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.

Limerick

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

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.)

Building Tangents

Here’s a way to find tangents to an ellipse from a point outside it, say E. Use E to draw any two chords CD and FG. Now lines CF and DG will meet at H, and CG and DF will meet at J. The line HJ intersects the ellipse at A and B, and EA and EB are the tangents we sought.

In 2001 David Bloom of Brooklyn College wrote, “I owe the above to a course I took in 1958, taught by O. Zariski. The result seemed so beautiful that I’ve never forgotten it.”

(“Miscellanea,” College Mathematics Journal 32:4 [September 2001], 317-318.)

Sure Enough

Another time-lengthening effect, the ‘watched-pot’ phenomenon, has been studied by Richard A. Block. Actually using the old adage ‘a watched pot never boils’ as the impetus for his experiments, Block tested the subjective time experiences of observers watching a pot of water as it was heated slowly to the boiling point. One group of subjects, told that they would subsequently be asked for a time estimate, attended carefully to the passage of time. They felt that the time interval was long. A second group, instructed that the experiment involved visual perception, attended to time less carefully and therefore estimated the duration to be shorter. One of Block’s conclusions is that attention to time has a strong influence on perceived length.

— Jonathan D. Kramer, The Time of Music, 1988

(Richard A. Block, Edward J. George, and Marjorie A. Reed, “A Watched Pot Sometimes Boils: A Study of Duration Experience,” Acta Psychologica 46:2 [1980], 81-94.)

A Lofty Honor

The names of 72 French scientists, engineers, and mathematicians are engraved on the Eiffel Tower, under the first balcony, in letters about 60 cm high:

Petiet • Daguerre • Wurtz • Le Verrier • Perdonnet • Delambre • Malus • Breguet • Polonceau • Dumas • Clapeyron • Borda • Fourier • Bichat • Sauvage • Pelouze • Carnot • Lamé • Cauchy • Belgrand • Regnault • Fresnel • De Prony • Vicat • Ebelmen • Coulomb • Poinsot • Foucault • Delaunay • Morin • Haüy • Combes • Thénard • Arago • Poisson • Monge • Jamin • Gay-Lussac • Fizeau • Schneider • Le Chatelier • Berthier • Barral • De Dion • Goüin • Jousselin • Broca • Becquerel • Coriolis • Cail • Triger • Giffard • Perrier • Sturm • Seguin • Lalande • Tresca • Poncelet • Bresse • Lagrange • Belanger • Cuvier • Laplace • Dulong • Chasles • Lavoisier • Ampère • Chevreul • Flachat • Navier • Legendre • Chaptal

Gustave Eiffel added the names when artists had protested against the tower on aesthetic grounds. But the choice of the honorees is itself open to criticism: None of the 72 are women, and none has a name longer than 12 letters.

Etna’s Rings

Periodically Mount Etna emits rings of steam and ash. Not much is known as to how they form — perhaps a vent has assumed a particularly circular shape, so that emitted gas forms vortex rings — but they can be hundreds of feet wide.

Naturalist filmmaker Geoff Mackley captured these in June 2000, but they’ve recurred as recently as 2013.

Ghosts in Color

From Reddit: Stare at the red dot on this woman’s nose for 30 seconds, then look at a white wall and blink.

I was surprised, and agreeably amused, with the following experiment. I covered a paper about four inches square with yellow, and with a pen filled with a blue colour wrote upon the middle of it the word BANKS in capitals, and sitting with my back to the sun, fixed my eyes for a minute exactly on the centre of the letter N in the middle of the word; after closing my eyes, and shading them somewhat with my hand, the word was distinctly seen in the spectrum in yellow letters on a blue field; and then, on opening my eyes on a yellowish wall at twenty feet distance, the magnified name of BANKS appeared written on the wall in golden characters.