eπi+1 = 0
Euler's equation, above, is genuinely surprising.
- Both the special number e (see "Exponentially..") and π (the ratio of a circle's circumference to its radius) are numbers that cannot be represented as exact fractions (their decimal representation goes on forever without repeating).
- The symbol i stands for the square root of -1, a so-called an imaginary number because there is in reality no real number that when squared gives -1. (Every positive number when squared gives a positive number, and every negative number when squared also gives a positive number.)
- 1 is the first natural number and so rather special as the foundation stone of counting.
- The idea of 0 was one of the greatest of all mathematical discoveries. (Try doing long multiplication in Roman numerals if you do not believe this.)
- The equality symbol, =, denotes a type of assertion that is fundamental to all mathematics.
Hence, some people claim, this equation contains the seven most important symbols in maths and expresses a uniquely surprising, satisfying and in some sense a beautiful idea. How does raising the never ending number e to a number that does not really exist multiplied by another never ending number produce such a simple result?
This actually has a close connection with our investigations of epicyclic motions - but you will have to wait and see why. The next bit is a proper bit of maths (though I have skimped on some of the real technical detail). Skip if you wish.
Maths students meet this equation as a bon-bon somewhere in an early calculus course in which they have learned that the exponential function which is defined so that exp(x) computes the value of ex, can also be represented by adding up infinitely many terms of the form:
exp(x) = 1 + x + x2/2! + x3/3!+ x4/4! + x5/5! + x6/6! ....
where 3! means 3x2x1 and 4!=4x3x2x1 etc. and the "....." means and so on, in this pattern for ever and ever. (A lot of advanced mathematics is effectively about infinity or zero.)
This is already a pleasingly regular pattern. Students also learn that they can represent sine and cosine functions with the infinite series:
sin(x) = x - x3/3!-+ x5/5! ....
cos(x) = 1 - x2/2! + x4/4! - x6/6! ....
These are actually very surprising results. Think about it! A sine wave (or cosine wave - the same shape shifted along a little) varies smoothly between -1 and +1 in a very regular repeating manner and goes on like that forever as x gets larger and larger. However, terms like -x3/3! go screaming off to negative infinity as x gets larger and larger, it is just that the next term, +x5/5! , with the opposite sign starts screaming off in the other direction shortly later and brings things back to Earth before it starts to scream off to positive infinity, until -x7/7! turns it around again... and so on and on and on. I never cease to be surprised that this works at all.
Now you notice that if we put i.x into the exponential function (that is, we use an imaginary number):
exp(x) = 1 + i.x +( i.x)2/2! + (i.x)3/3!+ (i.x)4/4! + (i.x)5/5! + (i.x)6/6! ....
which, because i is the square root of -1 this means i2=-1, and i3=-i and i4=1 and so on, so, if we regroup terms
exp(x) = 1 +x2/2! + x4/4! + x6/6! ... + i(x -x3/3! + x5/5!...).
We can now see that this is just the same as if we had added together the term-by-term series expansion of:
exp(i.x) = cos(x) + i.sin(x)
Now let us put x = π, which means cos(π) = -1 and sin(π) = 0, giving eπi = -1, which is the same as eπi+1 = 0.
(The mathematically adept will know that in this sketchy demonstration I have skipped over some important caveats about rules for re-arranging infinite series - but then, I am not trying to teach you calculus.)
There it is, a surprising connection between elementary trigonometry and numbers that only exist in the mathematicians imagination!