Tuesday, March 03, 2015

Math is a way of thinking

kw: book reviews, nonfiction, mathematics, mathematical thinking, mathematical games

In the realm of the English and Americans being "divided by a common language" (widely attributed to Shaw, but author not known), the abbreviation for "mathematics" is "maths" in England and "math" in the U.S. The term itself can be colloquially translated "learnèd techniques". Note the accent; thus, mathematics are techniques of those who are learnèd.

Matt Parker wants to make math—he writes "maths", being British—enjoyable. For most people, "Math is hard," to quote the talking Barbie doll. The funny thing is, we use math all the time. To make us more aware of our penchant for mathematical thinking, and to show us some ways to play in a mathematical way, he has written Things to Make and Do in the Fourth Dimension. He bills himself as a stand-up comic and mathematician. The book is subtitled "A Mathematical Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More".

Well, how can mathematics, which encompasses much more than mere number-work, be made enjoyable? Can it be FUN? In my case, Parker is preaching to the choir. I was the kind of geeky kid who did enormous long division problems for fun. The kind who angered a series of calculus instructors by correcting them during class (It took me decades to learn sufficient tact to brace a fellow with his errors in the privacy of his office).

To anyone who has survived the standard American curriculum and graduated from High School, we started with "four banger" arithmetic (add, subtract, multiply, divide), went on to just a bit of exponents and roots (in my day we learned to extract a square root with pencil and paper), then geometry and algebra (in either order), trigonometry, and, if you were a High School senior after about 1966, introductory calculus.

Once you'd been schooled in algebra and plane geometry, did anyone bother to tell you they are equivalent? that one can solve with straight edge and compass the same problems that are presented with X's and Y's and such? I thought not. Probably because they were taught by different teachers; the algebra teacher probably didn't know geometry all that well, and vice versa: nobody told them either!

OK, what's fun about math anyway? Do you remember π? That odd number a bit larger than 3 that has something to do with a circle? For everyday purposes we can use 3.14 or 3 1/7 or 22/7. If you get familiar with it, you can win bar bets and get the occasional free drink. Here's how. You make a bet with someone that the glass he or she is drinking from is bigger around than it is tall. Make sure to use the word "around" not "across". Most people will say, "No way!" If they take the bet, hand them a piece of string. Have the person wrap it around the glass, and mark the length, then hold it next to the glass. The mark will nearly always be above the rim. Why do I say, "nearly always"? Some drinking glasses are quite tall and thin, but not the kind you'll find beer in. So do this for preparation. Get some string and do the comparison using all the different kinds of drinking glasses you find around the house. It is likely that only a really skinny iced-tea glass will be taller than it is around. In a bar, just eyeball that the height is less than three times the width, and you'll be OK.

But fun with math is more than just bar bets. Parker's stand-up routine is based on math, and he writes of a number of card tricks that use mathematical methods. One well-used card trick bases its "clairvoyant" result on the fact that 27 is 3x3x3…and here you thought the deck a stage magician was using had all 52 cards in it! And there are the numbers for lovers (Parker calls them "amicable numbers"). The smallest "loving" pair is 220 and 284. All the factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Add those 11 numbers: the sum is 284. All the factors of 284 are 1, 2, 4, 71, and 142. Add those 4, and the sum is 220. You can sometimes buy a little "puzzle heart", in two pieces with 220 on one half and 284 on the other. There are other (mostly much larger) pairs of amicable numbers, if you want a more geeky puzzle heart made to order.

