Euler’s number

Why things don’t grow forever

PUT a pound in the bank. If the yearly interest were 100 per cent, then a year later you would have £2. That’s simple enough. But what if instead of calculating the interest at the end of the year, the bank worked it out more regularly? It turns out this question leads us to one of the most subtle numbers in mathematics.

Say your bank paid interest twice a year but halved the rate to 50 per cent. That would take your £1 to £1.50 after 6 months, and at the end of the year you would get another 50 per cent, making £2.25 – a nice gain. If you got interest monthly but scaled down the rate accordingly, you’d end up with £2.61. Do the same thing daily, reducing the interest rate in the same fashion, and you’d get £2.71. The improvements get ever smaller as this process continues, and the most you could have turns out to be about £2.71828.

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This number is actually a special irrational, which, like π, keeps on going forever after the decimal point. It’s called Euler’s number (or simply e), after the Swiss mathematician Leonhard Euler.

Euler’s number doesn’t just appear when computing compound interest. For instance, mix together the imaginary number i (see “The imaginary number”) and e and, with a little mathematical nous, you can derive one of the most famous…