Math to me is not quickly computing the tip on a $56 dinner or figuring out if one loan is better than another. Those things I consider arithmetic. Mathematicians are notoriously bad at arithmetic because mathematics is about charting new ground in the sea of mathematical logic and knowledge. To chart new ground means to understand the old ground first, and move on in search of new mysteries. Most mathematicians are pretty awful at small time arithmetic (“Hey! What’s 114 minus 87?”) because once they understood how one would figure out such a thing in general, they moved on without bothering to stick around to master that.
To many people math is thought of as a dull field, and I think this often due to the fact that grade school math can indeed be quite boring. Then, a family vacation or a flu comes along that takes you out of school for a week and now, not only is it dull, but you are lost and starting to get bad grades. All of a sudden, “I hate math” is ringing out, and most people, once they escape the clutches of their math requirements, never turn back to look at it again.
Which is unfortunate because math, as you will see, rather than being dull, is full of strange surprises, strange beauty, and demonstrates astounding human reasoning. Math is a discipline which can make us smarter. I like to hope – and I admit that this might be wishful thinking – that as the human race evolves it gets on the whole smarter and we will eventually rectify the numerous problems that we have created: the massive inequalities across the globe and across the cities regarding wealth and opportunities, the daily widespread destruction of life on earth due to manipulation of habitats and the environment for our near-term convenience. We’ve come far. We have built impressive things. But we have a long way to go regarding how we treat each other and how treat the plants and animals around us. What does all this have to do with math? In my optimistic hours, I like to believe that logic is a key component to both ethics and to social policy making, which are critical components to making the human race a more respectable lot. One can argue that math by itself will solve none of these. But, I think that to learn to be logical and to examine all the ifs, ands and buts that go into designing complicated systems and making large scale decisions, one requires practice, and math is a fantastic way to exercise the brain.
I also accept the position made famous by the 20th century English mathematician G.H. Hardy, which is math for the sake of math. Hardy saw math similar to other endeavors such as music and art which should be pursued do to their inherent beauty and interest. Of course, it would be great if we humans could fix the major human failings, the ones that provide content for our newspapers every day (and the frustrating part is that we can solve them, we have all the technologies we need, but the social and political institutions we have built up prevent and even reverse progress in this arena), but I don’t think that we should put on hold all other endeavors like art, music, math, literature, and sports until we do so. We’d be waiting a long time.
Math is the purist form of human reason. But strict adherence to reason alone wouldn’t suffice to make math interesting. What is interesting is that starting with only a few primitive accepted ideas, such as numbers, and a few rules, such as how to add them together, logic, plus creativity and pulling together ideas from differing disciplines, can lead us to mind bending truths.
Here’s a simple statement: . Easy enough to check: 9 + 16 = 25. Can we find another three numbers that do the same thing? Yes, we can find lots of numbers that do it. 5, 12 and 13 work: . There are endless combinations of whole numbers that make this equation work. But, if we simply take the “squared” and turn it into a “cubed”, that is, we raise the numbers to the power of three, can we find three positive whole numbers such that the first one cubed plus the second one cubed equals the third one cubed. The answer, which is not at all obvious from the question, is a firm, resounding: NO. This should be slightly intriguing since one might think if one kept trying on outward to bigger and bigger numbers, one would eventually stumble on a trio of numbers where this simple equation is fulfilled. But no one will. Ever. And even though we cannot possibly check them all, since the whole numbers never stop, we still know this to be true. There are early ones, like 5, 6 and 7, that get close: and , but there will never be a set that does it. And here is the real whopper. Instead of the third power, what if we took this equation and used the 4th power, or the 5th, or any other higher number. We would never find a solution to any of these equations. The only time solutions exist is when the power is 2. Now, not only would it take an infinite amount of time to check one of the powers, say 3, but there would be infinite powers to check. How are we sure?
