Estimating Payments

In 1996 Peyman Milanfar, a reader of Mathematics Magazine, presented a quick way to estimate monthly payments on a loan, passed down from his grandfather, who had been a merchant in 19-century Iran:

\displaystyle \textup{Monthly payment} = \frac{1}{\textup{Number of months}} (\textup{Principal} + \textup{Interest})

The interest is calculated as

\displaystyle \textup{Interest} = \frac{1}{2} \textup{Principal} \times \textup{Number of years} \times \textup{Annual interest rate}.

The exact formula given in finance textbooks is

\displaystyle C = \frac{r(1 + r)^{N}P}{(1 + r)^{N} - 1},

where C is the monthly payment, r is the monthly interest rate (1/12 the annual interest rate), N is the total number of months, and P is the principal. Rendered in that notation, the folk formula becomes

\displaystyle C_{f} = \frac{1}{N} \left ( P + \frac{1}{2}PNr \right ).

“In many cases, Cf is a surprisingly good approximation to C,” particularly when the principal is fixed, the monthly interest rate is sufficiently low, and the total number of months is sufficiently high, Milanfar writes. For example, for a four-year loan of $10,000 at an annual rate of 7% compounded monthly, the precise formula gives a monthly payment of $239.46, while the folk formula gives $237.50.

“While its origins remain a mystery, the method is still in use among merchants all around Iran, and perhaps elsewhere.”

(Peyman Milanfar, “A Persian Folk Method of Figuring Interest,” Mathematics Magazine 69:5 [1996], 376.)

The Tonal System

In 1859, far ahead of its application in computing, engineer John W. Nystrom proposed that we adopt base 16 for arithmetic, timekeeping, weights and measures, coinage, and even music.

“It is evident that 12 is a better number than 10 or 100 as a base, but it admits of only one more binary division than 10, and would, therefore, not come up to the general requirement,” he wrote. “The number 16 admits binary division to an infinite extent, and would, therefore, be the most suitable number as a base for arithmetic, weight, measure, and coins.”

He named the 16 digits an, de, ti, go, su, by, ra, me, ni, ko, hu, vy, la, po, fy, and ton, and invented new numerals for the upper values. Numbers above this range would be named using these roots, so 17 in decimal would be tonan (“16 plus 1”) in Nystrom’s system. And he devised some wonderfully euphonious names for the higher powers:

Base 16 Number Tonal Name Base 10 Equivalent
10 ton 16
100 san 256
1000 mill 4,096
1,0000 bong 65,536
10,0000 tonbong 1,048,576
100,0000 sanbong 16,777,216
1000,0000 millbong 268,435,456
1,0000,0000 tam 4,294,967,296
1,0000,0000,0000 song 1612
1,0000,0000,0000,0000 tran 1616
1,0000,0000,0000,0000,0000 bongtran 1620

So the hexadecimal number 1510,0000 would be mill-susanton-bong.

The system was never widely adopted, but Nystrom was confident in its rationality. “I know I have nature on my side,” he wrote. “If I do not succeed to impress upon you its utility and great importance to mankind, it will reflect that much less credit upon our generation, upon scientific men and philosophers.”

His book is here.

Finger Painting

In the last decade Iris Scott has completed nearly 500 canvases, mostly in oils, using her fingers rather than brushes. “When I see an artwork that makes me gasp — a painting by Artemisia Gentileschi, Klimt, or Picasso, for example — my head exits time, space melts, and the moment stretches into a new dimension of hyper-reality,” she writes. “That is a very important sensation: it is the awe of understanding that a human did this, and it empowers you to believe you can do something profound, too.”

More at her website.


A Maths Master, teaching at Rye,
Bought his pupils a succulent π.
But we’re sorry to state
That 3/8
With 6=7 knows why.

Punch, Sept. 29, 1937, via William R. Ransom, One Hundred Mathematical Curiosities, 1953

(I read this as “three overate, with sick sequels, heaven knows why.”)

Venn Primes
Image: Wikimedia Commons

The classic three-circle Venn diagram on the left has threefold rotational symmetry, and the more complex five-ellipse diagram on the right (discovered by Branko Grünbaum in 1975) has fivefold symmetry. Pleasingly, it turns out that a Venn diagram with n curves having an n-fold rotational symmetry exists if and only if n is prime.

(The diagram below has four curves and fourfold symmetry, but properly speaking it’s not a Venn diagram because it doesn’t represent all possible intersections of the sets.)

(Stan Wagon and Peter Webb, “Venn Symmetry and Prime Numbers: A Seductive Proof Revisited,” American Mathematical Monthly 115:7 [2008], 645-648; Frank Ruskey, Carla D. Savage, and Stan Wagon, “The Search for Simple Symmetric Venn Diagrams,” Notices of the AMS 53:11 [2006], 1304-1311.)

“It Is Not Enough to Mean Well”

Maxims of Theodore Roosevelt:

  • A bad man of ability is worse than a bad man of no ability.
  • It is almost as irritating to be patronized as to be wronged.
  • Timid endurance of wrongdoing may often be to commit one of the greatest evils that one can possibly commit against one’s fellows.
  • The lives of truest heroism are those in which there are no great deeds to look back upon. It is the little things well done that go to make up a successful and truly good life.
  • Our system of government is the best in the world for a people able to carry it on. Only the highest type of people can carry it on.
  • No one ought to submit to being imposed upon, but before you act always stop to consider the rights of others before standing up for your own.
  • The wicked who prosper are never a pleasant sight.
  • It is hard to fail; but it is worse never to have tried to succeed.
  • Don’t let practical politics mean foul politics.
  • For almost every gain there is a penalty.
  • There is grave danger in attempting to establish invariable rules.
  • Woe to all of us if ever as a people we grow to condone evil because it is successful.
  • Remember that the shots that count in war are the ones that hit.
  • What every man needs is robust virtue, that will enable him to go out into the world and remain true to himself.
  • Capacity for work is absolutely necessary, and no man can be said to live in the true sense of the word if he does not work.
  • In doing your work in the great world, it is a safe plan to follow a rule I once heard preached on the football field: Don’t flinch; don’t fall; hit the the line hard.

(More here.)

A Fitting Mascot
Image: Wikimedia Commons

This is almost comically American: Between 1830 and 1836, a bald eagle lived at the Philadelphia Mint. Named Peter, he would roam the city by day and roost in the mint at night. Fatally injured in a coining press, he was stuffed and mounted and is currently on display in the lobby.

He is said (uncertainly) to have been the model for the eagle on U.S. silver dollars issued between 1836 and 1839 and the Flying Eagle cents of 1856-1858.