Alex’s Adventures in Numberland by Alex Bellos

This book is as close as it can get to a concise encyclopedia of mathematical concepts. The best way to introduce it would be to show the notes I made for each chapter. These are not necessarily the best concepts that are mentioned in each chapter but are those that I thought were most interesting and worth remembering.

CHAPTER 1

Is math learned or intrinsic to humans? It seems like addition is learned. The book provides support for arguing that humans have innate propensity for working with ratios.

CHAPTER 2

Platonic solids: 3D figures whose sides are all regular polygons. There are only five platonic solids in total. Proof for it was made by Euclid. To have a solid you must have three sides always meet at a point.

CHAPTER 3

Arabic numbers – a misnomer – it really first started in India, where the symbol for zero got first developed due in part to their culture for saying numbers out loud.

These two different ancient ways of multiplying do the same thing:

• Renaissance) finger multiplication: you number both hands 6 to 10 starting with the thumb as 10. Connect the two fingers representing the numbers you want to multiply. Then:
  • the first digit of the multiplication is the number represented by the finger minus the number of fingers above the connected finger on the opposite hand.
  • The second digit of the multiplication is the multiplication of the number of fingers above the linking fingers.
• Vedic math) Using the different representations of a number by using 10.
  • for example: try multiplying 85*77
    • write: 85 -15 (for the first part: cross add 85-23=77-15=62)
    • 77 -23 (for the last part: 15*23=345, add 3 to the first part)
    • =============
    • 65 45 (take the last two digits of the last part)
  • “cross” multiplication: 234*384
    • 2 3 4 (1. multiply the two last digits 16 – leave 6)
    • 3 8 4 (2. cross multiply the last two digits: 3*4+8*4=44 +1)
    • (3. cross multiply first and last two digits and add second columns: 2*4+3*4+3*8=44+4 from before)
    • (4. cross multiply first two digits: 2*8+3*3=25+4 from before)
    • ==============(5. multiply the first digits: 2*3=6+2 from before)
    • 8 9 8 5 6

Zero as a non-existent entity. But could it be something and nothing at the same time? asks the Shankaracharya asks. Bellos interprets this as saying as in the example the zero in 10 surely does exist. But then could just zero exist by itself?

CHAPTER 4

This chapter is about the evolution of pi. Two facts mentioned in the chapter were most memorable to me. With pi, we see nice solutions to the following problems:

1. The probability of getting 50 percent heads and 50 percent tails is 1/sqrt(n*pi) , if you toss a coin 2n times.
2. Buffon’s Needle Problem: On a floor with strips of wood of width l, what’s the probability that, if you drop a needle of length l, it crosses the boundaries of strips?

The answer to the second problem is actually discussed also in the book How Not to Be Wrong by Jordan Ellenberg. He suggests that it can be solved nicely with the method of finding expectations. For the problem, find the expected number of crossings. Since the width of the strips and length of the needle are the same, it crosses at max one time. So, the expectation of X (for X = number of crossings) = p (p=probability of crossing the boundary once). Pretend the needle’s length is extended by 2. Then by the linearity of expectation, E[2X] = 2p. Actually, for any needle length n, E[n*X] = n*p. If n is the circumference of a circle with length l, then it equals l*pi*p. But a circle always crosses the boundaries at two points. So l*pi*p = 2. Solving for p, p = 2/ (l*pi).

CHAPTER 5

Before the invention of calculators, to approximate the multiplication of large numbers, a slide-rule was used. It had measurements for logs and utilized the log property that the log of a product is the sum of the log of its divisors.

CHAPTER 6

A chapter on recreational math.

CHAPTER 7

A chapter on sequences if numbers and infinity. A paradox harder than that of Achilles and tortoise paradox(also by Zeno):

For a distance that a man reaches, he must first have reached half of that distance first. But to have reached that distance he must have reached half of that distance. There are infinitely many of such distances, but he has finite time. So he can never leave where he stands.

The paradox hinges upon the unresolved confusion of infinite intervals summing to a finite interval. A similar paradox arises when we write 0.9… with 9s extending to infinity. Does this eventually become 1? When exactly does it actually become 1?

