top and bottom are limited to three-digit numbers the best fraction is 878/323.
This second fraction is a sort of palindromic extension of the first one –
mathematics has a habit of offering these little surprises. A well-known series
expansion for e is given by
The factorial notation using an exclamation mark is handy here. In this, for
example, 5! = 5×4×3×2×1. Using this notation, e takes the more familiar form
So the number e certainly seems to have some pattern. In its mathematical
properties, e appears more ‘symmetric’ than π.
If you want a way of remembering the first few places of e, try this: ‘We
attempt a mnemonic to remember a strategy to memorize this count...’, where
the letter count of each word gives the next number of e. If you know your
American history then you might remember that e is ‘2.7 Andrew Jackson
Andrew Jackson’, because Andrew Jackson (‘Old Hickory’), the seventh president
of the United States was elected in 1828. There are many such devices for
remembering e but their interest lies in their quaintness rather than any
mathematical advantage.
That e is irrational (not a fraction) was proved by Leonhard Euler in 1737. In
1840, French mathematician Joseph Liouville showed that e was not the solution
of any quadratic equation and in 1873, in a path-breaking work, his countryman
Charles Hermite, proved that e is transcendental (it cannot be the solution of any
algebraic equation). What was important here was the method Hermite used.
Nine years later, Ferdinand von Lindemann adapted Hermites’s method to prove
that π was transcendental, a problem with a much higher profile.
One question was answered but new ones appeared. Is e raised to the power
of e transcendental? It is such a bizarre expression, how could this be otherwise?
Yet this has not been proved rigorously and, by the strict standards of
mathematics, it must still be classified as a conjecture. Mathematicians have
inched towards a proof, and have proved it is impossible for both it and e raised
to the power of e^2 to be transcendental. Close, but not close enough.
The connections between π and e are fascinating. The values of eπ and πe are