract notation (x, y) [Eves, p. 99, Swetz, p. 715], which he introduced in 1833. This of course was not yet the last word in the evolution of our understanding of the complex numbers. Complex numbers can be seen as Hamilton's abstractions, either points or vectors in the plane, vector operators and, hence, matrices of a specific form. They serve as a basis for a powerful and beautiful analytic function theory with applications from hydrodynamics to the number theory. The road to the acceptance might have been bumpy, but, one cannot overestimate the important role played by complex numbers in modern mathematics. As J. Hadamard has put it [Kline, p. 626],

The shortest path between two truths in the real domain passes through the complex domain.

References

H. Eves, Great Moments in Mathematics After 1650, MAA, 1983
M. Kline, Mathematical Thought From Ancient to Modern Times, v. 1, Oxford University Press, 1972
M. Kline, Mathematical Thought From Ancient to Modern Times, v. 2, Oxford University Press, 1972
F. La Nave, Deductive Narrative and Epistemological Function of Belief in Mathematics: On Bombelli and Imaginary Numbers, in Circles Disturbed, A. Doxiadis, B. Mazur (eds), Princeton University Press, 2012
D. E. Smith, History of Mathematics, Dover, 1968
D. E. Smith, A Source Book in Mathematics, Dover, 1959
F. J. Swetz, From Five Fingers To Infinity, Open Court, 1996 (3^rd printing)

Complex Numbers

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