The queen of mathematics Book

Mathematicians do not always state the obvious.


I hadn’t planned on reading this book all the way through. I brought it downstairs thinking it might be a useful book for browsing randomly next to the armchair. But it’s so interesting I’m already fifty pages in. This is a bathroom reader for fans of the movie π. Each entry is a short description of what makes that number interesting.

Pi itself comes in on page 48, “the most famous and most remarkable of all numbers”. In seven pages—one of the longest entries in the book—he describes how the Greeks attempted to determine more and more accurate representations of π, as well as the Chinese and then Ludolph van Cuelen.

Many numbers feature something to do with the Fibonacci sequence. In the number five, with ten pages it is probably by far the longest entry, six of the pages are taken up by the Fibonacci sequence.


It is one of the curious coincidences that occur in the history of mathematics that a problem about rabbits should generate a sequence of numbers of such interest and fascination. Rabbits, needless to say, do not feature again in its history.


The book is filled with tricks that will be difficult to remember unless you remember what number they’re associated with. The casting out of nines is probably an easier one, since it appears under the number 9.


Arithmetical sums may be checked by the process called ‘casting out nines’. This came to Europe from the Arabs, but was probably an Indian invention. Leonardo of Pisa described it in his Liber Abaci. Each number in a sum is replaced by the sum of its digits. If the original sum is correct, so will the same sum be when performed with the sums-of-digits only.


Some of the tricks make Darren Aranofsky’s movie look tame. To test if a number is divisible by 11, start from either end, and add and subtract the numbers alternately. That is, 121, 1-2+1 = 0, or 2781, 2-7+8-1 = -4. If the result is divisible by 11 (which includes zero, as in 121), the original number is also divisible by 11. So, 121 is divisible by 11 and the somewhat randomly chosen 2781 is not. The reason this works is “because 11 = 10 + 1”.

Among the reasons that the numbers 12 and 13 are interesting is that they can be reversed to get the reverse of their squares. That is, 12 squared is 144; 21 squared is 441.

Part of the reason there are so many interesting numbers is that mathematicians have come up with so many reasons for them to be interesting. “Abundant” numbers also come in under the number 12; it is the first abundant number, which is to say, the first number that “is less than the sum of its factors excluding itself.”

All numbers, in fact, are interesting up to 39:


39
This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.
It is therefore also the first number to be simultaneously interesting and uninteresting.


It is the third of I suspect nine integers that are equal to the sum of their digits added to the product of their digits. Which is, admittedly, uninteresting.

While 42 has an entry, he does not mention that 42 is the meaning of life, though the calculation probably was spurious. He does include many predictions that a number is prime and explains when those predictions were falsified. The entry for 42 entry also describes Catalan numbers, because 42 is the fifth Catalan number.

There are a couple of numbers that repeat powers when divided into 1, a feature that would have had Aronofsky’s protagonist drilling into his head in the first ten minutes of the movie. For example, 1/97 repeats the powers of three, and appears to do so for a long time, in a very strange way.


0.010309 278350 515463 917525 773195…


If you look closely, 1, 3, 9, 27, are obvious. But 83 is in fact 81 (3^4) overlapped with 243 (3^5). The 2 of 243 adds to the 1 of 81, to give 83. Likewise, the 7 of 729 (3^6) adds to the 43 of 243 to give 50.

1/98 does the same with powers of two, 1/96 the powers of four, and so on.

I have a lot more sympathy with people who believe God has left secret messages in strange numbers after reading this book.

Some of the entries include interesting history that you’d never be able to find without already knowing it. For example:


641
Euler found the first counter-example to Fermat’s conjecture that 2^2^n + 1 is always prime, when he discovered in 1742 that 2^2^5 +1 is divisible by 641.


The number 1,089 includes a potential magic trick. “If a 3-digit number is reversed and the result subtracted, and that answer added to its reversal, the answer is always 1089: 623-326=297 and 297+792 = 1089.

This book came out in 1986. On page 204, very near the end, he talks about factorizing large numbers; at this point, the numbers are so large that he doesn’t reproduce them, and uses, in brackets, the number of digits. The CRAY at this time could factor 100-digit numbers “in about 40 seconds”. This is important because “in 1975, Whitfield Diffie and Martin Hellman invented the trapdoor function, and shortly afterwards, Rivest, Shamir and Adleman showed how to make it a practical proposition.”

He’s almost certainly talking about the RSA algorithm used in public key cryptography.

From that point on, modern computers get a lot more mention, though always big ones. Three pages later, on page 207, he mentions a personal computer for the first (and only) time, the Apple II.

This is an amazing book for browsing through. There is a lot of mathematical history literally hidden among its numeric entries, and a lot of the superficially, at least, strange behavior of numbers.

A Python script for calculating Pi, based on the method on page 55:

#!/usr/local/bin/python3
# calculate pi using David Wells, p. 55

from sys import argv, exit
from decimal import Decimal, getcontext
from argparse import ArgumentParser

parser = ArgumentParser(description="calculate pi")
parser.add_argument('precision', type=int, nargs='?', default=5)
parser.add_argument('--verbose', action='store_true')
arguments = parser.parse_args()

getcontext().prec = arguments.precision

d1,d2,d4,d6 = Decimal(1), Decimal(2), Decimal(4), Decimal(6)

a=d1
x=d1
b=d1/d2.sqrt()
c=d1/d4
previousPi = d1

while True:
y=a
a=(a+b)/d2
b=(b*y).sqrt()
c=c-x*(a-y)**d2
x=d2*x
pi = (a+b)**d2/(d4*c)

#leave loop if pi has stopped changing
if pi == previousPi:
break
previousPi = pi

if arguments.verbose:
print(pi)

print(pi)