# Your Daily Equation #05: 1=.9999…

#YourDailyEquation  with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way.

The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality.

Today Brian gives a bite-sized daily equation–a simple formula that makes you think twice.

Speaker 1:

Hey everyone, welcome to this next episode of Your Daily Equation. I’ll begin with a couple of quick announcements. A number of you have asked if I would do these sessions live and here’s the thing, I’d love to do these live. I love that engagement directly on the fly. The challenge is that when we’ve done tests, there’s a sort of lag and asynchrony between the movement of the lips and the words that I’m actually saying. So it kind of looks like you’re watching some kind of foreign film that’s being dubbed. So it’s kind of less than satisfying. So I don’t know. Let us know whether that would bother you or not. And I’m happy to try a live session if we want to give that a roll. Also, a number of you asked if I would do some Q&A sessions. I am happy to do that.

So send in your questions. Many of you put questions in the comments and we’ll see. Maybe I’ll do the Q&A live or maybe I’ll extract some of the more fruitful questions that are listed in the comments that you send to me directly and I can address those in perhaps a series of sessions that will be interlaced with the Daily Equation episodes that we have been creating. Okay, good. So today, I’m going to do something a little bit different. You no doubt have noticed that I’ve grouped the early equations. They’re all having to do with Einstein’s special theory of relativity motivated by our first equation, equals MC squared. How could you not begin a series called Your Daily equation without starting with E equals MC squared? And there are a handful of other equations in special relativity that I’m looking forward to discussing with you.

But just to mix it up a little bit. Today, I’m going to do a different equation, a kind of simple equation that, I don’t know, sort of surprisingly mind slapping in a way. It’s a little bit, maybe it’s even surprising that it’s surprising, but it is one of those equations. I just tried it out on my son. He thought it was kind of cool. It’s an old one. Many of you no doubt have seen this, but let’s just take a quick look and see this little analysis for today’s Daily Equation. So what is the equation that I have in mind? Let me bring up the iPad. Good. So the equation is dead simple. It’s simply this, right? It’s one equals 0.9999999, and it just keeps on going with the nines. And the claim is that this is a true equation that we have here.

And it is surprising, right? I mean, when you just look at that, many people would say, “Hey, the thing that you have there on the right hand side.” Yeah, it may go on forever, but the right hand side is clearly always going to be less than when it gets close to one, ever closer to one, but it never quite reaches one. But the claim is that if you take those sort of three dots at the end seriously there, and imagine this thing does go on forever, then that really is an equality. It really does equal one. Now there are a number of ways of establishing this depending on the level of mathematical rigor that you’d like to use. My goal here is just to kind of give you a sense of how to show that it’s true and I’ll leave it for the mathematicians among you to fill in all of the subtleties. So I’ll say two things about this.

One is you can approach this question by just writing down some finite versions of the infinite decimal expansion and seeing how they differ from one. So the difference of 0.9 from one, of course, is just 1/10th. If you were to go to 0.99, then the difference from one, of course, would be 1/100th. And of course, you see the pattern. If you go to the Nth decimal place, so you have 0.999, finite number, but N of these guys, then you’ll differ from one by one over 10 to the Nth power. And this is just formalizing what you already know by looking at the expression even naively that the more nines you have, the closer you get to one. And you can formalize that a little bit. You can say, “Hey, if X is equal to 0.999 and go on forever with that.” If X itself were less than one, so if X were less than one, then you say to yourself, “Well, there’s some difference between one and X.” Let’s call that difference Epsilon.

Clearly, it’s going to be pretty small if there is an X that is not equal to one. And you then recognize from what we have above, that by choosing N sufficiently large, you can make the difference between one and the finite expression to be arbitrarily small. One over 10 to the N gets smaller and smaller as N gets bigger and bigger. So whatever Epsilon there is, if it’s a finite non-zero value, then you run into a contradiction because you just choose N large enough so that N large enough will ensure us that one minus X, whoops, sorry about that. One minus X, this guy over here will actually be less than Epsilon because one minus X is just one over 10 to the N. And make N large enough, you can make one over 10 to the N smaller than any finite value of Epsilon. So that pretty much establishes right there that X has to be equal to one.

There is no finite non-zero value of Epsilon, which would be the difference between one and X. Good. Okay. That’s sort of one way of establishing it, but that doesn’t… I don’t know, that just sort of formalizes this idea that 0.99999 gets closer and closer to one. It might still leave you feeling dissatisfied. And when I gave that little argument to my son just a little few minutes ago, he was a little dissatisfied with it. But he was satisfied with the following argument, which it’s kind of a nice way of thinking about it. So let’s play this game. So let’s let X equal 0.999 and imagine this thing just keeps on going forever. Well, let’s now multiply both sides by 10. So we got 10X equals, you all know, multiplied by 10, you just move the decimal over by one. So you got 9.99 and this thing just keeps on going forever. Good, that’s nice because now let’s do 10X minus X.

