Numeracy instruction

kw: book reviews, nonfiction, mathematics, mathematical thinking, instruction, learning methods

When I saw A Mind for Numbers: How to Excel at Math and Science (Even if You Flunked Algebra) by Barbara Oakley, PhD, I was intrigued. The main title was in the form of a cutesy equation, which I supposed was the editor's conceit. Having read the book, I am not so sure.

Dr. Oakley knows what she is talking about, because she began as a math-phobic, but learned to love it. Throughout the book she has 1-page items by people with a similar background, who now are comfortable with mathematics, at one level or another.

The book is a breezily-written compendium of learning techniques and tips gathered into 18 chapters. It is not intended to be taken in wholesale. Different people learn in different ways, and in many cases, a single chapter, or even one point in a chapter, can unlock the math potential for someone. But I wonder…

At a certain level, to be human is to be mathematically adept at some level. Very young children, asked to choose one of two piles of coins, will pick the one that is spread out rather than a neat stack of the same number of coins. They equate spatial area with quantity, and don't realize that the two piles are equivalent. But, I suspect they have not yet learned to count, and it takes this further level of sophistication before they have the mental equipment to fairly evaluate the two piles.

I think that is analogous to an experience I had at about age 12. Someone had showed me a few Algebra equations. I saw something like 10x=5 and wondered, "How can that be?" I thought the "x" was supposed to represent a digit, like the stuff on the left would be a number from 100 to 109. So I thought something else had to be going on. This caused quite a delay in my getting the point when I began Algebra class later that year. But I think, a month or so into the school year, when it all began to "click", that my brain had simply grown up enough to have the right tools for doing algebra.

We all do a certain amount of calculation. Most of us can quickly evaluate the change we're given at the store (if we used cash). People who bowl soon learn to keep score without writing down their calculations. When we drive (without a GPS), and we see "Chicago, 95 miles", we check the odometer, note what it shows, and can then glance at it later and know how many miles we still have to go. Many times, we can even be told a "problem" like this:

John and Mary ride bicycles toward each other at 5 mph, from 1 mile apart. Their pet bird, which flies 20 mph, flies back and forth from one to the other until they meet. How far does the bird fly?

Most of us, by age 10 or so, can figure that John and Mary each ride half a mile, because they are going the same speed. The bird flies four times as fast as either of them, so its total flight is two miles. This kind of "figuring" is actually algebra, without the equations. In fact, it would take me longer to write down the equations for solving this using "traditional" algebra, than it did to write the two sentences above. What is funny is when someone tries to tackle the problem as a series of flights of decreasing length by the bird. The equations to get that to work are gnarly!

So every one of us has some amount of math built in. Standard equipment. But that "some amount" varies a great deal from person to person. Not everyone can learn algebra, no matter how it is taught nor how hard they try. But most can, and by "most" I mean "more than half but not a great deal more". Some kids who had no trouble with algebra never, ever get the point of Trigonometry. My senior year of high school, we got done with the ordinary curriculum for the year a few days early, so the teacher did an experiment. He got out a few copies of a basic text in Calculus and taught it to us. In just a couple of weeks, I learned enough Calc so that I pretty much breezed through the Calc 101 course the next year in college, which used that same textbook!

I was a working mathematician, at a certain level, for decades. But there are branches of math that have never made sense to me, and others that I can puzzle out with desperate levels of effort. I had to take Differential Equations three times to pass it. I'm still not comfortable with it, but the explanation of how to use it takes only two pages in my old CRC Handbook of Chemistry & Physics, and actually contains most of what one needs to do nearly any Diff Eq problem!

And so it goes. Each human brain has a certain mathematical limit. With luck, we might grow mathematically to our full potential, but it is time consuming. Most of us never need all that stuff. But we also grow into certain abilities over time. Just as the brain doesn't finish emotional maturation until about age 25, it must be true that certain math circuits only get set up at certain ages. It may be that the ten years between my second and my third try at learning Diff Eq made more difference in my ability, than the exposure I'd had during the first two attempts.