The answer lies in the idea of a mathematical proof. And the proof for this one is hefty. The claim was made in the 1600s by a French mathematician named Pierre de Fermat, but it wasn’t proven until 1995, by a mathematician named Andrew Wiles. Wiles holed himself away for about a decade in the attic of his Princeton house. His colleagues gave up on him; they had no idea that he was about to solve one of the largest outstanding math problems in existence. He finally produced a proof of about 200 pages. Mathematicians had to pour over it for months to verify its accuracy. We won’t be covering that proof, obviously; only a very small fractions of mathematicians understand it. But the point is, with only logical steps, possibly a very large number of them which are very complex, we can prove things about the infinite, even though we can never check all of the infinite possibilities. In fact, if we are talking about infinity, once we have explored even a huge set of numbers, we are no closer to accomplishing our task. Because unlike “a drop in the ocean,” which is actually a small fraction of the ocean, any search over an infinite set, like numbers, leaves the reader no closer to completion: what you have looked at compared to what there is left is always the simple amount: 0%. Such is the power and dizziness of infinity.
Sometimes I imagine what it would be like if someone handed me an envelope and inside it was a piece of paper that contained the explanation of the universe. It would likely have to be abbreviated, but the contents would be true. Right away, some philosophically inclined persons might object and say that there is no explanation that could be put in human terms, or even stranger sounding complaints such as the very notion that there is an explanation to the universe is wrong. But “That’s OK,” I’d respond, for if that is true the paper will tell us so and that will be the end of it, but even in that case we will learn something. There is some piece of paper that could be written that would have the truth, at least in part, on it. Of course, this piece of paper is not readily available.
I used to think about this a lot, and at some point in my mind it became connected with the idea that the number line, the simple mathematical object that appears on grade school walls and helps students figure out things like 5 minus 9, contains every possible written document. Huh? To explain, let’s limit ourselves to the numbers from 0 to 1. When asked what types of numbers are in there, the answer — fractions! — comes to mind. Mathematicians need to be precise when they make their definitions and assertions, so let’s take fraction to mean one whole number divided by another whole number. These are also called rational numbers. Halfway between 0 and 1 lies the most famous of all fractions, ½ and then on either side you can find as many more as you like. And right there in that innocent sounding phrase as many more as you like opens up the door to infinity. Can we really put as many as we like, could anyone ever fill up the number line? No, because we can always sneak another fraction in between any two that we have already written in, and that number will be a fraction too. By taking the average of these two, for example (4/9+10/21)/2 we get the fraction 29/63. Since we can do this forever, we have infinite fractions between 0 and 1.
And if we have infinite fractions, we have infinite stories because it is easy to turn a fraction into a story. Step 1: write the fraction as a decimal. Step 2: take the digits after the decimal point in pairs and turn them into letters: 01 will become the letter A. 02 = B, and so on up to 26 which is Z. This gives us a lot of leftovers which we could use as punctuation, or maybe lowercase and uppercase, whatever.
Thinking about this another way, we could imagine taking a really sharp ax down onto a number line at a random spot, reading off the fraction, converting it into a string of letters, and reading the story. And if we got lucky, we could chop at just the right spot to land right at the story that is the true explanation of the universe. The true explanation of our universe lies somewhere between 0 and 1.
Of course, everything else does too. Explanations that seem possible but are wrong, instructions on how to make a peanut butter and jelly sandwich, what Ralph Waldo Emerson ate for dinner on July 2, 1842. And most of the time, the vast majority of the time, the ax will fall on rubbish. Things like kSud7b38ah- ;glYhnryYiru… and onward like that. But still it is somehow haunting that all works, even the as yet unwritten ones, are in there. Every shipwrecked captain’s lost diary, every letter from every mother to child. Every Sears catalog that was every published. The other day standing under an awning avoiding the rain, I thought: the position and shape and velocity of every raindrop that has every fallen. The time and trajectory of every leaf that has ever floated down a stream. The infinite is inconceivable.
But the stories that arise when the ax comes down and lands on a fraction eventually repeat themselves, over and over again, because every decimal version of a fraction eventually repeats itself. To see why this is, imagine the tedious process of long division applied to a fraction. At some point, no matter how big the numerator and the denominator are, you will get to a point where you are dividing the denominator into a chunk you have already seen before, and at that point the long division begins to repeat itself.
