In The Joy of X: A Guided Tour of Math, from One to Infinity, mathematician Steven Strogatz provides an entertaining refresher course in math, starting with the most elementary ideas, such as counting, and finishing with mind-bending theories of infinity--including the idea that some infinities can be bigger than others.
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IRA FLATOW, HOST:
Get out a pencil and paper and your graphic calculator, because it's time for a little math review. And we'll warm up with some algebra, move on to imaginary numbers, quadratic formula, finish up with a bit of vector calculus and probability theory. No, no, just joking. Don't turn the radio off.
Math can actually be fun - if you have the right teacher, that is, like my next guest, mathematician Steven Strogatz. His new book, "The Joy of X: A Guided Tour of Math, from One to Infinity," it literally breezes through every one of these scary-sounding math subjects and makes it quite digestible and fun. And it's entertaining reading, covering how math relates to even zebra stripes and sunsets and dragonfly wings, even your dating life. Now I have your attention.
Steven Strogatz is the author of "The Joy of X: A Guided Tour of Math, from One to Infinity." He's also professor of applied math at Cornell in Ithaca. He joins us from WBUR. Welcome back to SCIENCE FRIDAY.
STEVEN STROGATZ: Thanks very much, Ira. It's great to be here.
FLATOW: It's great to have you back. One of the most surprising things you write in your book is that, one, the number one is actually equal to .9999999 - well, forever. How is that possible?
STROGATZ: Yeah, that's a really sensitive and controversial subject. You know, if you look on the Web, you'll see some flame wars, people arguing back and forth about that. You wouldn't think that a number could provoke that kind of reaction, but it does.
So I remember hearing about it in middle school. I had a friend who told me it was true. You know, I thought it wasn't true. I thought you could write as many nines as you want, and I'll write one followed by as many zeros, and then I'll subtract your number from mine, and it's going to be a lot of zeros and then a one. So I thought they can't possibly be the same.
But what my friend said that convinced me was, no, they are the same because if you have infinitely many nines - you know, .999-forever - then there's no room for any number between that number and one.
FLATOW: So they're functionally equivalent, or totally equivalent?
STROGATZ: I mean, because - they're the same number. They're two names for the same number. Because if two numbers are truly different, there's always a number between them, like their average, or infinitely many other numbers, too. But no, they're really the same number. And, I don't know, maybe it shouldn't be so shocking.
There are other things, like, in ordinary language, sometimes we have synonyms. So it's sort of like that. It's just two - it's sort of a deficiency of the decimal system, in a way, that there just happen to be two names for this same point on the number line.
FLATOW: Now, we're going to take a break and come back and talk more with Steven Strogatz, because a lot of the stuff that's in your book is what you think should be right and then what's different about it when you actually take out your pencil and paper. And there's something wrong with your intuition sometimes when you hear a problem, and it comes out different than when you actually work the problem. We'll get into those.
Steven Strogatz, who's author of "The Joy of X: A Guided Tour of Math, from One to Infinity." Our number: 1-800-989-8255. Is there something about math you'd like to talk about? You can also tweet us @scifri, @-S-C-I-F-R-I. Maybe there's a concept you could never understand, and we can figure it out for you. Stay with us. We'll be right back after this break.
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FLATOW: I'm Ira Flatow. This is SCIENCE FRIDAY, from NPR.
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FLATOW: This is SCIENCE FRIDAY. I'm Ira Flatow. We're talking about math this hour. Don't be afraid. We have Steven Strogatz, here. He's author of the new book "The Joy of X." He talks about all these interesting concepts and fun ways of talking about that. Our number is 1-800-989-8255. You can also tweet us @scifri, S-C-I-F-R-I.
Steven, you write about how words and word problems can actually fool you when you're thinking about math problems, and I'm thinking of one in particular. You write something very simple: Suppose the length of a hallway is Y when measured in yards and F when measured in feet. Write an equation that relates Y to F. And most people get it wrong, a simple thing like that. What do they do? What's their conceptual problem here?
STROGATZ: It's - we would tend to think that you would translate the common-sense idea that one yard is three feet into something that sounds just like that in symbols. That is, if you ask students - or even, I'm afraid, sometimes their teachers - to write that, it's an easy trap to fall into.
