## Thursday, 25 September 2014

### e

 Leonhard Euler is widely considered one of the greatest mathematicians. (Photo credit: Wikipedia)
In Our Time on Radio 4 introduces us to e, the irrational number that underpins so much of our modern mathematics.

e is an infinite decimal, like pi, and is approximately 2.718. It is irrational, transcendental and a part of what is often described as the most beautiful equation ever written.

Pi has been known about since the times of the ancient Greeks, but e was not discovered until the 17th century, because the mathematics required simply did not exist. The ancient Greeks' fear of the infinite was a part of the reason they did not stumble across it.

Jakob Bernoulli was the first mathematician to discover e cropping up in his studies of compound interest. A little later, John Napier invented logarithms, using a base approximately equal to 1/e. These huge tables of numbers took Napier 20 years of his life to devise and were designed to make multiplication easier and more accurate. Logarithms were later refined by Henry Briggs.

e also appears a lot in modern calculus. The function e to the power x has a gradient which always has the same value as the function itself. When Leibniz and Newton discovered calculus they used infinite sums to form the derivative of the exponential function.

Later the natural logarithm was found to be the area under the curve y=1/x.

e was named by Euler, the great mathematician of the 18th century, possibly the greatest mathematician of all time and probably the most prolific in terms of publications. But he did not, we are told, name e after himself. Instead, he thought e to be the first letter of the alphabet not widely in use in mathematics. Euler showed e was the sum of infinite series and introduced his famous Euler identity, which many mathematicians consider the most beautiful equation ever written.

To give just a few applications, e crops up in radioactive decay, in the normal distribution in statistics, and in the prime number theorem, which tells us roughly how many primes there are below any given integer.