With all that in mind, I find no sense in delving into what Dr. Oakley has to say. The book is a fantastic resource. Someone who needs encouragement and help in how to learn math and science will do well to read the book quickly, then return to read over certain sections with more care: those sections that seemed to make the most sense the first time through. The first two chapters will be helpful to everyone. As for the others, while the author attempts to make them generally applicable, each will actually be best suited to people with a certain kind of mind, one way or another.

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.

The seductive power of mathematics

kw: book reviews, nonfiction, mathematics, mathematical thinking

We are nearly two weeks into the new year, and this is my first post of the year. It is not because the book was extra-hard to read, but that the year itself has begun extra-busy! Actually, though the book was long (437 pp + 15 pp notes), I spent less time reading it than many shorter ones because math is of great interest to me.

More than 2/3 of those who read that first paragraph will respond, "But not to me", and be tempted to stop right there. I hope you will continue anyway, because the author's design is to show how we all use mathematical thinking and can benefit from a better acquaintance with it. Theoretical mathematician Jordan Ellenberg has written How Not to be Wrong: The Power of Mathematical Thinking.

Contrary to popular thought, Mathematics isn't mainly about numbers. If you break the word down it means "The Studies of Learning". Note the "s" on "mathematics" and on "studies". The field has hundreds of branches, thus where an American would, in our streamlined way, speak of "math", the English speak of "maths". Being American, I'll go the American way. Only two of the many disciplines under the "math" umbrella explicitly involve numbers.

For most of us, our introduction to math began with arithmetic and the "plus table" and "times table". Though even grade schoolers are now permitted to use calculators in class, it is useful to know how to do simple sums and multiplications in one's head. At the very least, when you punch in some numbers and get a result, you are more likely to detect a punching-in error if your mind is at least estimating the result in the background.

The second numerical branch of math is Number Theory, which deals with properties of whole numbers. A big sub-field is Prime numbers, which we will return to later on. But most people who might read this have been exposed to additional branches.

At least in Western and Westernized societies, facility in basic arithmetic was needed to advance through Plane Geometry, Algebra, Trigonometry, Analytical Geometry (sometimes just called Charting), and Calculus. Before the early 1960s Calculus was not introduced to high school students, but the teacher of my senior class in Analytical Geometry was one of the first to finish the school year with a few weeks of instruction in basic Calculus. Now at least half a year is taught to most high school seniors.

So if you had all those courses, think back: most of the work was learning to use certain symbols and sets of symbols in a consistent way. Working out problems using numbers was less important than the proper use of those symbols. That's why the teacher kept harping on "Show Your Work!". Also, particularly in Geometry, formal proof methods were introduced, primarily because visual proofs are easier to comprehend than the symbolic proofs that are the stock in trade of "higher math" (that is, stuff for college juniors and beyond, and only in technical disciplines).

Most of us shudder at that word, "proof". Few understand it. It takes a certain kind of mind to construct a useful proof. My brother, a working mathematician for some years, whose name I shall call Rick, had two friends at college; call them Tom and Harry. They all took some rather gnarly "higher math" courses together, and did lots of formal proofs. Another friend described them thus:

"Send Tom into a room with a mysterious machine in it having several large gears, a big flywheel and other bulky items of unknown import. He is requested to make its wheel turn. By putting a shoulder to the largest gear and pushing very hard, he is able to make it turn, slowly. He leaves and Rick enters. He noses around a bit and finds, behind the machine, a crank with a long handle. Fitting the handle into a convenient socket, he is able to turn the wheel more easily. He leaves and Harry enters. He looks around further, sees the crank, but keeps looking until he finds a button. He presses the button and a motor somewhere makes it all run."

"Pushing the right button" represents concocting a useful proof. I like visual proofs, and you can see one that proves the Pythagorean Theorem here. Remember the Pythagorean Theorem? It pertains to a right triangle, one for which one angle measures 90°. If the two sides that meet at that right angle have lengths represented by a and b, their relationship to the third side, of length c is c² = a² + b². In words, we say that the sum of the squares of the lengths of the two legs of a right triangle equal the square of the length of the hypotenuse (the third side). Pythagorean triples are sets of three whole numbers that can be used to produce a right triangle, such as 3, 4, 5 (3²=9, 4²=16, 5²=25, and 25=9+16). Try with 5, 12, 13 and 8, 15, 17.