The common-sense move would be to say Y=3F. Now that's what it sounds like you would say. That's what you'd think if you translating the idea one yard is three feet. It just doesn't work, though, because if you plug in a number, like, say, well, plug in F=1, that would mean one foot. If you said Y=3F, that would say Y would be 3*1, which would be three. You'd say one foot is three yards, and suddenly it's backwards.
FLATOW: Right, so it should be - yeah. So is the lesson here to sit down and not let your intuitive part take over, but sit down and think it out on paper?
STROGATZ: You can certainly think it out and realize by just trying numbers into a formula that there's something wrong. But there's a more systematic way in this case, which is the concept that science teachers like to teach of a conversion factor, that what we're really doing is trying to multiply by the number one, but we're writing one in a funny way as one yard divided by three feet. You know, that's the same thing: one yard equals three feet.
So, in fact, the correct formula is F=3Y, where the three is not just the number three. It's the three with units that three feet, you know, is one yard.
STROGATZ: It's a little tricky. The main point is that the three is not a naked number. It has units on it.
FLATOW: Yeah, it's interesting. And they always teach you the right way in school was to write out all those units so that they cancel out, and you get - you know if you're getting the right answer or not.
STROGATZ: Excellent. Yes, see? That's it. That's the way - in fact, that's why your old high school science teacher who told you to write the units was really a good teacher, because that is the way you can check. The units have to cancel, as well.
FLATOW: That took so much more chalk, though, to do that.
FLATOW: Let's go on to another example. You have another example. Suppose three men can paint - here it is, everybody. Here's one, now. Listen. Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence? Three men, three fences, three hours - one man, one fence, how long? Hands up, yeah?
OK, now, of course, everybody says if it's three, the answer is one hour, right? Three men, three hours - one man, one hour. Conceptually wrong again.
STROGATZ: Yeah. I'm afraid that it's wrong. And that's an especially sneaky word problem, because from part of our education, we're taught to value parallel construction. That is, in English class, parallel construction is often what you're looking for. And so three men, three fences, three hours, it sounds like - just by the drumbeat of that sentence - that you want to say one man, one fence, one hour.
But it's wrong. It's actually one man, one fence, three hours, because you've got to sort of visualize it. That is, picture each man painting his own fence. And so the guy on the left, it took him three hours to paint his fence. And the guy in the middle, it takes him three hours to paint his fence. And the guy on the far right, it takes him three hours to paint his fence.
So you had three guys painting three fences in three hours, and it took each guy three hours to paint his one fence.
FLATOW: Do really talented mathematicians have problems like this also, conceptually, moving from English to symbols? Or do they just have it in their mind, and they can visualize it?
STROGATZ: I - that's an interesting question. I'm not sure. Sometimes the words are seductive, and it's possible for us to fall into those traps, too. Sure, I would say we're not immune to it, but we often think a little more symbolically or visually using geometry.
So, I don't know. That's kind of an interesting empirical question. I'd like to have some psychologist do a study on that and see if we're more likely to make that mistake than anyone else. Certainly, there are some famous puzzlers that famous mathematicians have fallen for. And so I know we're not immune to it.
FLATOW: We started out talking about infinite nines, and one equal to .999-infinite. And you write that the concept of infinity was actually banned for some time.
STROGATZ: Mm-hmm. It's true. Yes.
FLATOW: Tell us about that.
STROGATZ: Well, infinity was a big no-no for a long time, starting with the days of Zeno's paradoxes back in - Zeno was - well, let's see now. I think Zeno might be a little bit older than Socrates. I mean, we're talking a long time ago. And Zeno had four paradoxes that were all designed to show that motion was impossible, contrary to what our senses tell us.
People may be familiar with some of these. He has one about that if you move half the distance to the wall, and then half the distance remaining and keep going by half the distance, you'll never get to the wall. So, but, of course, for all practical purposes, you will. And he has others. He had three other paradoxes.
But anyway, they confounded his contemporaries to the extent that people thought that infinity was paradoxical and best left alone and avoided, really. So it was not used for, really, several hundreds - maybe something like two millennia. If, you know, you date him at 500 B.C., infinity comes back in a powerful resurgence around the beginning of calculus in the 1600s. So it's about 2,000 years.
