## The Mathematical Experience |

by Philip J Davis & Reuben Hersh |

4.Unity within Diversity (p198)

UNIFICATION, the establishment of a relationship between seemingly diverse
objects, is at once one of the great motivating forces and one of the great
sources of aesthetic satisfaction in mathematics. It is beautifully illustrated
by the formula of Euler which unifies the trigonometric functions with the
"power" or ''exponential" functions. The trigonometric ratios and the sequences
of exponential growth are both of ancient origin. Recall
the legend of the magician who was to
be paid in grains of wheat-one on the first square of the chessboard, with
the number of grains doubling with each square. No doubt the sequence 1,
2, 4, 8, 16,. . . is the oldest exponential sequence. Now what on earth do
these ideas have to do with one another?

It would be a nice piece of mathematical history to trace the growth of these
notions until they fuse. We would see the extension of the sine and cosine
functions to periodic functions, the switch-over to* e ^{x}*
as the basic exponential, where

**sin x = x - x^{3}/3! + x^{5}/5! + .
. .,**

**cos x = 1- x^{2}/2! + x^{4}/4! - . . .,**

*e ^{x}*

leading to the final unification, Euler's formula,

** e^{ix }= cos x + i sin x**, where

Thus, the exponential emerges as trigonometry in disguise. Reciprocally,
by solving backwards one has

**cos x = ^{1}/_{2}(e^{ix }+
e^{-ix}),**

**sin x = ^{1}/_{2i} (e^{ix }-
e^{-ix}), **

so that trigonometry is equally exponential algebra in disguise. The special
case where *x *= p = 3.14159. . . leads
to

** e^{pi }=
cosp + i sinp = -
l, **or

There is an aura of mystery in this last equation, which links the five most
important constants in the whole of analysis: 0, 1, *e*,
p = , and *i*.

Within the same story one moves forward from this mystic landing to Fourier
analysis, periodogram analysis, Fourier analysis over groups, differential
equations, and one comes rapidly to great theories, great technological
applications, and always a sense of the actual and potential unities that
lurk in the corners of the universe. (See Chapter 5, Fourier Analysis.)

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The Mathematical Experience