So if math isn't primarily about numbers, it sure uses them a lot. But the power of most branches of math lies not in the use of numbers, but in the core concept of math: Operators. To illustrate, when we learn the Plus Table, we are actually learning to use an operator, the +, the addition operator. With a little more thought and practice, we also learn the – operator, the subtraction operator. Similarly, the Times Table helps us learn the ×, the multiplication operator, and later the ÷, or division operator. Even later we learn the exponentiation operator, which has several symbols, but the ² is the special one for squaring (multiplying a number by itself). And, we soon learn the √, the square root operator, and allied symbols for taking other roots. And on and on it goes. In the middle of learning Algebra, we learn of Polynomials, and how the + and – and × seem to attain superpowers to add and subtract and multiply these groups of many symbols, as though they were unitary in themselves. Calculus adds further superpowers, while adding a further set of operators. Sure, these operators work on numbers, but that is baby steps compared to the symbols and sets of symbols (and so forth) that they also work on.

Very few have a mind like Harry's. Most of us don't need one, just as most of us don't need to be an automobile mechanic to be able to drive a car. However, a certain amount of mechanical smarts can make us a better driver. Dr. Ellenberg's notion is to make us a little better at thinking in operational terms, like a mathematician. Then we might be "less wrong" about many things. And the title provides a clue to the author's aim. The kind of mathematical thinking that underlies most of the examples is Statistical thinking.

The book has five sections. First is Linearity. The most amusing example is found in its third chapter, "Everyone is Obese". A soberly-written article came out a few years ago that can be summarized thus:

• In about 1972 half of Americans had a BMI of 25 or greater. (Body Mass Index over 25 is "overweight" and beyond 30 is classified as "obese", at least in government literature)
• Twenty years later, the number of overweight Americans was 60%.
• By 2008 just under 75% had a BMI of 25 or more.
• At this rate, all Americans will be overweight by 2048.

If you chart these three points and project a straight line through them, it will cross 100% at 2048. But do you see the fault in this reasoning? Firstly, the "line" one wants to project isn't very straight. The percent of overweight first goes up 10 points in 20 years, then another 15 points in 16 years. Do get from 2008 onward, do you project the next 25 points (100% - 75%) over 50 years, or closer to 25 years? The authors of the study projected an average of the two shorter-term rates and got there in 40 years. But why didn't they say, "Well, the rate of obesity increase has nearly doubled more recently. Maybe it will continue to speed up, and double again. Then the (now curved) line will hit 100% in just 12-13 more years, and we'll all by fat by 2020."

The real case is that, while many people are prone to gaining more weight as their prosperity increases, it isn't so for everyone. I seem to be like the majority, easily gaining weight; my wife is not, and has weighed between 98 and 108 pounds for the whole 40+ years I have known her. And she never diets. If my wife and I are still around in 2048 (we'll be over 100), I am pretty certain that she, at least, will not be obese. My BMI stays around 28-29, and is more likely to go down than up as I exceed the age of 85 or so. And our very fit son, who will almost certainly be alive in 2048, is very, very unlikely ever to have a BMI greater than 24.

The Earth is round, but we treat it as flat for most everyday uses. Straight lines serve us well. But look at a survey of Sections in the central plains. A Section is a square mile, very hearly. On a perfectly flat Earth, every Section would have exactly 640 acres. But on U.S. Geodetic Survey maps you'll see a correction every six miles further north you go. Only the southern row of Sections has something close to the full 640 acres. The northern row of a 6-by-6 Section Township has Sections with about 639 acres, because the curvature of the Earth has drawn together the meridians used to lay out the survey, by five feet near 40°N.

The takeaway point of the first section: Very few phenomena in nature proceed in a straight line forever. Keep that as a maxim in your mental bag of tricks.

The second section is titled "Inference". Here is the largest mass of material related to proofs. But it is presented in a much more entertaining way than you'd find in a college math course (or even your Middle School Geometry class). He begins with the legendary Baltimore stock broker, something I call the Binary Scam.