But one could imagine a story in there that never repeated itself. Do those stories exist? Are there numbers from 0 to 1 that are not fractions? It seems like there are: as long as the decimal never repeats it’s not a fraction, so we just have to write down a decimal that never repeats. But can identify such a number? Do we have names for such numbers, are there lots of them?
There are lots of them. In fact, just as there are infinite fractions, there are infinite “non fractions”, which are called irrational numbers. Irrational. Not rational. And if you could “randomly” pick a number between 0 and 1, it is 100% likely that you would get an irrational number. There are infinite numbers of both types, but there are many more irrationals than fractions. It took a long time in the history of mathematics to be able to defend such seemingly paradoxical sentences, and we’ll get there, but first, let’s find ourselves one of these irrational numbers. And prove that it is irrational. And we’ll discuss some historical stories.
According to legends, not everyone was happy when the existence of such numbers was claimed. One says that Hippasus of Metapontum, when he demonstrated the existence of an irrational number, was tossed from a ship, sometime in the 5th century B.C., by his fellow Pythagoreans, because this violated their notion of the simplicity and beauty of numbers. Another version had him exiled. Whatever the truth is, the discovery of such numbers threw a wrench in Pythagorean mathematics.
Here’s a simple drawing that we will use for two things: to prove the Pythagorean theorem and shortly thereafter show the existence of an irrational number. Showing the existence of an irrational number, a number that there is no way we can write as a fraction (one whole number divided by another), no matter how high into the reaches of infinity we go, should seem strange at this point. Numbers like that really exist? But first let’s tackle the Pythagorean theorem.
We are looking at a square that has a side length of two. The area of this square two times two which equals four. The goal is to find the side length of the inner square that is tilted so it looks like a diamond. Since we don’t know its value yet we give it a name. In the drawing we call the length c. The area of this little diamond is c times c, which is written as . All we are going to do is say that the BIG AREA is equal to areas of the four corner triangles plus the diamond inside. If you think about combining two of those triangles into one shape, which would be a square, and that square would have an area of one times one, which is one, then we have figured out that the area of each little triangle is ½. And there are 4 of them. So we have:
In other words, . This is the essence of the Pythagorean theorem. [to consider: doing it for a,b, c right away, or right after]. One could also write the equation for c as . Having quickly finished Pythagorous, we move back to finding an irrational number.
This number c is our culprit. This number, the square root of 2, or in other words, the number which multiplied by itself gives 2, is irrational. At this point in the demonstration Hippasus of Metapontum is close to getting tossed into the Ionian sea. Because he is about to prove that there is no fraction out there, no two whole numbers, call them a and b such that .
Before we blast right into proving this statement, it’s worth taking pause to consider that what we are about to do is prove that no two whole numbers, out of the whole infinite slew of them, will divide to give you exactly . To get a feel for the magnitude of this statement, let’s take a closer look at this number. Typed into a calculator, but who owns those anymore…typed into Google (type sqrt(2) into the search bar) you get 1.41421356237. It ends there when you ask for it, but it never really ends. Anyhow, one would think you could find two numbers out there that give you this number. For example, try 577 divided by 408. You get 1.41421568627. The Babylonians and the ancient Indian mathematicians liked this one. It’s close, and it would seem that if you searched long enough, there would be two numbers that you could divide to get the length of that diagonal of a square with side length equal to 1. But there are not. We are about to jump in and do the proof. But one of the problems I think with the way math is taught is that the mystery and ‘breathtakingness’ of the whole venture is lost. For example, a uninspired math teacher might start a class with “Today we are going to prove the irrationality of root two” and by the end of that sentence, half the kids are already asleep. Even though, if it is carefully explained exactly what will be proved, it should stir people up.
At this point, in one last pause before diving in to the proof, think about the fact that two pages from now you will know how to prove this, but right now, you likely have no idea. You have no idea where to start, you have no idea if you have the mathematical tools, you are just sitting there, pencil and empty page in hand, and nothing is happening. And that’s fascinating. The proof of this deep fact is right around the corner, is short, anyone can understand it with a little time (some five minutes, some an hour), and sitting there with a pencil and paper, almost no one would even have a clue of where to start to show this. Mathematical proofs are a sequence of logical steps, but knowing where to step often requires a stroke of genius. Take a pause. Then, we’ll do it.
