## Compound InterestNovember 9, 2015 My introduction to mathematical exponentiation was in Irving Adler's, "The Giant Golden Book of Mathematics," an excellent mathematics book for children.[1] I still have this over-sized (10-1/2 inches wide by 12 inches tall) book on my bookshelf, and the state of its binding is an indication of how much it was used. One chapter of that book, "The Puzzle of the Reward," (page 21) poses the following puzzle. A king was saved from drowning by a poor farm boy, and he offers the boy a choice of two rewards, each paid over the course of thirty days. The first reward had a payout of $1 on the first day, $2 on the second day, $3 on the third day, etc.$1 + $2 + $3 + $4 + $5 + ... $30
The second reward had a payout of 1¢ on the first day, 2¢ on the second day, 4¢ on the third day, 8¢ on the fourth day, etc.
1¢ + 2¢ + 4¢ + 8¢ + 16¢ + ...
The first series, an arithmetic series, starts strong, at $1, but has a payout of only $465. If the poor farm boy studied his math, he would realize that the second series, a geometric series, while starting small at just one cent, will have the larger payout. In this case it's $10,737,418.23, or 2^{30} -1 cents.
where r is the annual (or monthly) interest rate (as a decimal value), and n is the number of years (or months). So, if you get a two year loan of $1,000 at 4% annual interest, after those two years you will need to repay ($1000)(1.04)^{2} = $1081.60. If you were unlucky enough to get a credit card interest rate of 16% per annum, your debt at the end of two years would be $1,345.60.
per annum over a period of seven years. Not surprisingly, Umma could not pay this debt, and war ensued.[2]
The principal of the loan, 5,20,0,0, sìla is somewhat abstruse, since the Sumerians used the sexagesimal number system, but we can calculate the percentage of the principal needed to be repaid after seven years; viz.
## References:- Irving Adler, "The Giant Golden Book of Mathematics," Illustrated by Lowell Hess, Golden Press (New York, 1960), 92 pages, via Amazon.
- Kazuo Muroi, "The oldest example of compound interest in Sumer: Seventh power of four-thirds," arXiv, September 17, 2015.
- Popeye: Spree Lunch (1957, Seymour Kneitel, Director).
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