For any real number φ,
Euler's formula is
|Euler's formula illustrated in the complex plane.
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 eiπ 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]
eiφ = 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
eiπ = cos π + i sin π,
and since cos π = -1 and sin π = 0, this reduces to
eiπ = - 1
eiπ + 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
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
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 eternal, recurring sinusoidality of nature
itself, deriving from the identity eix = 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'
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 necessarily constant) unit of
counting actually '-eiπ', and not '1'...?
* 'Data and datoids,' Comment: Letters, Physics World,
June 2005, p. 21.