And, you know, there were also theological reasons that infinity was taboo, that only God could be infinite. And so for human beings to even dare think about it was approaching blasphemy. And, in fact, people were burned at the stake for violating that edict.
Giordano Bruno, the Italian monk, you can see a statue in his honor in Campo De'Fiori in Rome. Giordano Bruno was executed for proposing that God might have made infinitely many different worlds.
FLATOW: Wow. He stuck to his opinion.
STROGATZ: Yeah, he really - well, I think, you know, he was religious. This was a monk. He was - he just happened to believe that God, in God's infinite power, why couldn't God make infinitely many worlds? But, of course, it's very disruptive if now we're not the only game in town.
FLATOW: So Galileo came after him, right?
STROGATZ: Exactly. Galileo knew...
FLATOW: He learned something from that experience.
STROGATZ: Yeah, they meant it. They were - they - when they were going to kill you, they will kill you. So Galileo was smart about recanting.
FLATOW: Mm-hmm. And there's also the concept that some infinities are bigger than other infinities. I remember studying Georg Cantor as probably the person I know most recently, but it goes back further than that.
STROGATZ: Well, I think I would put it at Cantor. Yeah, Cantor's the man. That's right. Cantor gets the credit for realizing that there could be different types of infinity. And so, first of all, as we say, you really weren't supposed to think about infinity to begin with. And when he did, several other great mathematicians used words like a disease. You know, they said: What is this disease that Cantor is introducing into mathematics?
FLATOW: Yeah, he was really harshly criticized.
STROGATZ: He was. Yes, he was. And he also suffered from mental illness and had, it seems, an unhappy ending to his life. He ended up in a mental institution and was very severely depressed, I think partly because of the lack of recognition he was getting.
But other great mathematicians realized that he had opened up fantastic vistas with his ideas about infinity, and they're pretty nearly universally accepted today.
FLATOW: Let's go to the phones, Heidi in Longmont, Colorado. Hi, Heidi.
FLATOW: Hi, there.
HEIDI: I love your problem. I'm thrilled to be on this. So excited, but I'm nervous. My 10-year-old actually came up with a little proof that one does equal .99999. We all know that you divide a pie into thirds, you get one-third. Well, if you represent one-third as .33333, you multiply it times three, you get .99999 and so on, equals one. I just - I thought it was a lot of fun that he had come up with this on his own.
HEIDI: So thank you for this program. I'm really enjoying it.
FLATOW: Well, thank you. And son, I think, you know, you should encourage him...
HEIDI: Oh, he's a fifth-grader in eighth-grade math class.
HEIDI: So it's in his genes.
FLATOW: Well, Steven - stay on, Heidi. Who is that childhood prodigy who was asked when he was three years old, the class was given an assignment as a punishment to add all the numbers from one to 100.
STROGATZ: Yes. That's Gauss.
FLATOW: Gauss, right?
STROGATZ: Gauss is often considered the greatest mathematician in history, and that is...
STROGATZ: ...a legendary story about Gauss. But your son - this is a phenomenal insight that your son had, and...
STROGATZ: I don't...
STROGATZ: You know, this is some real talent to recognize that argument. That's a very nice argument.
HEIDI: It's a - I have a quick question for you. I'd like to find a biography of Einstein for - geared for someone his age or Newton or Gauss. Do you have books by any author?
STROGATZ: Mm-hmm. Yes, I do. Maybe Ira has one in mind too.
FLATOW: No, go ahead.
STROGATZ: OK. My suggestion would be, as far as Einstein, a book by Banesh Hoffman called "Albert Einstein: Creator and Rebel." I remember reading it as a junior high school student, maybe a freshman in high school. But given how advanced your son is, I think he might be ready for it. And it not only talks about Einstein the scientist, but it explains, at a level that a young student could understand, what the basics of relativity are. And it's really a nice book. It's from early 1970s. I hope it's not out of print, but a good library will have it.
FLATOW: And he actually knew Einstein, Banesh Hoffman.
STROGATZ: Banesh Hoffman worked with Einstein.
