Euler's formula

Euler's formula illustrated in the complex plane. Source: Wikipedia

The equation shown at the top right describes the circle in the complex plane. When φ = 0 the equation evaluates to zero, i.e. it's at 1 on the x-axis. When φ = π, the equation evaluates to -1 on the x-axis. So e^iπ simply refers to the point (-1,0) in the plane. The mystery of the formula goes away entirely if instead of thinking of imaginary numbers we simply use (cos φ, sin φ) as the locus of the circle. When φ = 0 we get the point (1,0), when φ is π/2 we get (0,1), when φ = π we get the point (-1,0) and when φ = 3π/2 we get (0,-1). [Caption credit: Andrew Barker]

For any real number φ, Euler's formula is

     e^iφ = cos φ + i sin φ

where e is a fundamental constant (the base of natural logarithms) and i = √-1. If we now put φ = π, we get

     e^iπ = cos π + i sin π,

and since cos π = -1 and sin π = 0, this reduces to

     e^iπ = - 1

so that

     e^iπ + 1 = 0.

This most extraordinary equation, called Euler's identity, first emerged in Leonhard Euler's Introductio, published in 1748. It is remarkable because it links the most important mathematical constants, e and π, the imaginary unit i, and the basic numbers used in counting, 0 and 1. In describing the equation to students, the Harvard mathematician Benjamin Peirce said: "Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore, we know it must be the truth."

In a letter to Mathematics Today, Dec. 2005 (p. 197), Stephen Castell writes:

This expression underlines that it should never cease to astonish us, mathematically, that raising a transcendental to the power of another multiplied by something totally imaginary has the same value as a perfectly real number – and a negative one to boot. One might have 'expected' the value to be some other – positive – transcendental?

Physically, this beautiful identity links real-world ratios of key circle metrics with natural logarithms, and with the 'intrinsic exponentiality' of the growth of many physical and biological processes; and ... with the entirely 'imaginary' (dark matter?). Yet it is all grounded in an implicit statement about the ternal, recurring sinusoidality of nature itself, deriving from the identity e^ix = cos x + i sin x.

But ... the latest thinking in physics is that 'universal constants' (such as the speed of light), may not be 'constant' after all – they may perhaps only be examples of what I recently called datoids.^* It is therefore interesting for mathematicians to ask: does this beautiful identity still hold true in a part of the universe where (through gravitational contortions in the space-time continuum) the circle metrics give a value for π which is not 3.14159... – but, say, precisely 4.14 [and similarly for the value of e]?

One would suppose that the answer to that question is "No", since this would lead to the conclusion that i and/or 1 could not, in that strange region, be exactly 'integers'. Such a conclusion would present a profound mathematical / logical / philosophical problem, since the 'intrinsic' phenomenon and process of number, and (ontologically integer) counting, would be threatened – it may not even 'work' at all.

But why should gravitationally-twisted space-time result in the breakdown of integer and counting? Could it be, rather, that intrinsically integers themselves are also mere datoids? If so, what is the more fundamental, integral, truth of which whole numbers are only a highly-useful-on-Earth approximation? Is the 'universal' (and not necessarly constant) unit of counting actually '-e^iπ', and not '1'...?

^* 'Data and datoids,' Comment: Letters, Physics World, June 2005, p. 21.

Related category

• COMPLEX NUMBERS

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