close but it is easily shown (without actually calculating their values) that eπ > πe.
If you ‘cheat’ and have a look on your calculator, you will see that approximate
values are eπ = 23.14069 and πe = 22.45916.
The number eπ is known as Gelfond’s constant (named after the Russian
mathematician Aleksandr Gelfond) and has been shown to be a transcendental.
Much less is known about πe; it has not yet been proved to be irrational – if
indeed it is.
Is e important?
The chief place where e is found is in growth. Examples are economic growth
and the growth of populations. Connected with this are the curves depending on
e used to model radioactive decay.
The number e also occurs in problems not connected with growth. Pierre
Montmort investigated a probability problem in the 18th century and it has since
been studied extensively. In the simple version a group of people go to lunch
and afterwards pick up their hats at random. What is the probability that no one
gets their own hat?
It can be shown that this probability is 1/e (about 37%) so that the probability
of at least one person getting their own hat is 1 –^1 /e (63%). This application in
probability theory is one of many. The Poisson distribution which deals with rare
events is another. These were early instances but by no means isolated ones:
James Stirling achieved a remarkable approximation to the factorial value n!
involving e (and π); in statistics the familiar ‘bell curve’ of the normal distribution
involves e; and in engineering the curve of a suspension bridge cable depends on
e. The list is endless.