And on the subject of amicability, or better, there is that "optimal dating algorithm" of the subtitle. An algorithm is a recipe, for cooking up the solution to a problem. In this case, the problem is finding a compatible spouse. In an early chapter, Parker refers us to a few gents (very few early mathematicians were female) who showed that the "optimal testing proportion" of a string of dates is the square root of the total number of dates (with different people) you are prepared to embark upon. Thus, if you plan to allow up to two years for the search, and have time for one date weekly (Friday or Saturday, your choice), that is about 100 maximum dates. The square root of 100 is 10, so you use the first 10 dates to gather information, make your lists, compile the strong and weak points, and determine which person you dated is the most compatible potential spouse. Then, you continue dating new people until someone comes along who is better than the best of the first 10. Stop your search and propose marriage. Suppose you get to the end of the two years, and nobody beat "good old #7"? You can't go back, #7 probably already married someone else. And the chances are about 5% that the 2-year search will fail, statistically. Now what? You can shrink the chance of such failure this way. After the next group of 10, you drop your standard a little, say to better than the second-best of the first 10. There's more statistics one can do, but you'll probably get swept off your feet by someone unexpected long before you reach the 100th date anyway!

Number tomfoolery and some mapping stuff (like the 4-color problem) take up 9 chapters, and then we get into higher dimensions. The 4th dimension is just the beginning. Though it takes a while, we eventually read of a conjecture that requires the use of a space with nearly 200,000 dimensions! The fact that we are alive is sufficient proof to me that no 4D space exists, at least not one that can contact our 3D space. An entity who lives in a 4D space could reach inside us and stop our heart, or remove it for our inspection, as Regina and Rumpel do in episodes of "Once Upon a Time". Doing so would be as easy for them as it is for us to touch the middle of a circle drawn on paper. There is a bigger reason, though, that he mentions as an aside. Orbital mechanics won't work in 4D, not even a little bit. You can't get a planet to orbit a star in any dimension higher than 3. And this is why I deny "string theory", which requires either 10 or 11 dimensions for the math to work.

A lot of the ground the book covers is in the field of topology. Mathematical I may be, but topology is an area I have shunned. The author did more to give me at least a glimmer of topological understanding, than shelves of math books by others. But not more than a glimmer. It really depends how your mind works.

Clearly, in dimensional and topological math, Parker is a genius compared to me. I do find that he comes up short in other areas, however. For one, he mentions at one point his computer idling along at 2.7 GHz, and follows with a parenthesis and a footnote:

The parenthesis: "(2.7 gigahertz is a measurement of how many times its logic gates can be run every second).*"

The footnote: * Actually, this is how many times the processor performs commands in a second, each of which could involve more than one calculation. So this is a low estimate for comparison. A more dedicated me would research how many actual calculations it does per second, aka FLOPS.

The italics in the footnote are mine, and point out an error. The original parenthesis is correct. 2.7 GHz is the rate at which the processor's clock runs, and the clock controls the logic gates. Some hardware operations (what he loosely calls "commands") take one clock tick to run, others take more than one, usually two to four, but perhaps even more. So the basic hardware instruction rate is slower than 2.7 GHz, and 2.7 GHz (for the CPU in his computer) is the highest rate, and thus is a high estimate, not a low one as he states. Furthermore, FLOPS refers to FLoating-point Operations Per Second, where floating-point refers to the calculation of numerical quantities. A 2.7 GHz processor includes a special floating-point processor, these days called a math unit, and it tops out at several hundred MFLOPS (millions of FLOPS).

Back to areas in which our author shines. He presents a geometric proof that an infinite series can have a finite sum, using one based on Zeno's Paradox (though he doesn't say so). Zeno asked, if a runner (he called him Achilles) has two miles to run, first he runs a mile, then a half mile, then a quarter mile, and so forth: does he ever arrive? Of course we know that the second mile is run in about the same time as the first. But it is stated as 1 + 1/2 + 1/4 + 1/8 and so forth, a series that goes on forever. We know in our gut that the sum is 2. Here is the geometry:

It is easy to see that you can continue dividing by 2 as long as your patience holds out. The little blue square holds all the pieces I didn't have patience to draw.

This is the most ancient (known) example of a converging series. Most series diverge, and the one that is right on the edge is the sum of all reciprocals: 1 + 1/2 + 1/3 + 1/4 and so forth. The book has a very clear proof on page 289 that this sum grows without bound (I was careful not to use the word "infinity". That is for later).