The concept of convergence is also involved in analyzing the harmonic series, which surprisingly never converges. Three interesting facts about harmonic series: 1) architecture application- you can theoretically build tiles that overhang each other forever (it’s basically a sum of fractions of one over consecutive multiples of two), 2) subtracting all terms including any number from the harmonic series tames divergence. A corollary from this is that an infinite series of all terms including a certain number in the harmonic series diverges since infinity minus a convergent series is again infinity, and 3) prime harmonic series is divergent.

CHAPTER 8

Golden numbers(phi) and Fibonacci numbers(ratio of consecutive Fibonacci numbers converge to phi) are the most irrational of irrational numbers. Phi can be expressed as a continued fraction of one.

CHAPTER 9 and 10

This was my favorite chapter because op probability is my favorite mathematical subject. Most of the content was a reminder of what I already knew, but some were not. Most of the time pattern recognition is usually a smart thing to do in math. But when it comes to randomness, people tend to avoid finding a pattern too much that it becomes not random anymore. A funny quote by Steve Jobs illustrates the idea: he had to make the iPod shuffle less random to make it appear random. This idea is also funny because randomness happens very often and people do not recognize randomness very well. Every sporting event result happens randomly, each event is independent. If random events really do regress towards the mean and result in sports are governed by random events more than most people expect, every miraculous result in sport should tend to be followed by a sequence of ordinary events since people are not so lucky all the time.

CHAPTER 11

I never knew how essential Euclid’s fourth postulate about a line and a point is: given that there is a point and a line, there is at most one point that is parallel to the line and goes through the point. An attempt to overthrow this established and cherished belief has not really succeeded but gave rise to a whole new set of discoveries with hyperbolas without which we could not have the advanced field of mathematics and physics. That geometry as they knew of at the time was true only given that they were working in a Euclidean space and did not work in hyperbolic space. I especially found it very exciting when the text began talking about how hyperbolic space cannot be described by formula and, henceforth, cannot be produced by a computer. I liked how geometry got imaginative. But then it soon moved on to say that although it can not be visualized by a computer, a pair of paper and scissors or just knitting needles and yarns will do just that. As someone who likes mathematics the more it becomes abstract, it was a disappointing discovery but, after a pause, it was so interesting. It’s probably one of the few things that humans can do that computer cannot do without an algorithm since there is no formula to produce it. So then, basically, is what differentiates humans from machines algorithms? Is this a truth that artificial intelligence has understood far in advance in trying to bridge the difference between humans and machines?

Hilbert Hotel is such a fun way of remembering the cardinality of infinity. Let’s pretend the hotel had infinitely many rooms and they are all occupied when a bus with intently many occupants arrive there asking for rooms. The host finds this is no hard problem as he decides to move every person in the hotel to move to a room that is twice the number of their current room number. Now arrives more difficult set of bus occupants. Let’s say infinitely many number of buses with infinitely many occupants arrive and ask for a room for each person. The ingenious hotelier arranges so that the existing customers move to twice the number of their current room number again. Then, the hotelier arranges all bus occupants in a table with once occupant in each row and column so one can count all the occupants by going diagonally up and down the table. This is equivalent to counting all rational numbers with two numbers in the table combining to form a rational number.

The two scenarios illustrate that infinity and infinity times infinity have the same cardinality. But this finally scenario illustrates infinity with cardinality c. Let’s say the bus occupants arrive with a tshirt each of which lists unique decimal expansion of a number in between zero and one. Cantor’s diagonalization proof shows there can’t be enough hotels for each of these customers. Another interesting fact about cardinality c is that it has the cardinality of decimal expansions in between zero and one is equal to that of the entire number line from negative infinity to positive infinity. This can be proved by forming a correspondence between a number on the number line outside the interval and a number inside the interval by drawing a circle, which I thought was one of the coolest proof I saw in the book(409). The book also states that the next cardinality is d which is the cardinality of curved lines. So cool, beyond imaginable.