You get a piece of junk mail (these days, spam e-mail) with the bold statement, "Using my special stock evaluation system I predict Apple stock will rise tomorrow." The next day, Apple's stock price indeed rises, and soon another missive arrives: "See it at work. The stock will rise again the next day." It does so, and a third message now predicts a drop, which indeed happens. After a couple of weeks— and the messages now include a "Click here to invest" button—the fellow has been right ten times out of ten. You are ready to invest!!

What don't you know? You don't know that the first message went to more than 100,000,000 people. Half of them got a message saying the stock would go, not up, but down! Those 50 million or so never got the second message, but half of those who did, got one saying the opposite of the one you received. And so it goes. After 10 "predictions", the field has been cut by a factor of about 1,000. (Strictly speaking, by exactly 1,024, the tenth power of two). This leaves 100,000 or so people who tend to think this guy has a system that really, really works. If even 1% of them invest with him, that could be millions of dollars. And on day 11 he might just be in Switzerland or somewhere with those millions, and a "dead" address with no forwarding.

There is a variation of this, in which, even though half the people on day 5 got a "prediction" that "failed", they get a special message: "As you can see, nothing is perfect, but I think you will be pleased when the system continues to produce a high rate of correct calls." Guess what? Our psychology is such that a larger number of those folks will invest!

Inference is all about doing your best to gather more information, and when you have done so, remembering what Donald Rumsfeld said (I paraphrase), that we make decisions based on what we know, and try to take account of what we don't know, which is in two parts: the Known Unknowns and the Unknown Unknowns. The more "wonderful" an opportunity seems, the more likely it is that the unknown unknowns are so much bigger than what you know and what you know you don't know, that you are at best guessing while wearing a blindfold.

He closes the section with a cogent explanation of Bayesian Inference, which is quite a bit different from ordinary statistical thinking. Though it is more powerful than the kind of inference used in a typical scientific journal article, it takes a different kind of thinking, and I confess I can't use it numerically without having a text open to guide me. This is evidently true of scientists in general.

I promised a return to prime numbers. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Prime numbers have no divisors, no factors (1 doesn't count). You can see that 4 of the first 7 natural numbers are primes. Then they start to thin out: 4 of the next 12, then 2 of the next 10. Something called the Prime Number Theorem states that the number of prime numbers, P, less than some large number N, is equal or less than N/ln(N), where ln refers to the Natural Logarithm. Look it up if it interests you. Here we can test it with the 10,000th prime number, 104,729. P=10,000 and N=104,729. N/ln(N)=9060.28 and some more digits. The millionth prime is 15,485,863, and the calculation on these numbers yields 935,394 (and some decimals), about 6.5% lower than a million. For really big numbers, the theoretical number gets quite close.

What the Prime Number Theorem tells us is that prime numbers thin out steadily, and somewhat predictably, the further out we go on the number line. But they never die away completely. There may not be a high density of primes between 100 quadrillion and 101 quadrillion, but there are still a lot of them, roughly 25 trillion. However, this is very thin indeed, with only one number out of 40 being a prime at this level, on average.

Why should this be useful? Prime numbers are at the core of modern encryption, which is used by your bank to send a secure message or payment to another bank whenever you make a credit card transaction or write a check. Your password is also encrypted. The encryption method uses a long number made up of two or more long prime numbers. The rarity of long primes means there are lots of long numbers to choose from, that are hard for a computer program to figure out whether they are prime or not, and what their factors are. 101 quadrillion is only an 18-digit number, and your bank is using numbers of 85 digits or longer. Just cracking an 18-digit "composite number" (in the industry this means a long number with only two prime factors of roughly equal size) requires doing several million divisions. Today's computers can do that in a few seconds. But an 85-digit composite? No machine yet built can determine its factors in less than a few billion years. And when machines get millions of times faster? We'll just go to 200- or 400-digit encryption.

Well, there are three sections of the book to go. "Expectation" is about using probability methods to figure out how likely something is. The weather forecaster uses an expectation method to say that the chance of rain tomorrow is 40%. But particularly for weather, expectation is not like it is when rolling dice or playing roulette. If a 6-sided die is make properly (most are pretty close), each number will come up 1/6 of the time if you roll it many times. Of course, if you make only 12 trials, you are very likely to find three instances of a particular number and only one or none of another. The 1-out-of-6 expectation starts to get accurate only for a few hundred rolls at least. And here is a key point. If you roll a 2. How likely is it that the next roll will be a 2? The same as the first time, 1 out of 6. But we don't think that way, which leads to all kinds of grief at the craps table! We think a 2 is less likely than it was the first time. Not so.