For those who aren't afraid of exponents, the sum of reciprocal numbers to a power, where the exponent is close to one, has a finite sum as long as the exponent is greater than one, but grows without bound if it is one or less. Thus, 1/1ⁿ + 1/2ⁿ + 1/3ⁿ + 1/4ⁿ is finite even if n is 1.00000000001 (or add as many zeroes as you like, but keep that last 1 ).

OK, let's talk about infinity. A late chapter is called "To Infinity and Beyond" (nods to Buzz Lightyear). Do you recall the different kinds of numbers? For review:
  • Natural numbers: 1, 2, 3, etc. Also called Counting Numbers.
  • Integers: the Natural numbers plus zero and negatives of the Natural numbers.
  • Rational numbers: Ratios of any two integers such as 1/2, 19/14, 32768/4195.
  • Irrational numbers: All non-Integers that have unending, nonrepeating decimal parts. The most familiar examples are √2 and π, and most people remember at least 1.414 for the one and 3.1416 for the other.
As it happens, there are two kinds of Irrational numbers, but not everyone hears of them even in high school math classes. Firstly, Algebraic numbers are also called Computable numbers, because they are the solution to certain computations, primarily involving polynomials, such as square roots. Secondly, Transcendental numbers are a great deal trickier. Some of them such as π are found in trigonometric equations, and others such as e (2.71828...) in logarithmic and exponential expressions. But they are not "computable" the way square roots are.

With that under our belt, Algebraic irrational numbers are abundant and comparatively familiar. Transcendental numbers are difficult to deal with, and the ones that are known to be so are rather few. It is very difficult to prove that a certain quantity is a transcendental number. The odd thing is, it is not hard to prove that there are a lot of them lurking in the number line. In fact, the Transcendental numbers infinitely outnumber all the rest! A paradoxical phrase I learned in graduate school states:
Between any two transcendental numbers, there exists at least one algebraic number. Between any two algebraic numbers, there exists an infinite quantity of transcendental numbers.
Parker demonstrates this with an amusing analogy called the Hilbert Hotel, attributed to Georg Cantor (Hilbert and Cantor were math geniuses of roughly 120 years ago). Infinite busloads of several kinds of "guests", meaning several kinds of algebraic numbers, are accommodated in the hotel and can always be fit in. Then a bus with just the transcendental numbers between 0 and 1 shows up, and the hotel cannot hold them all. The proof is on page 413, and makes sense while I am reading it, but escapes me immediately thereafter!

This shows that there are at least two kinds of infinity, now called Aleph-0 (or -null) and Aleph-1. But it is not known if there is a different Aleph that is "larger" than Aleph-0 but "smaller" than Aleph-1.

I think Matt Parker genuinely believes that anyone could love and enjoy math, given the right approach. I'd agree only if we recognize that mathematical thinking of certain kinds may be universal among us humans, but that a great many branches of the math tree are forever beyond the reach of many people, no matter what kind of schooling or inducement is offered. Certain kinds of minds are required to do certain kinds of thinking. As I get older, I realize more and more the immense diversity of humankind. A political scientist, a journeyman carpenter, and a medical technician, all regularly think thoughts in realms that will forever be beyond my understanding. They can think thoughts I could never learn to think. That's OK. I think I have a few thoughts of my own that many other folks will never comprehend.

I'll go further. Look at your automobile. The days are long gone that a single person can design and build an entire auto, the way Carl Benz did in 1885. It takes about 8 different kinds of engineer to do so now. Even 40-50 years ago I could take out the motor and rebuild it (did so, 3 times). Now I couldn't get it out without a set of tools I can't afford.

But don't let my quibbles and quandaries discourage you from reading the book. Matt Parker writes delightfully, with a clarity that gets around the defenses we might have against allowing any more math to get into our overstuffed head. Reading this book is like looking through a microscope or telescope. It shows a new landscape, and you may not comprehend it all, but the view is worth it anyway.

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