November 9, 2015
My introduction to mathematical exponentiation was in Irving Adler's, "The Giant Golden Book of Mathematics," an excellent mathematics book for children. 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 230 -1 cents.
This puzzle is a retelling of the wheat and chessboard problem in which a single grain of wheat is placed on the first square of the chessboard, two grains on the second, four grains on the third, etc., for a final number, 18,446,744,073,709,551,615. Not surprisingly, since mathematics was well developed in the Arab world, the source of the chess problem is Persian, contained in the Shahnameh (c. 1000 AD).
Exponentiation appears, also, in compound interest. Although it's rightly disputed, Albert Einstein is quoted as saying that "Compound interest is the most powerful force in the universe;" or, "Compound interest is man's greatest invention;" or, "Compound interest is the eighth wonder of the world." I would have attributed the latter to Benjamin Franklin, but there's no evidence for that.
Compound interest, of course, is interest on interest. This makes sense, since the amount of earned interest adds to the original value, so this new total should get interest. The basic compound interest formula is as follows:
Future Value = Present Value x (1+r)n
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.
While compound interest works in your favor in a savings account, a person's usual exposure is when the money goes in the opposite direction, as in paying an automobile or home loan. In those cases, you don't pay everything at the end; rather, you pay off the present value (principal) a little at a time, so the interest doesn't accrue to such large values. Formulas for these calculations can be found on this Wikipedia page, although there are a myriad of online calculators to simplify the process.
Compound interest has been with us for a long time, long enough that it was outlawed as a form of usury in Roman law more than 2,000 years ago. How far back can we trace compound interest? A recent arXiv paper by Kazuo Muroi finds a calculation of compound interest in a 4,400 years old Sumerian inscription.
Muroi writes that the earliest mention of compound interest goes back to the Old Babylonian period (c. 2000-1600 BC), since "interest on interest" is mentioned in Akkadian texts, and there are even mathematical problems about it during that period. Interest was also calculated in the Pre-Sargonic period (c. 2600–2350 BC), and the Sumerian words for "interest" and "interest-bearing loan" were used in that period.
This inscription is likely also the earliest example of the problems of repressive Third World debt. Inscribed on a conical monument called the Enmetena Foundation Cone is the story of a loan of a large quantity of barley from the ruler of the Sumerian city of Lagash, Enmetena, to the neighboring city, Umma. The interesting part of the loan is a compound interest rate of 33% per annum over a period of seven years. Not surprisingly, Umma could not pay this debt, and war ensued.
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.
Future Value/Present Value = (1.3333...)7 ≈ 750%
- 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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