And on the left hand side, of course, 10X minus X is nine times X. And on the right hand side, well, the 0.999, subtract that out from 9.999, it just kills everything after the decimal point. So we get 9X equals nine. And lo and behold right there, we have X equals 1. And that’s kind of, I don’t know, for many people, that’s psychologically maybe a little bit more convincing or satisfying than the previous argument. And there you have it. So you do have this equality that is today’s equation, that is 1.99999 and you just keep on going. Now as I was saying, there’s a way in which this is both surprising because you kind of stare at it and you think that it can’t quite really be true, but we just showed that it is. But there’s another way of thinking about it where it’s more surprising that it is surprising.

And what I mean by that is nobody would blink an eye if I wrote down one third equals 0.3333 and so forth. Let those guys go on forever. That one is almost second nature to most of us. But now if I multiply both sides by three, well, three times a third is one, and three times 0.333 and so on is 0.9999 and so on. So from that perspective, the surprising equation at the top over here is nothing but this boring equation over here that nobody would take second notice of multiplied by three. Now you’d think that a boring equation, this one over here, would stay boring if you just multiply both sides by three. In fact you might say, “Well, if it’s boring, then three times, it is going to be three times as boring, right?” But yet kind of surprisingly, you multiply it by three and you get an equation.

The one that we started with, which is for most people, at least at first glance, kind of surprising. So this is just one of those interesting mathematical niceties, mathematical facts that you can try out on your friends and your family. Probably at the moment, more on your family than your friends. Although, you can do it digitally of course. And it is a kind of unexpected equality between two mathematical terms, each of which is straightforward in its own right and it’s just unexpectedly that they are equal to each other. But as we just saw, very easy to establish, at least at the level of convincing this and rigor of convincing this is a word that we’re doing here. Okay. So that’s just a little bite size version of Your Daily Equation and we’ll take it up again tomorrow with another equation. Again, as I do these, I kind of figure out what I’m going to do on the fly, so I’m not precisely sure what I’m going to do.

But again, send in your suggestions, keeping track of those. In fact, this one I should say really did come from a suggestion from one of you guys. So I appreciate, I forget who it was, but I was familiar with this curiosity that someone brought it up as something that would be worth a moment of explanation. So thank you very much. And also, if you want to do some live Q&A, just send in your questions and I’m happy to go through them and either answer them in one of these sessions or in a live Q&A itself. Okay. That’s all for today. That is Your Daily Equation. Look forward to seeing you next time. Until then, take care.

# Your Daily Equation #05: 1=.9999…

#YourDailyEquation  with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way.

The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality.

Today Brian gives a bite-sized daily equation–a simple formula that makes you think twice.

### Transcription

Speaker 1:

Hey everyone, welcome to this next episode of Your Daily Equation. I’ll begin with a couple of quick announcements. A number of you have asked if I would do these sessions live and here’s the thing, I’d love to do these live. I love that engagement directly on the fly. The challenge is that when we’ve done tests, there’s a sort of lag and asynchrony between the movement of the lips and the words that I’m actually saying. So it kind of looks like you’re watching some kind of foreign film that’s being dubbed. So it’s kind of less than satisfying. So I don’t know. Let us know whether that would bother you or not. And I’m happy to try a live session if we want to give that a roll. Also, a number of you asked if I would do some Q&A sessions. I am happy to do that.

So send in your questions. Many of you put questions in the comments and we’ll see. Maybe I’ll do the Q&A live or maybe I’ll extract some of the more fruitful questions that are listed in the comments that you send to me directly and I can address those in perhaps a series of sessions that will be interlaced with the Daily Equation episodes that we have been creating. Okay, good. So today, I’m going to do something a little bit different. You no doubt have noticed that I’ve grouped the early equations. They’re all having to do with Einstein’s special theory of relativity motivated by our first equation, equals MC squared. How could you not begin a series called Your Daily equation without starting with E equals MC squared? And there are a handful of other equations in special relativity that I’m looking forward to discussing with you.

But just to mix it up a little bit. Today, I’m going to do a different equation, a kind of simple equation that, I don’t know, sort of surprisingly mind slapping in a way. It’s a little bit, maybe it’s even surprising that it’s surprising, but it is one of those equations. I just tried it out on my son. He thought it was kind of cool. It’s an old one. Many of you no doubt have seen this, but let’s just take a quick look and see this little analysis for today’s Daily Equation. So what is the equation that I have in mind? Let me bring up the iPad. Good. So the equation is dead simple. It’s simply this, right? It’s one equals 0.9999999, and it just keeps on going with the nines. And the claim is that this is a true equation that we have here.

And it is surprising, right? I mean, when you just look at that, many people would say, “Hey, the thing that you have there on the right hand side.” Yeah, it may go on forever, but the right hand side is clearly always going to be less than when it gets close to one, ever closer to one, but it never quite reaches one. But the claim is that if you take those sort of three dots at the end seriously there, and imagine this thing does go on forever, then that really is an equality. It really does equal one. Now there are a number of ways of establishing this depending on the level of mathematical rigor that you’d like to use. My goal here is just to kind of give you a sense of how to show that it’s true and I’ll leave it for the mathematicians among you to fill in all of the subtleties. So I’ll say two things about this.

