172 // What is e, and Why?

What is e, and Why? The number e, which is approximately 2.7182, is the `base of natural logarithms', a term that refers to its historical origins. One way to see how it arises is to see how a sum of money grows when compound interest is applied at increasingly fine intervals. Suppose that you deposit £1 in the Bank of Logarithmania--

No, no, no. This is the twenty-first century. People don't deposit savings in banks, they borrow.

OK, suppose you borrow £1 on your Logarithmania credit card. (More likely it would be £4,675.23, but £1 is easier to think about.) Once the 0% balance transfer deal has lapsed ­ about a week after you sign up for the card ­ the bank applies an interest rate of 100%, paid annually. Then after one year you will owe them

£1:00 borrowed þ £1:00 interest ¼ £2:00 total

If instead you paid 50% interest every six months, compounded (so that interest becomes payable on previous interest) then after one year you would owe

£1:00 invested þ £0:50 interest þ £0:75 interest ¼ £2:25 total

This is ð1 þ 1Þ2 , and the pattern continues like that. So, for

2 example, if you paid interest of 10% at intervals of one-tenth of a year, you would end up owing

1 10

1þ ¼ 2:5937

10

pounds. The Bank likes the way these sums are going, so it decides to apply the interest rate ever more frequently. If you paid interest of 1% at intervals of one-hundredth of a year, you would end up owing

1 100

1þ ¼ 2:7048

100

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