### Compound Interest

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.[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.

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.[2]

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.[2] 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.[2]

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.[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.

**Future Value/Present Value = (1.3333...)**^{7} ≈ 750%

### 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).