The number e, growth and decay
See why (1 + 1n)n settles at e, lay the graph of y = a×ex on a given curve, and build growth and decay models.
- UK: Year 12
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- Germany: grade 10
- Iran: grade 11
- Afghanistan: grade 11
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- Russia: grade 11
Guide
Suppose £1 earns 100% interest a year. Paid once, it becomes £2. If the year is split into n parts with interest each time, the end-of-year amount is (1 + 1n)n. As n grows this number rises, but it never goes past about 2.718. We call this number e.
The graph of y = ex is the exponential function with base e. In y = a×ex, a is the height of the curve on the y-axis, because e0 = 1.
In the model P = A×bt, b > 1 means growth and 0 < b < 1 means decay. A fall of 20% a year multiplies by 0.8 each year, not by 0.2.
Common slips: rounding e to 3; thinking the number goes to infinity as n grows; adding instead of multiplying in an exponential model.