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Point Nine Repeating Equals One! 9.999... reasons in 9.999... minutes. Bonus points if you can name all 9.999... lords a-leaping. Dear YouTube, wouldn't it be nice if I could include the full script with this video? A larger character limit would not be unreasonable. Created by Vi Hart.
Video transcript
99.9 repeating percent of mathematicians agree, 0.9 repeating equals 1. If because I said so works for you, you can go ahead and do something else now. Maybe you're like, 0.9 repeating equals one, that's this 0.9 repeating-derful! Otherwise, on to reason number 2, or reason 1.9 repeating. See, it's weird, because when we think of the number 1 or 2, in most contexts we mean it as a natural number, like 1, 2, 3, 4, 5. In the sense then, the next number after 1 is 2. 9.9 repeating may equal 10, but you wouldn't say you have 9.9 repeating lords a leaping, in the same way you wouldn't say you had 9.75 lords a leaping plus 1/4 lord a leaping. Lords, leaping or otherwise, come in natural numbers. So what does this statement mean? 0.9 repeating is the same as 1? It looks pretty different, but it equals 1 in the same way that a 1/2 equals 0.5. They have the same value, You can philosophize over whether, if 1 is the loneliest number, 0.78 plus 0.22 is just as lonely, but there's no mathematical doubt that they have the same value, just as 100 years of solitude is exactly as long as 10 plus 40 times 2 years of solitude, or 99.9 repeating years of solitude. So reason 2 is not a proof, but a reason to stay open minded. Numbers that look different can have the same value. Another example of this is that, in algebra, 0 equals negative 0. Reason 3. 0.9 repeating is a decimal number, a real number. See, if you want 0.9 repeating to be that number infinitesimally close to 1, but not 1-- and let's face it, some of you do-- then you're writing down the wrong number when you write 0.9 repeating. That number infinitely close to 1, but less than 1, is a number, but it's not 0.9 repeating or any real number. OK, let's do more 3.9 repeating-mal proof for reason 3.9 repeating. According to this 3.9 repeating-mula, 3.9 repeating is 4. First step, say 0.9 repeating equals x. Then multiply each side by 10. Third, subtract 0.9 repeating from this side, which equals x, which we subtract from the other side. So 9.9 repeating minus 0.9 repeating is 9. And 10x minus x is 9x. Divide by 9, and you get 1 equals x, which you might notice also equals 0.9 repeating. There's no tricks here. It's simple multiplication, subtraction and division by 9, which are all allowed because they are consistent. When something is inconsisten-- 9.9 repeating --t, we just throw it out of algebra altogether. For example, in algebra if you try to divide by 0, you get this problem where anything can equal anything. I mean, if you want to say everything is equal, fine, but your algebra sucks. Normal, everyday elementary algebra, the one they shove down students throats as if it were the only algebra, doesn't allow dividing by 0. So it stays consistent and suspiciously practical. We also could have shifted the decimal point twice, multiplying by 100 to prove that if you have 99.9 repeating bottles of beer on the wall, 99.9 repeating bottles of beer. Take one down, pass it around, 99 bottles of beer on the wall. All right, moving 3.9 repeating-ward. Reason number 5, there's infinite 9s. If anyone ever thinks they have the biggest number, well, they don't, because just add 1, or multiply by 2, or whatever, and it's even bigger. Infinity, though, is not a number you can add 1 to to get a bigger number. Adding 1 is an algebra thing that you do with real numbers. Subtracting doesn't work either. Infinity bottles of beer on the wall minus 1 is still infinity bottles of beer. When we did this decimal shift to multiply by 10, unlike un-infinitely many 9s, there's no last 9 that got shifted over to create a 0. Infinite 9s plus another 9 is still infinite 9s, the kind of infuriating property that makes infinity not a real number and makes that proof work. If you're the type of person who is discon-- 9.9 repeating --t with the idea that 9.9 repeating equals 10, you might also feel that 1 divided by 0 should be infinity. And, as it turns out, there is other systems or calculation besides elementary algebra where it does. That's right, mathematicians figured out how to divide by 0 a long time ago. But elementary algebra can't deal with infinity. If you allow infinity in your algebrizations, once again, you get contradictions. Infinity may not be a real number, but it is a number, a hyperreal number. Hyperreals, like infinity and the infinitesimal, follow different rules. And while algebra can't handle them, some people thought they should be numbers, and you should be able to use them. And so they figured out how, and bam! you get something like calculus. Reason 6. Take the number 1, and subtract 0.9 repeating. It's pretty clear that it's infinite 0s, but you might be tempted to think there's some sort of final 1 beyond infinity. Let's write that down as 0.0 repeating 1. Of course, if the 