Without further ado, here it is:
\[e^{\pi i} +1 = 0\]

It’s usually called Euler’s Identity, after the great Swiss mathematician Leonard Euler (1707 – 1783). It can be seen on tee shirts and coffee mugs, and several polls of mathematicians and physicists have bestowed on it titles such as “the greatest equation ever” (Crease, 2004).

This feeling that that the identity is beautiful and elegant comes from the fact that it combines in a simple form the five most important numbers in mathematics: e, the base of natural logarithms, i, the square root of -1, \(\pi\), 1, and 0. Looking at it closer, most people would wonder about the exponent \(\pi i\) : what does it mean to raise a number to an imaginary power? Patience, patience; we will get to that.

To explain where the formula comes, we must first derive a more general formula that Euler discovered, then show that our identity is just a special case of that formula. The general formula is quite marvelous in itself, and it has far-reaching applications in mathematics, physics, and engineering.

The first step in our journey is to understand that most functions in mathematics can be defined in the form of an infinite sum of powers of the argument. This is an example:

\[\sin(x)\: =\: x\: -\: \frac{x^{3}}{3!}\: \: +\: \frac{x^{5}}{5!}\: -\: \frac{x^{7}}{7!}\: +\: \cdot \cdot \cdot \ \tag{1}\]

Here x is in radians, not degrees. We can get a good approximation of sin(x) for a particular x value by just using the first few terms of the series. This is an example of a Taylor Series, and it is fairly easy to derive this formula using calculus. In this post, I am not presuming a knowledge of calculus, so the reader is asked to take it on faith.

The corresponding formula for the cosine function is:

\[\cos(x)\:=\: 1\: -\: \frac{x^{2}}{2!}\: +\: \frac{x^{4}}{4!}\: -\: \frac{x^{6}}{6!}\: +\: \cdot \cdot \cdot\ \tag{2}\]

Finally, there’s:

\[ e^{x}=\: 1\: +\: x\: +\: \frac{x^{2}}{2!}\: \: +\: \frac{x^{3}}{3!}\: +\: \frac{x^{4}}{4!}\: +\: \cdot \cdot \cdot \tag{3}\]

e is the constant 2.71828…, and Euler had been the first to recognize its fundamental importance in mathematics, and to derive (3) ((1) and (2) were discovered by Isaac Newton). Books have been written about the number e (Maor, 1994), and I may do a future post about it.

Sometime in the year 1740, Euler looked at these three formulas, laid out as we have them here. It’s immediately obvious that every term in (3) also appears in either (1) or (2). However, half the terms in (1) and (2) are negative, whereas every term in (3) is positive. Most people would leave it there, but Euler saw a pattern in all this. He first added (1) and (2):

\[ \sin(x) + \cos(x)\, =1 + x - \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \frac{x^{5}}{5!}- \frac{x^{6}}{6!} - \frac{x^{7}}{7!} + \cdot \cdot \cdot \tag{4}\]

Notice the sequence of signs in this series: ++  - –  ++  - –  ++ … , repeating in groups of 4. Euler recognized that this is the same sequence of signs obtained when we raise the imaginary number i to integer powers:

i⁰ = +1,    i¹ = +i    i² = -1    i³ = -i    i⁴ = +1    i⁵ = +i …

This meant he could replace x in (3) with xi to obtain:
Revised 9/12/2010:

\[ e^{xi}\, = 1 + xi - \frac{x^{2}}{2!} - \frac{ix^{3}}{3!} + \frac{x^{4}}{4!} + \frac{ix^{5}}{5!}- \frac{x^{6}}{6!} - \frac{ix^{7}}{7!} + \cdot \cdot \cdot \tag{5}\]

Now the signs match those in (4), and the new series equals (4), except that the terms for sin(x) are multiplied by i. That is, (5) is exactly the same as:

\[ e^{xi}\, = \cos(x) + i\sin(x) \tag{6}\]

This is a surprising and mysterious result; it says that there is an intimate connection between the number e and the sines and cosines of trigonometry, even though e was know only from problems that did not involve geometry or triangles. Besides its elegance and strangeness however, it would be hard to overstate the importance of this formula in the mathematics developed since its discovery. It appears everywhere, and a recent book of some 400 pages was published (Nahin P. Dr. Euler’s Fabulous Formula, 2006) which is devoted to describing some of the formula’s applications.

Notice that the earlier question about imaginary exponents is now resolved: to raise e to an imaginary power, just insert the imaginary number into (6) . If the base is a number other than e, only a slight modification is required.

Now, back to the magic identity. We can substitute any real number x into (6), and the result will be some complex number. One propitious choice for x is x = \(\pi\) . Recall from trigonometry that \(\pi\) radians is 180 degrees. The cosine of 180 degrees is -1, and the sine is 0.

Therefore:

\[ e^{\pi i}\, = -1 + i0, \: \mathrm{or }\; {e^{\pi i}\, +\, 1\, =\, 0} \]

All this provides a hint of the power and creativity of Leonard Euler, and why he is sometimes called the foremost intellect of the eighteenth century. I’ll be doing further posts about him and some of his results, in a series I’ll call Euler’s Greatest Hits.

Bibliography

Crease, R. (2004). The Greatest Equations Ever. Retrieved January 31, 2010, from http://physicsworld.com/: http://physicsworld.com/cws/article/print/20407

Maor, E. (1994). e, the story of a number. Princeton University Press .

Nahin. (2007). An Imaginary Tale: The Story of i. Princeton University Press.

Nahin, P. (2006). Dr. Euler’s Fabulous Formula. Princeton University Press.

Posamentie, A. S. (2004). Pi: A Biography of the World’s Most Mysterious Number. Prometheus Books.

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