To prove it will require us something smarter than “start looking”. We could look for 100 years on the best computers in existence and if we came up with nothing, we still would not know for sure. We’ll use the concept of “proof by contradiction”. In this approach, you assume the opposite of what you are trying to prove, and then show that, after a series of logical and irrefutable steps, you arrive at a contradiction. Since each step was bullet proof logical, the only rescue is to reverse the original (false) assumption, and then the proof is finished. Here we go.
So, we assume that there are two whole numbers somewhere out there that fulfill the equation:
We’re also going to assume that we have chosen a and b so that they have no common divisors. For example, if 3 could go into both of them, we’ll assume that we have already done that step to reduce the fraction to its simplest terms. Therefore, there are no numbers that go into both a and b. Let’s get rid of that square root sign by doing a valid algebra step: square both sides of the equation (if two things are equal, then the square of those two things are equal too). This gives
and rearranging this gives:
Looking at this equation for a few seconds should convince you of the following: is an even number. After all, it is 2 times some other number, so it has to be even.
Up until now the proof is proceeding pretty reasonably. Here comes the first little bump to slow us down. If is an even number, then a is an even number. One way to see this is to imagine that a was an odd number (this is a mini proof by contradiction within a proof by contradiction) and then imagine squaring it. Try 3, 5, 7, etc, you see that every time you square an odd number it comes out odd. So if we start to think: “well, if is an even number it might be that a is still odd….” Not so! Because if a were odd then would also be odd!
So, a is even. To recap, we’ve shown that if square root of 2 is equal to a/b then a is an even number. Now for a tiny dose of algebra. If a is even then we can write it as 2 times some other number. This is always true, it’s the definition of being even. 12 is 2 times 6. 30 is 2 times 15. So we write, instead of a, 2 times some other number, let’s use the letter d. Then, where we see a in the above equation, we’ll substitute 2d.
And now we have nearly arrived at the end of the proof. Using exactly the same argument as above, when we had and thus concluded that a was even, we now have and hence can conclude that b is even.
And we are done then because having a be even and b be even is a contradiction to our early assumption that we were writing this fraction with no common divisors. And so our initial assumption, that we could find two values a and b such that is wrong! The length of the diagonal of that nice little triangle can not be written as a fraction. The simplicity and powerful result of this proof deserves pause.
The square root of two (mathematicians say root two for brevity) is our first irrational number we have found. Another famous irrational number is π. Proving that one is irrational though requires pages and pages of hard math. We won’t do that one! But we will look at probably the most famous mathematical constant out there, π, for a bit.
Most people forget what π is for a simple reason: we don’t need to know what it is for daily living, and so after it is learned, it is forgotten. But it also is forgotten because for most people, it is never really learned. In the rush to cram for exams and get good grades, people store in their short term memory things like “π r squared” but forget the big picture. π is very simple: it is a circle’s circumference divided by its diameter. This is pretty powerful: take any circle and measure all the way around it and write that number down. Now measure straight across it and write that number down. Divide, and you get π. Any circle, no matter the size. And that simple number, the number which relates the circumference to the diameter, π, is irrational.
One consequence of forgetting that the circumference and the diameter of a circle are related easily through π is the following puzzle, which is fun to give people. Imagine that you have been assigned by some strange king to put a hose around the equator of the Earth. You go ahead and procure the hose, you buy just enough and voila you are done. Then the king says that he actually wants the hose to be 1 foot off the earth all the way around. The question is, how much more hose do you need. To me, it seems like you might need a lot more hose, and that without knowing the diameter of the earth, you might be stuck.
The answer to the question of how much more hose you need though is: just over 6 feet. More precisely, it is 2 times π. And it does not matter if the Earth is the size of a pea or the size of the sun. This is very simple from an algebra point of view, one you accept that circumference = 2 π r. When the king said he actually wanted it one foot off the ground, he said, instead of r, he wants (r+1). So instead of needing 2 π r feet of hose you need 2 π (r+1), which equals 2 π r + 2 π. That is, there is now an additional 2 π, which is just over 6 feet. Sometimes our brains deceive us on simple matters.