STROGATZ: That's right. He was one of his assistants and collaborators. But also, you know, the recent book by Walter - and I'm suddenly forgetting...
STROGATZ: Oh, thank you. Walter Isaacson's book is marvelous about Einstein. But it may be a little more at a grown-up level, and it's not so much about the science, so...
FLATOW: But if you go to the library, there are also these juvenile-level books about him. I used to read them when I was a kid. Yeah.
HEIDI: Yeah, (unintelligible) young, real simplistic. Well, thank you very much.
FLATOW: You're welcome. That was - tell the story of how you add numbers from one to 100 and what he did in the class there.
STROGATZ: Well, yes. The legend - and, you know, I guess it's true - I'm not sure - is that Gauss and the rest of his classmates were - I'm not sure, so much, that it was a punishment as that maybe the teacher needed a break and wanted to keep the students busy for a while.
STROGATZ: So the teacher said, add up all the numbers from one to 100 and make sure that you get that answer right. Be very careful and do it correctly. And so Gauss realized that there was a shortcut, which is that instead of adding them by adding one plus two and then, you know, three to that, you could add them sort of from both ends, that as you add one to 100, that would make 101. Then you could add two to 99. That would also make 101. Then add three to 98 - again, 101.
And so you would start to realize that if you do this, you're going to be adding 101 to itself 50 times. And so that's the answer - 50 times 101, which is 5,050. So he realized that very fast. And more or less by the end of the teacher's assignment coming out of his mouth, Gauss wrote it on a slate and walked up to the front and handed it to him.
FLATOW: Yeah. That's genius. 1-800-989-8255. Let's go to John(ph) in Gadsden, Alabama. Hi, John.
JOHN: Hi. Thank you for taking my call. Enjoy your show.
FLATOW: Thank you.
JOHN: When I was in grammar school, we always had - this is going back to elementary and adding numbers. And I went to a parochial school, and the nuns always wanted to make sure your totals were correct. Well, obviously, one way they said you could do that was to take the numbers that you're adding and subtract them from the total, and you'll get to zero and you'll be correct.
But another way they showed us was something called casting out nines. And I never understood it, but it always worked. Where you take each line of numbers that are in the, you know, in the problem, and you add them across. And they - if they equal nine, and then you - I don't know - you add them all the way down, and then your total, it equals nine. It always works, and I was wondering how. How do nines make this work? That's my question.
FLATOW: All right. John, hang on. Let me remind everybody that this is SCIENCE FRIDAY from NPR. I'm Ira Flatow, here with Steven Strogatz, author of "The Joy of x." Steven, you know the answer to that?
STROGATZ: You know, I'm going to admit that I don't. I've heard of casting out nines. I never learned it, never been curious about it, and I don't know the answer to your question. Sorry.
FLATOW: Yeah. Well, look up for the next book. Yeah. Get that.
FLATOW: Let me move on to another subject because you talk about a lot of mathematical words actually come from Arabic - like the word for algorithm, for example - and there's a long history of mathematics that way, isn't there?
STROGATZ: Absolutely. The Islamic world was the dominant source of learning, really, in the period that in the West is sometimes called the Dark Ages, after the fall of the Roman Empire. A lot of the great wisdom of the Greeks was brought through traders, you know, traveling across Asia to Baghdad and the Middle East, and also in Alexandria. So - and also great insights were developed by Arabic mathematicians.
So the man that you mentioned, who gives us the word algorithm, is al-Khwarizmi, was a scholar around the time of, say, 800 A.D. in Baghdad. And he's the first one to figure out how to systematically solve the quadratic formula - the quadratic equations that we sometimes have to endure in high school algebra.
What's interesting, actually, about the history of algebra - the word algebra, too, is an Arabic word, al-jabr, which means something like restoring, having to do with taking a term that's on one side of an equation and restoring the balance to the equation by moving it to another side.
But the interesting thing here is that - I didn't realize this till I was writing the book - that what motivated algebra was Islamic inheritance law, that if a man had, say, two sons and a daughter, and then he dies and needs to leave some money, and maybe his wife is no longer alive, the rule in the law is that he has to give twice as much to his sons as to his daughter, but the sons have to get the same amount as each other. And so you can start to see it. It's like a little algebra word problem, and that's where algebra was created, to solve fair division problems under Islamic law.