0 repeats infinitely, then you never get to the 1, so you might as well leave off the number. Thus the difference between 0.9 repeating and 1 is 0. There is no difference. Here's another "there's no difference" proof. Remember how the next higher natural number after 1 is 2? What's the next higher real number? The game, is for any number you claim is the next real number, I can find one that's even closer to 1. Of the many delightful things about real numbers is that for any two numbers, no matter how close, there's still an infinite amount of numbers between them, and an infinite amount of numbers between those, and so on. There is no next higher real number after 1. Likewise, there's no next lower number. If 0.9 repeating and 1 were different real numbers, there would have to be infinite other real numbers between them. If you can't name a number higher than 0.9 repeating, but lower than 1, it can only be because 0.9 repeating is 1. If you don't like it, well, go to college and learn about hyperreals, or better yet, surreals. That's a system where you can have a number that's infinitely close to 1, but not 1. But even weirder, there's infinity more numbers that are infinitely even closer. Anyway, on to reason number 8, another common proof. Take 0.3 repeating, a repeating decimal equal to 1/3. Multiply it by 3. Obviously, by definition, 3/3 is 1, and 0.3 repeating times 3 is 0.9 repeating, which you might have noticed is also 1. The only assumption here is that 0.3 repeating equals 1/3. Maybe you don't like decimal notation in general, which brings us to reason number 9, this sum of an infinite series thing. 9/10 plus 9/100 plus 9/1000. And we can sum this series and get 1. But I can see why you might be unhappy with this. It recalls Zeno's paradoxes. How can you get across a room, when first you have to walk halfway, and then half of that, and so on. Or, how can you shoot an arrow into a target, when first it needs to go halfway, but before it can get halfway, it needs to go half of halfway, and before that, half of half of halfway, and half of half of half of halfway, and so on. Therefore, it can never start to move at all. Anyway, it's 1/2 plus 1/4 plus 1/8, dot, dot, dot, dot, dot, to get 1. Each time, you fall short of 1. So how can you ever do anything? Luckily, infinity has got our backs. I mean, that's like the definition of infinity, a numbers so large, you can never get there, no matter how many steps you do, no matter how high you count. This way of writing numbers with this dot, dot, dot business, or with a bar over the repeating part, is a shorthand for an infinite series, whether it be 9/10 plus 9/100, and so on to get 1. Or 3/10 plus 3/100, and so on, to get 1/3. No matter how many 3s you write down, it will always be less than 1/3, but it will also always be less than infinity 3s. Infinity is what gets us there when no real number can. The binary equivalent of 0.9 repeating is 0.1 repeating. That's exactly 1/2, plus 1/4, plus 1/8, and so on. That's how we know a dotted, dotted, dot, dot, dot half note equals a whole note. The ultimate reason that 0.9 repeating equals 1 is because it works. It's consistent, just like 1 plus 1 equals 2 is consistent, and just like 1 divided by 0 equals infinity isn't. Mathematics is about making up rules and seeing what happens. And it takes great creativity to come up with good rules. The only difference between mathematics and art is that if you don't follow your invented rules precisely in mathematics, people have a tendency to tell you you're wrong. Some rules give you elementary algebra and real numbers, and these rules can't tell the difference between 0.9 repeating and 1, just like they can't tell the difference between 0.5 and 1/2, or between 0 and negative 0. I hope you see now that the view that 9.9 repeating does not equal 10 is simply un-- 9.9 repeating --able. If you started this video thinking, I h-- 7.9 repeating that 7.9 repeating is 8, I hope now, you're thinking, oh, sweet, 4.9 repeating is 5? High 4.9 repeating! If you're having math problems, I feel bad for you, son, I got 98.9 repeating problems, but 0.9 repeating is 1. Here's the moral of the story. The idea of a number infinitely close to but less than 1 is not stupid or wrong, but wonderful, and beautiful, and interesting. The true mathematician takes "you can't do that" as a challenge. If someone tells you can't subtract a bigger number from a smaller number, just invent negative numbers. If someone tells you can't multiply a number by itself to get a negative number, then invent imaginary numbers. If someone tells you can't multiply two non-zero numbers together to get 0, or raise one non-zero number to the power of another and get 0, you should probably say, I'll do both at once, and in 8 dimensions. And if you ignore them telling you that numbers aren't 8-dimensional, and that inventing fake numbers is a useless waste of time, and then actually try to figure it out, the next thing you know, you've got split octonions, which besides being super awesome, just happen to be the perfect way to describe the wave equation of electrons and stuff.