Footnotes on Number Theory

I love to watch the odometer. It is not always convenient to stop and take a picture. 

186285 miles were 1 Light Second

Kaprekar’s Constant

There’s a lot of these out there on YouTube. It is very popular in India, of course. 

Numberphile has it. I was disappointed that Mathologer did not. 

Take any four digits, as long as all four are not the same.

Order them highest to lowest and then lowest to highest. Subtract the smaller from the larger. Do it again. Soon, it reduces to 6174. Every time. I wonder how it works in different bases. 

Hidden Circle in Pi?

That is also a challenge that I have never pursued from Carl Sagan’s Contact.  He posits that if you extend pi out far enough-as I recall, in Base 11—you get a string of 1s and 0s that can be arrayed to display a circle. There’s always something else to do… but a computer program seems easy enough and it can run all night and probably not much longer with today’s MacBook, Dell, etc. 

Kaprekar's Constant from Numberphile

10 REM MIKE MAROTTA. FEB 5, 1987. NEWTON'S ALGORITHM FOR SQUARE ROOTS

19 LIMIT= .0001

20 PRINT "ENTER A NUMBER"

21 INPUT X

56 XN = X/2

60 R1 = (XN + X/XN)/2

70 IF ABS(XN-R1) < LIMIT THEN GOTO 80

75 XN = R1

76 GOTO 60

80 PRINT "THE SQUARE ROOT OF ";X;" = ";R1

 

Fast Square Root.
(She only does it for Primes.)

I first wrote this in the winter of 1976-1977, cadging time at the Michigan State University computer center when I was between terms at Lansing Community College. I used it again when I was employed (briefly) at the MSU campus bookstore. They had a Data General Nova for which the vendor delivered a “Business Basic” without algebraic functions. (“You don’t need them for business.”) I wanted to project economic order quantities, which does require finding a square root. I also wrote numerical expansions for natural and common logarithms and made the set into a library of callable functions.

Square Roots

YouTube has a ton of videos on calculating square roots by hand. In this day of cellphones, it is seldom necessary. But I found this video interesting and wrote a sticky note to hang on a monitor for a while. 

 

Another trick I use is based the binary search. Two or three iterations are usually enough. 

Sqrt(97) < 10. Guess 9.3. Square 9.3. Guess again. Richard P. Feynman has a story about beating a Japanese  abacus salesman in a bar in Brazil figuring like that in his head. “More digits! … More digits!”

 

PREVIOUSLY ON NECESSARY FACTS

 

Number Theory as an Adventure 

Nerd Nation: Natalie Portman, Danica McKellar, and Felicia Day 

The Man Who Loved Only Numbers 

Contradictions in the Patentability of Numbers