FLATOW: Why was calculus invented?
STROGATZ: Calculus had - I'd say the primary impetus was to understand the motion of the planets. That is people, in the time before we had artificial light, you know, before Edison, it was dark at night. And people would sit outside and look at the sky and noticed that the planets were these odd objects that moved throughout the year, and they moved in complicated ways. They didn't just seem to go forward. Sometimes they'd go backward. So that's where the world planet comes from. It's a wandering object, the planet, the wanderer.
So anyway, people knew about the planets, but they couldn't really understand their motion very well. There was the old Ptolemaic system that was good enough for making calculations, but it didn't really - actually wasn't the right picture of what was going on. So to understand the detailed motion of the planets and also motion of things on Earth, projectiles, calculus solved both of those problems in one stroke.
FLATOW: We're going to come back - great explanation — come back and talk more with Steven Strogatz, author of "The Joy of X: A Guided Tour of Math, From One to Infinity." We'll take your tweets @SCIFRI. 1-800-989-8255 is our number. Stay with us. I'm Ira Flatow. This is SCIENCE FRIDAY from NPR.
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FLATOW: This is SCIENCE FRIDAY. I'm Ira Flatow. We're talking about the fun way to study math by reading "The Joy of X: A Guided Tour of Math, From One to Infinity." My guest is Steven Strogatz. He is also a professor of applied math at Cornell University up there in Ithaca, New York, in the Finger Lakes district. Beautiful - it must be beautiful up there, Steven, this time of the year?
STROGATZ: It's a very pretty fall we're having right now, yeah.
FLATOW: Yeah. Well, how did the ancients figure out what the constant pi was? Pi has been around a while, right?
STROGATZ: Mm-hmm. Yeah. Well, early on, it was an experimental thing. You could take rope and just wrap it around a wheel or something that is always a pretty good approximation to a circle. And so it was known - just from pretty careful measurements with strings or ropes, that it was a number a little bigger than three. Maybe we should remind your listeners that pi is the ratio of how far it is around a circle, a circumference, to how far it is across a circle, the diameter. And so it was known to be around a little bigger than three. But Archimedes gave the first systematic solution to the problem of approximating pi by replacing a circle by a shape that is almost a circle but has straight sides. That is like picture a hexagon or the shape of a stop sign, that's and octagon. You know, that's - those are six- or eight-sided shapes. They're not perfectly round, of course, there you have corners.
But if you put an octagon, let's say, or a hexagon inside of a circle so that its corners are on the circle, it turns out it's quite easy to figure out the perimeter of the hexagon. And then you can use that as an approximation to the circumference of a circle, but it's not really right because it has corners. So you have to - well, Archimedes' idea was that a circle should be thought of as a polygon that has more and more sides. Like, instead of just six, he did - first he looked at a six-sided figure, then he did one with 12 then 24. And by the time he got to a 96th-gon, he was able to prove that pi , whatever that number is, is somewhere between three and one-seventh, and three and tenth-seventy-first.
FLATOW: Wow. It's pretty good.
So it's really close. It's somewhere in there. You know, it's like, we know you're in there, but I don't know what you are.
Mm-hmm. I have a tweet coming in from Samuel Arbesman(ph) who says: Ask Steve what his favorite, most unexpected part of knowledge or the world where math can be found.
STROGATZ: This is a sneaky question from a former student of mine, Sam Arbesman.
FLATOW: He must know the answer and it must be a good one.
STROGATZ: I don't know what he has in mind.
STROGATZ: Let's see. Well, you know, certainly, romance is an interesting area to think about. Most people assumed that the romantic love is about the farthest thing from the cold logic of mathematics that you could imagine. But there are cases where it might be helpful to think about love or affairs of the heart, mathematically. And I did do that myself at one point my - is really my first girlfriend when I was in college. I was a sophomore. She was a freshman. I couldn't really understand what was going on in this relationship between us, because it seemed like whenever the more ardent I became, the more she was backing away. But then when I would give up, suddenly, she got interested.
