CTK Insights

Augustus De Morgan's On the Study and Difficulties of Mathematics was first published by the Society for Diffusion of Useful Knowledge in 1831. I am in possession of the 2005 Dover edition, which - in my view - is a delightful reading.

It is well known that the imaginary numbers had a hard time entering the mainstream mathematics. According to Florian Cajori, the earliest mathematician seriously to consider imaginaries and to introduce them in the expression for the roots of equations was. H. Cardan (Ars Magna, 1545). Cajori notes that Cardan doubted the validity of arithmetic operations performed upon imaginary numbers. Euler was the first to use symbol i for √-1. It was adopted by A. L. Cauchy in 1847, but not yet by A. De Morgan in 1842.

Even in 1949 De Morgan spoke [Florian Cajori] of "the introduction of the unexplained symbol √-1". And further on: "the use, which ought to have been called experimental, of the symbol √-1, under the name of an impossible quantity ... the intelligible results (when such things occurred) of the experiment were always true, and otherwise demonstrable."

Almost 2 decades earlier - in the above mentioned book - De Morgan set out to proof the nonexistence of imaginary numbers. I shall reproduce his argument below.

Having established that the square root of positive and negative numbers is always positive, De Morgan takes a to be a real number and continues:

It is therefore absurd to suppose that there is any quantity which x can represent, and which satisfies the equation x² = — a², since that would be supposing that x&sup2:, a positive quantity, is equal to the negative quantity —a². The solution is then said to be impossible, and it will be easy to show an instance in which such a result is obtained, and also to show that it arises from the absurdity of the problem.

Let a number a be divided into any two parts, one of which is greater than the half, and the other less.

Call the first of these x, then the second must be a - x, since the sum of both parts must be a. Multiply these parts together, which gives

(a - x)(a + x) = a² - x².

As x diminishes, this product increases, and is greatest of all when x = 0, that is, when the two parts, into which a is divided, are a/2 and a/2, or when the number a is halved. In this case the product of the parts is a/2 × a/2, or a²/4 and a number a can never be divided into two parts whose product is greater than a²/4.

After an extra effort put into a more general quadratic equation, De Morgan concludes:

We have shown the symbol √-a to be void of meaning, or rather self-contradictory and absurd.

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