998,001 and its Mysterious Recurring Decimals - Numberphile

Key Takeaways

so the latest mathematical fact taken the Internet by storm uh well you may have seen it let's check this out I don't know if you have seen it it's been all over the internet I'm going to take one divided by uh 998,199 not not not that's zero it goes not not one there's one not not two not not three not not four not not five and it'll give you every single 3digit number except for one it doesn't give you 998 every single three digigit number except 998 okay so let's let's go on so it goes uh six seven skip a few then it will go 996 997 there is no 998 it goes 999 and then it starts again it'll go back to 0000 0 and then 00 01 and go starts again it repeats so we had lots of people emailed us about this and uh you know one of our favorite YouTubers V Source gave us a little shout out and asked us about it as well and so we had to make a video about this we have seen this before it's not as remarkable as you may think so this is a repeating decimal uh for a start I want to show you that this is part of a family of similar facts so one thing to notice is 9981 this is actually 999 squared now this is why we've got a little family of facts here because if I took uh 1 over 9801 this is actually 1 99 squar same sort of idea and what you would get are all the uh numbers or the two digit numbers for that so you would get 0.00 01 02 03 04 so it's the same idea you get all the two uh digit numbers except for or 98 in that case very similar idea 9998 01 okay it's the same sort of idea again this is 1 over uh 9999 squar and you'll get all the four-digit numbers now that one over9 squ now that's the one I really want to show you cuz we want to make this a bit simple okay let's not go straight into it let's take the simplest example 1 9^ 2 so 1 over 81 that's what we're going to look at c 1 9^ 2 and that will give you all the one-digit numbers 0.0 1 2 3 4 5 6 7 it misses eight nine and then back to zero again it repeats that pattern so it's the same idea so let's see if we can work out why this is true and then this will this idea will work for the other examples that we've seen so far right do you want another piece of paper a piece of paper let's get you more [Applause] paper so the easiest explanation for this is that we could make any repeating decimal that we wanted to make uh so if you wanted to make say this and let's say it's going to start with not point now let's take some digits let's call it A1 that's the first digit A2 the second digit A3 and let's say it's I don't know n digits long and then it repeats so you get the same thing it goes dot dot dot you get that repeated over and over again you can make that quite easily all you have to do to make that is to take this little sequence A1 A2 A3 up to a n and divide it by uh the same number of nine you got n n 9 9 N whatever number that is so those two things would be equal for our example then uh for 1 over 81 uh we we wanted uh not Point not one 2 3 4 five 6 7 nine dot dot dot 0 1 2 3 4 5 6 7 nine over 9 n n n n n 9 N and that would give you that decimal representation and that does simplify that makes a nice 1 over 81 now now you could make any repeating decimal using that method but there is something special about that cuz it simplifies so nicely makes such a nice number so not all fractions like this would make a nice number so why is it so nice let me try and show you that so here's a little formula if you study mathematics here's a little formula that you will learn perhaps at some point anyway let me show you what it is a little little expression uh it starts with uh one + 2x uh + 3 X2 + 4 x cubed plus right and that can go on forever okay so imagine this little equation now here's a little cool thing if if your number X is less than one uh in fact this works for negative numbers as well if they're sort of sort of greater than minus one what I'm saying is if the size and that symbol means the size of x if it's less than one this actually really simplifies to something pretty cool it becomes 1 / 1 - x^ 2 and Isn't that cool that complicated thing that goes on forever actually if if your X is less than one it's the same as that will give you the same answer that's pretty cool that's pretty cool isn't it right so we're going to use this uh to work out our fraction okay so if I let uh let's say let X x = 1 10 right what do I get on this side I'm going to get I think I'll write it as a column okay I'm going get 1 2x so 2 * 1/10th that's 0.2 and you add them together this one is 3x^2 which is+ .03 plus 4 x cubed equal 0.4 and you keep going dot dot dot keep going forever all this on the left is equal to on this side if x is 1/ 10 well let's see 1 - 1110 is 9/10 squared 9/10 squared is 81 over 100 and because it's one over that fraction you flip it this is equal to 100 over 81 so all this is 1.23 456 equals 100 over 81 the one we're interested in was actually just the same thing but just 100 times smaller you just had to shift it across a couple of places so that's why you got that such a neat cool fraction out of that really simple but why is it missing the eight now that's interesting why is it missing the eight check this out okay let's go a few down let's do the eight not Point let me see I'm going to need six of those then eight let's do the next one .7 more then there's the nine and if I want to add 10 what it's going to mean zero try it again point now 0 0 0 0 0 z0 now I want to write 010 here uh but there is no symbol for 10 I can't write 0 10 I have to write 10 like that now if you add this all together and you just add up the columns just a column at a time can you see it carries on like you learned at school this carries on so this 1 + 9 the nine turns into a 10 so that carries on so you get 1 + 8 and that becomes a nine your eight becomes a nine your missing eight turns into a nine because of all this carrying on so all this is like a like a rare of dominoes you get a carry on carry on carry on your eight turns into a nine it disappears and you get that number rediscovered and the same sort of idea will work for all those other fractions that we looked at so the 1 over 98,0 And1 it's the same idea you can prove the same thing with the same idea it's really it's a cool fact isn't it it's a cool little little fact yeah I like the fact uh and it's not completely trivial uh but you can make any repeating decimal that you want to make so that's not particularly remarkable why is it suddenly beone on the internet why is it suddenly who knows who knows you get these things every now and again these things go viral uh you know mathematicians can go can roll their eyes and go oh no not this oh oh yawn boring but every now and again for no reason as far as I can see you get this happening