Construction-wise, I think if this story were told with a giant square instead of a sphere, it might be easier: wrap a hose around a square and then make the square one foot larger on each side and see how much more hose you need. So, say the square was 9 feet on each side. Original hose was 9*4 = 36 feet long, and now we make each side 10 feet so the new hose is 10*4 = 40 feet on each side. An extra 4 feet, no matter what the original side length was. Our brains get used to things that we see and work with often. The great mathematician, John von Neumann, once said “…in mathematics you don’t understand things. You just get used to them.”
I find this to be true in everyday words and language and ideas too. Anyone can throw “protein” into a conversation and no one will bat an eye, because people are very used to hearing that word. But 99% of us don’t know what protein is, really. We just hear it enough and the word gets ingrained into us. And so it is with mathematicians who can talk about prime numbers with such ease it makes the casual listener feel like a complete idiot.
There is a great story about an Indian mathematical prodigy named Srinivasa Ramanujan, who G. H. Hardy brought to England to collaborate with. Ramanujan taught himself mathematics, finding what books he could from nearby libraries, and went on to become one of the most famous, although short-lived, mathematicians of all time. One day, on taking a taxi over to see Ramanujan, Hardy remarked to Ramanujan that the taxi cab number, 1729, seemed a rather dull number. Ramanujan thought for a second and then said, no, it is an interesting number. It is the smallest number that can be written as the sum of two cubes in two different ways: and . For the rest of us, having this type of familiarity with numbers is completely foreign, and this story boggles the mind. Ramanujan lived with numbers.
This is a long way of saying: if our daily life consisted of wrapping hoses around spheres and cutting them to size, the hose problem would be no problem at all. So there is some hope for us all! We’ll talk about this again when we turn to probability, which humans seems particularly bad at when it’s anything more than “What’s the probability that there is a bear behind that rock that is going to eat me?”
We have shown that root 2 is irrational, and we have discussed π a bit, reminding us that it is the number that relates a circle’s circumference to its radius. Circumference = 2 π times the radius. Equivalently, since 2 times the radius is the diameter, it is even easier: Circumference = π times Diameter. Since π is close to 3, this gives a very quick way of estimating the circumference from the diameter: just multiply the diameter by 3. Maybe this will come in handy for you one day, although it never has for me.
Root 2 is irrational (as we proved) and π is as well (which we are accepting on trust here). Anyone of us could spend an hour writing down hundreds of rational numbers. Just write a bunch of decimals down, or fractions, and you have them. But so far we have only two irrationals. It’s time to return to the statement we made above that there are “more” irrationals than rationals, even though there are an infinite number of both.
One way to think about this is to think about the number line again, where we started, and shooting a very fine dart at it and asking, “What are the odds that we nail a rational number”. Stated another way, we could think about picking a random number between 0 and 1 the following way. Fill up a bag with 10 coins, numbered 0 through 9. Write down a dot on a piece of paper, reach into the bag with your eyes closed and pull out a random coin. Write that number down. Put it back and continue that way. So maybe you get .9568463520879084…. Imagine doing this “forever” and ask the question: what are the odds that this number that I write down will start over from the beginning and start repeating itself on and on, repeating itself in these big chunks forever, like:
I think its easy to see that the odds of this are 0. So that’s one way to make some sense of the statement that there are more irrationals.
Another way, and this is the standard proof used in college math classes the world over, is due to a man named Georg Cantor.
Rather than argue that there are more irrationals like we did above, by reasoning that the probability of getting an irrational if you “randomly” pick a number between 0 and 1 is 100%, Cantor went for a different approach, and invented some new fields of mathematics along the way.
His approach to the question of “Which infinity is bigger?” was to first think about how we compare sets of things that we can count, and then see if we can adapt that method to infinite sets. For example, Farmer Joe has a bag with 40 apples in it and Farmer Sally has a bag with 45 apples in it. Sally has more apples in her bag. One way to “prove” this is to pair up the apples – one from Joe, one from Sally, etc. At the end of this process, we can see that Joe is out of apples but Sally has some still left in her bag. This method works for infinite sets too, and this was a huge intellectual contribution of Cantor.