In weather, expectation works a bit differently. Weather systems are not usually solid lines of rain clouds, but storm cells with space between. If an advancing storm front is made up of storms 3 miles wide with 2 miles between them on average, then the 60% chance of rain really means there is a 100% chance of rain over 60% of the area. (Dr. Ellenberg doesn't state it this way. This is my example)

There is a very entertaining chapter on the lottery, and how certain lotteries can be beaten. But don't expect a how-to on getting rich at your state's expense. When a lottery is ill-conceived enough to be beaten, you still might have to fill out half a million lottery tickets to take advantage of the odd statistics, and thus risk half a million to a million dollars in the process. And there is always a chance that every one of those tickets will be a loser, even though if you play that lottery several times you are certain to come out ahead. There are easier ways to make a buck, for certain! Being the Baltimore stock broker, for example, if you don't mind exile at some point. But lotteries can be thought of primarily as entertainment for imaginative people, and as a tax on folks who can't do math. The state takes 30%-40%, so they only pay out 60-70 cents on each dollar taken in.

Fourth is "Regression", and this word has two meanings. One is a formal process of figuring out the best line to cast through a set of points that are correlated, but not perfectly so. One chapter talks about this kind of regression, but the main point in this section is that extraordinary results are usually not followed by more extraordinary results. The classic example is adult height in a family. Suppose a couple are both extra-tall; the man may be 6'-4" and the woman a 6-footer. Average heights in America are 5'-10" for men and 5'-4" for women. Knowing only this, if the couple has four children, when they are grown, do you expect all four to be extra-tall? While there is some chance that at least one boy might exceed the father's height, it is most likely that the four will be taller than average, but not extremely tall. Conversely, if a man and woman are very short, their children will also probably be shorter than average, but it is unlikely that they will be even shorter than their parents.

This is called Regression to the Mean. Human height is partly driven by genetics, but also partly by dietary factors, and partly by chance such as getting a disease that stunts growth, or conversely over-stimulates the pituitary leading to extreme height. There are numerous factors that influence height, and they are more likely to average against one another than cause additive extreme results. It is the same for sports performance. A basketball player who usually hits 55% of his free throws may hit his first 3. Does that mean he is likely to have a 100% season? Nope. There's that straight line again. We actually see that most ball players do better in the first half of a season than the second half, from a combination of tiredness and injuries coming in later on. Yet a few players will "rise through the months". Bookmakers make a lot of money from bettors who don't think through these things. In fact, a great principle is stated in a chapter on gambling: If you find gambling exciting, you're going about it wrong. Those who do best at gambling actually gamble the least. They find ways to make the largest number of sure bets and the fewest number of risky bets. You might want to read a book by Amarillo Slim on the subject before your next casino visit.

The final section is "Existence". Pundits predict a lot of things. It turns out, and clear numerical examples demonstrate, that such things as "public opinion" seldom exist. Voting seems a straightforward matter. It is, when there are only two candidates in a race, or only a yes/no question to be decided. Add a third choice, and it all goes out the window. Some lawmakers were wise enough to require a run-off election where no candidate gets a clear majority in a race with 3 or more. But even this doesn't guarantee you'll really get "the people's choice", and several entertaining examples, some historical and some theoretical, show what that means. Suffice it to say that, like the 3-body problem in astronomy—which is unsolvable!—3-way political races are impossible to craft into a perfect system. Just ask Al Gore…

The "power of mathematical thinking" is at its root a call to back off and think more broadly than a subject at first appears. For example, recall the tall family mentioned above. Suppose I told you an additional fact, that both the man and his wife were the tallest of several siblings, and the only one in each family who was taller than their parents? Would that change your estimate of their children's heights? If it would, you are thinking in a more Bayesian way, which isn't a bad thing at all!

And I find that I've written so much without looking at a single one of the pages I'd dog-eared. I like it when an article flows. Good way to start the year.