A Brief Guide to the Great Equations (14 page)

while cos
x
, whose pieces we’ll put in
this font
, is:

And Euler then used these functions to show how all the other trigonometric functions likewise could be expressed as infinite series.

Euler’s fluency at calculating now enabled him to arrange these trigonometric functions so that they added up to something identical to the exponential function with base
e
. He did so with the aid of the imaginary number
which he would later – years after he wrote the
Introductio
– symbolize as
i
. Although
i
is not a ‘real’ number – a number with a place on a number line – it is used in real mathematical operations and allows mathematicians to solve otherwise insoluble equations. If, for instance, you insert it in the exponent of
e
^x
it shows up in each term of the infinite series associated with it:

But
i
²
is −1, and therefore
i
³
= −
i
,
i
⁴
= 1,
i
⁵
=
i
, etc. So the series becomes:

Euler observed that if you group together the multiples of
i
, you obtain:

Or, as he wrote toward the end of
Chapter VIII
of the
Introductio
(using
i
where he used
as does the English translation),
¹⁰

This equation establishes the deep connection between exponential and trigonometric functions. When the great Indian mathematician Srinivasa Ramanujan (1887–1920) discovered this connection on his own while in high school, he wrote it down excitedly – and was so crestfallen to discover that he was not the first that he hid all
his calculations in the roof of his house.
¹¹

This equation is magic enough, but there’s more. Suppose
x
is π. The sine of π is 0, and the cosine of π is −1. Then
e
^i
π
= −1, or
e
^i
π
+ 1 = 0.

Another way to show the truth of this equation graphically is the following. Suppose we insert π for
x
. Then the formula a few paragraphs above becomes:

Mathematicians can add such a sequence as a series of vectors, with each one beginning at the tail end of the one preceding it, and with the imaginary number
i
rotating a vector 90 degrees counterclockwise.
¹²
If we start at 0, the first term (1) is a vector that takes us 1 unit out on the
x
-axis, to coordinate (1, 0). The second term (
i
π) takes the form of a vector that starts at (1, 0) and, rotated counterclockwise with respect to the first, goes straight up π units, ending at coordinate (1, π). The third term, (
i
π
²
/2!), takes the form of a vector that starts at (1, π) and – rotated another 90 degrees from the previous one – runs in the opposite direction from the first, going across the
y
= 0 line to the point (− (π
²
/ 2 − 1), π). The fourth term is a vector that runs downward, ending up below the
x
-axis, and so forth. Because the vectors keep rotating 90 degrees counterclockwise, and keep getting shorter because the denominator increases much faster than the numerator, the result is a polygonal spiral that converges on the point (−1, 0) (see diagram on next page).

Euler’s simple formula (according to some definitions, it is not an equation in this form, for it contains no variables) contains five of the most fundamental concepts of mathematics – zero, one, the base of the natural logarithms
e
, the imaginary number
i
, and π – as well as four operators – addition, multiplication, exponentiation, and equality – and each exactly once. It states that an irrational number multiplied by itself an imaginary number times an irrational number of times – plus one – equals exactly zero. The numbers π
^e
, 2
^π
, and
e
^π
are all thought to be irrational. But
e
^i
π
picks out that
special place in the architecture of numbers where rational, irrational, and imaginary numbers mix in a way that spookily ‘balances out’ to exactly zero. It has been said that all analysis is centreed here in this equation.
¹³
Among other things, Euler’s result here demonstrated that imaginary numbers, despite Descartes’ scorn, were not on the margins, but at the very centre of mathematics. They would play a greater and greater role in mathematics – and then, with the advent of quantum mechanics in the twentieth century, in physics and engineering and any field that deals with cyclical phenomena such as waves that can be represented by complex numbers. For a complex number allows you to represent two processes such as phase and wavelength simultaneously – and a complex exponential allows you to map a straight line onto a circle in a complex plane.

Polygonal spiral, showing how the infinite series converges to −1.

It may be true, then, that Euler’s formula
e
^i
π
+ 1 = 0 is but one implication, one step, in his exploration of functions. It is ‘only’ an equation, a single one of the thousands of steps in the ongoing process of scientific inquiry, a mere implication in Euler’s extended
exploration of functions. And yet, some steps in an inquiry acquire, and deserve, special status. Certain expressions serve as landmarks in the vital and bustling metropolis of science, a city that is continually undergoing construction and renovation. They preserve the work of the past, orient the present, and point to the future. Theories, equipment, and people may change, but formulas and equations remain pretty much the same. They are guides for getting things done, tools for letting us design new instruments, and repositories for specialists to report and describe new discoveries. They summarize and store, anticipate and open up.