This is another proof by contradiction. For the contradiction we’ll assume that we can pair up the positive integers in a one-to-one way with the irrationals between 0 and 1. The proof ends in a logical fallacy which then shows that we cannot pair them up and that no matter how we try, there will always be irrational numbers that we did not manage to pair off with an integer. And this is how Cantor proved there are “more” irrational numbers. [NOTE: Mathematicians don’t use the word “more” here since one would always need to put it in quotes. They invented a new term for this called cardinality, which refers to size of infinite sets. So, rather than “this infinite set is bigger than that one” you get “this infinite set has a larger cardinality than that one”.]
So, we assume we have the one to one pairing. Writing this down, integers on the left, irrational numbers on the right, would look something like
This table would of course go off to the right forever and go down forever, and the idea is that this table puts the integer numbers in one-to-one correspondence with the irrationals. For example, 4 is paired with a particular irrational number that starts of 0.590296291305203958675930132564. Now, to demonstrate that the original premise was wrong, Cantor now proceeds to show how to find an irrational number that is definitively not in the infinite list we just made. To do this, we rewrite the list and highlight the diagonal entries:
At this point we then make an irrational number as follows. Start with a decimal point, then for the first digit after the decimal, you are allowed to choose any number other than 4, the first number we highlighted. For the second digit, choose any number other than 9, for the third digit, and number besides 0. Keep going this way, maybe we get a number like 0.28755983… A number that is made by following this rule has the following property: it cannot be on the list we just made! Why is this? Well, it’s not the first number on our list, because we purposely chose the first digit of it to be different. It’s not the second number for the same reason. Same for the 3rd and 4th digits, and onward.
Cantor chose to handle the issue of comparing sizes of infinite sets by seeing if the two sets can or cannot be put into one-to-one correspondence with each other. If this is the definition used (the name of the definition being cardinality), than in Cantor’s terms, the size of the set of all numbers even numbers, for example, is the same size as the set of all whole numbers, since these two sets can easily be put into one-to-one correspondence:
And this might seem just wrong to say that these two sets are the same size: aren’t there only ½ as many even numbers as all numbers? In fact in our probabilistic version above, we would conclude that the set of even numbers is smaller, since you only hit them half the time, than the set of all integers. Yet Cantor says these sets are the same size.
If you don’t like this you are completely permitted, and encouraged, by the math world, to define it a different way. But, your way has to lead to interesting things, otherwise no one will listen. Cantor’s approach has emerged as the winner since it has given rise to a world of mathematical treasures. [to do: show that the number of rational numbers is the same size as the integers. The discuss that there are an infinite number of infinities, not just two.]
After Cantor introduced his diagonalization proof, a question that stumped mathematicians for decades after was: is there a set out there that is larger than the set of integers, but smaller than the set of all numbers between 0 and 1? This became known as The Continuum Hypotheses. It was not resolved until the 1950s when it was proven that the answer is: it is up to the mathematician! Meaning that one could decide that either such sets could exist or such sets could not exist, but either decision led to a self-consistent type of mathematics.
By Aurelien Guichard (Flickr: Borough Market) [CC-BY-SA-2.0 (http://creativecommons.org/licenses/by-sa/2.0)%5D, via Wikimedia Commons
About David Craft
David Craft is a researcher at Mass General Hospital in radiation treatment for cancer. He applies mathematics to the problem of finding optimal ways to irradiate tumors without overly irradiating the nearby unaffected organs. He has taught mathematics at Williams College, MIT, and Northeastern, and Brighton High School. After the Brighton High experience (long ago) he created this tool to make it easier for teachers to quickly create math worksheets to keep the wild students busy. He gave it to math.com under the agreement that it would always be free to use. David holds a B.Sc. in engineering from Brown University and a Ph.D. in Operations Research from MIT. He also enjoys music, foraging for wild edible plants, bike riding, and eating mainly vegetables.
Copyright © 2013-2014 by David Craft. All rights reserved. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without the prior written permission of the author.