One is you can approach this question by just writing down some finite versions of the infinite decimal expansion and seeing how they differ from one. So the difference of 0.9 from one, of course, is just 1/10th. If you were to go to 0.99, then the difference from one, of course, would be 1/100th. And of course, you see the pattern. If you go to the Nth decimal place, so you have 0.999, finite number, but N of these guys, then you’ll differ from one by one over 10 to the Nth power. And this is just formalizing what you already know by looking at the expression even naively that the more nines you have, the closer you get to one. And you can formalize that a little bit. You can say, “Hey, if X is equal to 0.999 and go on forever with that.” If X itself were less than one, so if X were less than one, then you say to yourself, “Well, there’s some difference between one and X.” Let’s call that difference Epsilon.

Clearly, it’s going to be pretty small if there is an X that is not equal to one. And you then recognize from what we have above, that by choosing N sufficiently large, you can make the difference between one and the finite expression to be arbitrarily small. One over 10 to the N gets smaller and smaller as N gets bigger and bigger. So whatever Epsilon there is, if it’s a finite non-zero value, then you run into a contradiction because you just choose N large enough so that N large enough will ensure us that one minus X, whoops, sorry about that. One minus X, this guy over here will actually be less than Epsilon because one minus X is just one over 10 to the N. And make N large enough, you can make one over 10 to the N smaller than any finite value of Epsilon. So that pretty much establishes right there that X has to be equal to one.

There is no finite non-zero value of Epsilon, which would be the difference between one and X. Good. Okay. That’s sort of one way of establishing it, but that doesn’t… I don’t know, that just sort of formalizes this idea that 0.99999 gets closer and closer to one. It might still leave you feeling dissatisfied. And when I gave that little argument to my son just a little few minutes ago, he was a little dissatisfied with it. But he was satisfied with the following argument, which it’s kind of a nice way of thinking about it. So let’s play this game. So let’s let X equal 0.999 and imagine this thing just keeps on going forever. Well, let’s now multiply both sides by 10. So we got 10X equals, you all know, multiplied by 10, you just move the decimal over by one. So you got 9.99 and this thing just keeps on going forever. Good, that’s nice because now let’s do 10X minus X.

And on the left hand side, of course, 10X minus X is nine times X. And on the right hand side, well, the 0.999, subtract that out from 9.999, it just kills everything after the decimal point. So we get 9X equals nine. And lo and behold right there, we have X equals 1. And that’s kind of, I don’t know, for many people, that’s psychologically maybe a little bit more convincing or satisfying than the previous argument. And there you have it. So you do have this equality that is today’s equation, that is 1.99999 and you just keep on going. Now as I was saying, there’s a way in which this is both surprising because you kind of stare at it and you think that it can’t quite really be true, but we just showed that it is. But there’s another way of thinking about it where it’s more surprising that it is surprising.

And what I mean by that is nobody would blink an eye if I wrote down one third equals 0.3333 and so forth. Let those guys go on forever. That one is almost second nature to most of us. But now if I multiply both sides by three, well, three times a third is one, and three times 0.333 and so on is 0.9999 and so on. So from that perspective, the surprising equation at the top over here is nothing but this boring equation over here that nobody would take second notice of multiplied by three. Now you’d think that a boring equation, this one over here, would stay boring if you just multiply both sides by three. In fact you might say, “Well, if it’s boring, then three times, it is going to be three times as boring, right?” But yet kind of surprisingly, you multiply it by three and you get an equation.

The one that we started with, which is for most people, at least at first glance, kind of surprising. So this is just one of those interesting mathematical niceties, mathematical facts that you can try out on your friends and your family. Probably at the moment, more on your family than your friends. Although, you can do it digitally of course. And it is a kind of unexpected equality between two mathematical terms, each of which is straightforward in its own right and it’s just unexpectedly that they are equal to each other. But as we just saw, very easy to establish, at least at the level of convincing this and rigor of convincing this is a word that we’re doing here. Okay. So that’s just a little bite size version of Your Daily Equation and we’ll take it up again tomorrow with another equation. Again, as I do these, I kind of figure out what I’m going to do on the fly, so I’m not precisely sure what I’m going to do.

But again, send in your suggestions, keeping track of those. In fact, this one I should say really did come from a suggestion from one of you guys. So I appreciate, I forget who it was, but I was familiar with this curiosity that someone brought it up as something that would be worth a moment of explanation. So thank you very much. And also, if you want to do some live Q&A, just send in your questions and I’m happy to go through them and either answer them in one of these sessions or in a live Q&A itself. Okay. That’s all for today. That is Your Daily Equation. Look forward to seeing you next time. Until then, take care.