If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Math for fun and glory>Unit 1

Lesson 8: Infinity . . .

# 9.999... reasons that .999... = 1

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.

## Want to join the conversation?

• So is 0.1111111111111111111111111111111111111111111111111111111111111111111111111111...
equal to 1/9?
Because 0.3333333333333333333333... = 1/3 divide both by three and 1/9 = 0.1111111111...
• You are absolutely right.
0.00000000000000000000000000000000000000000000000000.............. = 0/9 = 0
0.111111111111111111111111111111111111111111111111111111111111111............... = 1/9
0.2222222222222222222222222222222222222222222222222222222............... = 2/9
0.333333333333333333333333333333333333333333333333333............... = 3/9 = 1/3
0.444444444444444444444444444444444444444444444444444.................. = 4/9
0.555555555555555555555555555555555555555555555555555555.................. = 5/9
0.666666666666666666666666666666666666666666666666666.............. = 6/9 = 2/3
0.777777777777777777777777777777777777777777777777777777................. = 7/9
0.888888888888888888888888888888888888888888888888888................. = 8/9
0.9999999999999999999999999999999999999999999999............ = 9/9 = 3/3 = 1!
• Is there a way to write the closest number to 1 without being 1?

And not 0.999999999..., but some number or symbol.
It's like i, they both don't exist but can be represented.
• No, there is a way to express this. It is:
lim of f(x) = x
x→1+
Basically, it just means the number that gets REALLY close to 1 without being 1 as x gets bigger. Calculus Notation To The Rescue!!
• I just wanted to appreciate the fact that this video is exactly 9.999... minutes long :)
• So you're saying that 0.99999...1 is also equal to 1 because with th repeating 9s, you'll never get to the 1?
• The number 0.9999....1 is not a meaningful thing to write. The 9's go on forever, so you will never get to the 1 at the 'end' of your number, just repeating 9's. So it makes no sense to write anything after the 9's because it will never occur; it is not ever going to be a part of the number.
• so does that mean that 1.999....=2? I hope it does, i lost track after
• I think this was already answered for you, but I'm not sure so I will. Yes, 1.999.... is equal to 2. Think about it as (1 + .999....) = 2.
• Based upon the logic discussed at would the closest number to zero be one less than an infinite amount of zeroes, then a one at the end? But isn't that mixing hyperreals with integers? Where is the breaking point where you can start adding on zeroes instead of subtracting from infinity?
• There's no closest number to 0. As Vi said, if you can't introduce a number between two different numbers, they're not different. If you find a number that has no numbers between 0 and itself, its value is 0.
• why infinity numbers exsit??

• No one made infinity numbers. It's just... there. The reason infinite numbers exist is because you always can add one to the huge number. For example, My number is 9999999999999999999999999^999999999999999999999999999999. I could say that this is the biggest number. But someone could always come and add 1. 9999999999999999999999999^999999999999999999999999999999+1.
• There is also a video of Vi Hart with a title: Why Every Proof that .999... = 1 is Wrong

youtu.be/wsOXvQn3JuE

but the upload date is April Fools' Day
• OH MY GOODNESS THAT WAS A JOKE

now it makes sense that it is not on khanacademy thanks for telling me