And I started to think, this is a lot like things I was learning in physics class where there's pushing and pulling going on. And so I used the math of what are called differential equations to describe my behavior and her behavior, and forecast our - the progress of our relationship. But at some point, my math didn't work right, and I later realized why. And - there was a third variable that I didn't know about, which was her old boyfriend wanted her back.
FLATOW: It sounds like a Sheldon Cooper moment...
FLATOW: ...when the "The Big Bang Theory."
FLATOW: Where does math happen? And I'll - just try to think about that. Where does math happen if people don't expect it to be? Where in nature? There's a lot of math in nature, is there not...
FLATOW: ...in flowers and plants...
STROGATZ: That's true.
FLATOW: ...things like that.
STROGATZ: Yep. Absolutely. Sure. You know, you can look at something like a pinecone, and you'll notice that there are - as you rub your finger along those knobby little things, the fluorites on there that - if you count the spirals, you'll always get what are called Fibonacci numbers, that there may be 13 spirals or 21. But it's always one of these numbers that are in the series where you start one plus one makes two, then one plus two is three, two plus three is five. You take whatever numbers you've just produced, and then add them to get the next number.
And so the Fibonacci number start with one, two, three, five, eight, 13, 21. And what's amazing is that somehow nature - in pine cones, in sunflower heads - seems to know about the Fibonacci series, and it builds the architecture of plants using this pattern.
FLATOW: And does it know that these are efficient ways of doing things? Are they efficient ways?
STROGATZ: Yeah. This has been a question for about 100 years. Why is the Fibonacci sequence manifested in plants? And the best current thinking on this is that it has to do with biochemistry, that it's not a matter of anything but this - that when seeds form, there are certain biochemical inhibitors that are put out that tend to make it difficult for another seed to form near that. So the best place to form on the sunflower head is where the concentration of this inhibitor substance is low. And that tends to be sort of on the opposite side of the sunflower head, let's say, from where the given seed was.
STROGATZ: It's a sort of like, you know, roughly speaking, when people are standing in an elevator, they don't stand right next to each other. They try to get away to the most - but now if there are two people in the elevator, you have to get away from both of them if you - as you enter. And so these seeds are doing - it's not that they're calculating anything. They're not conscious. It's just that they're growing where there's the least biochemical inhibition. And when you work out the math of that, it ends up being tied to the Golden Ratio and Fibonacci numbers.
FLATOW: One last question. Richard Feynman used to talk about the joy of knowing, that he understood physics of nature and plants and things like that. It made him able to appreciate things more than people who didn't. Do you think that way about math also?
STROGATZ: Oh, yes. Absolutely. That's one of the most lyrical passages in, you know, all of science writing, when Feynman talks about going outside and, you know, when he looks at the stars, does he feel them any less than the poet? He thinks he feels them more because of the, you know, he understands the magnificent nuclear reactions that are happening in the sun to make the sun burn or - I mean, in his case, he is not so much talking about math. He talks more about understanding physics helps him appreciate the whole world.
But I'm sure he would agree that understanding math too, you start to see the unity of nature. You can see, like, say, in the case of the Fibonacci numbers, that plants are intimately connected to, you know, a geometry of other - I don't know - now I'm not going to come up with a good example.
FLATOW: OK. We understand what you mean.
STROGATZ: Yeah. Anyway, the point being that...
FLATOW: That's the joy of knowing. It's all about (unintelligible)...
STROGATZ: Yeah. Or let's say the spiral. I'm going to salvage myself here with the spiral. So the spirals on the sunflower head are, in many ways, the same mathematical pattern as the spirals in spiral galaxies or in DNA or the spirals on our fingertips.
FLATOW: Right. It's a beautiful thing, right? It's - as my math teacher used to say, it's elegant.
STROGATZ: Yes. It's like our favorite word.
FLATOW: Our favorite word.
FLATOW: Thank you, Steven. This is a terrific book. Steven Strogatz is author of "The Joy of X: A Guided Tour of Math, from One to Infinity." He's also professor of applied math at Cornell. Thanks, sir. Happy - happy fall to you.
STROGATZ: Thanks very much, Ira. It's great to be on.
FLATOW: Thank you. Thank you for joining us. Transcript provided by NPR, Copyright NPR.