maanantai 13. maaliskuuta 2017

Jean-Robert Argand: Essay on a manner of representing imaginary quantities through geometric constructions (1806)


Originally, one meant by a number something made out of several units - thus, unit itself was not yet a number, but the first number was 2. It goes without saying that all these numbers were what nowadays are called natural numbers, and in addition to 1, the important 0 was also missing from the list.

In addition to numbers, one also spoke of magnitudes or quantities. While numbers were sets of discrete units, quantities were more like continuous wholes, which divided always into further quantities. Still, there appeared to be a clear relation between numbers and quantities. One could express the relations between quantities through relations between number - or so it was thought at first. It became soon apparent that some quantities - such as a side and diagonal of a square - had a relation that could not be expressed through any numbers.

This development took European mathematics from natural numbers to fractions and then to certain irrationals. The set of irrationals known increased when mathematicians started to wonder about the relations between the circumference and diameter of a circle, but the inclusion of zero to the list of numbers or quantities had to wait until Europeans came in contact with this Indian invention through Arabs.

Although negative numbers are usually presented nowadays as the first increment to natural numbers, their introduction to mathematics in Europe happened rather late, during 15th century. Even then, it took centuries before they were fully accepted by mathematicians. Negative numbers did have obvious uses, for instance, when speaking of debts, but they seemed still problematic, because one apparently could not apply same operations to them as with positive numbers.

Particularly, taking roots of negative numbers appeared an absurdity. Yet, the calculations appeared to work in the sense that one could do calculations with these roots or imaginary numbers and results were still consistent. The only question was what meaning to give to these quantities.

An answer to this problem was given by a dilettante mathematician Argand in his work Essai sur une manière de représenter les quantités imaginaires par des constructions géométriques. Argands’s idea was to return the question back to geometry, but to add a new element to the equation, that is, direction.

Picture, Argand says, a positive quantity - say, 1 - as a line heading to a certain direction from some point of origin. Then, we could represent the corresponding negative number (-1) as a line of same size, but moving from the same point of origin to opposite direction. How to account then for the roots of -1?

Argand’s idea is simply to note that number one has the same relation to the square root of -1 as this root has to to -1, that is, if we use the i to represent the square root of -1, then i/1 = -1/i. In layman’s terms, i is is an sense just in the middle of -1 and 1. In terms of size, i is then represented by a line of same size as 1 and -1, but its direction, beginning from the same starting point, goes perpendicular to -1 and 1.

This simple move to two-dimensions and the inclusion of direction to the notion of quantity allows Argand then to represent imaginary numbers and their combinations with ordinary numbers of quantities - in effect, these are then lines moving to various diagonal directions.

The majority of Argand’s essay is spent on justification of this innovation - he attempts to show that e.g. some interesting trigonometric truths can be derived from this assumption. Yet, the major result of the essay is not any of these rather bland proofs, but the innovation itself. This is the first time, when an actual meaning is given to such a seemingly fictional kind of mathematics. We might take this as a beginning of period, when seemingly more and more useless and abstruse mathematical notions find their applications. It is also the first time when a mathematician has in sense proven that a consistent set of truths actually has a point of reference - a semantics is combined to mathematical syntax.

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