0.99 repeating = 1?

So I was browsing the internet the other day and came across a website that said 0.99 recurring = 1. Obviously, at first glance it doesn't seem that way, so I went a little deeper (twss) and found more proof. There is some debate, but most mathematicians agree 0.99 recurring = 1.

Here's the proof

For the sake of time 0.99r will be 0.99 recurring

Proof 1: Fractions

1/3 = 0.33r

2/3 = 0.66r

3/3 = 0.99r = 1

Proof 2: Subtraction

1 - 0.99r

= 0.00r

The thing here is that there is NO one. It is repeating, so it is just 0.000000000000000000000000000000... forever. There will never be a one at the end.

Main Proof: Algebra

x= 0.99r

10x = 9.99r

10x - x = 9.99r - x

9x = 9

x = 1

So what do you think about this? Do you find it cool? Do you disagree? Did you know about this already?

Anyway, I think it's pretty cool.

- Pip

who give a f***

ha ha you're a nerd

now if you'll excuse me, im going to go read some comic books

Cool.
It actually makes much sense since 0.99 continues forerver so the difference is minimal and we tend to round such numbers.

The problem is, that is not what people argue. They argue that .99 = 1; that there is no difference.

1/3 x 3 = 3/3 = 1.0. 1/3 = 0.33... so 0.33... x 3 = 0.99..., and therefore 0.99 = 1. That's one popular argument.

Never sat well with me, for the record. I'm just saying that it isn't about rounding up the number, but saying that 0.99 = 1.0. In other words: It is not the case that 0.99 < 1.0, but that 0.99 = 1.0.

Because 0.99r is secretly an infinite geometric series. You can break it up into pieces and add infinitely.

9/10 + 9/100 + 9/1000 + ... + 9/(10^(n+1))

Which is the same as in decimal form:

0.9 + 0.09 + 0.009 + ... = 0.99r

You can use a formula for an infinite geometric series. S = a0/(1-r) Where S is the sum of the whole series (if it were the sum of a partial series, it would be Sn or sum of the first n terms/elements) a0 is the first term in the series, and r is the common ratio. (This formula is said to only be used if -1 < r < 1 I really forgot why E: I think it was because if the common ratio were a number that didn't decrease the following term in a series, that series would just add to infinity)

Let's find r:

r = (an)/(an-1) or r = 0.09/0.9 (which is the same as any other ratio you plug in) so r = 0.9/9 = 0.1

-1 < 0.1 < 1 It fits. So plug in to the other formula: S = 0.9/(1- 0.1)= 0.9/0.9 = 1 Then 0.99r approaches infinitely close to 1 as its infinite sum is = 1

And a side note: in your first proof, you have 2/3 = 0.33r when it should be 0.66r.

Essentially, 0.99 is 1.

You can consider it as 0.99/9 = 0.11, which is 1/9.

Then 9/9 is basically 0.99, but 9/9=1, so 0.99=1.

I don't like it, LOL.

I suck at math

@shootingnova: In this case 0.11r (1/9) does not equal 0.11 because its not infinitely close to it. If you use those two numbers in the formula, you'd just get 1/9 = 1/9

@dr_harlequin: I was trying to explain things simply.

This might be a better source.

Hey i'm just trying to be like Dr Doom :3

@dr_harlequin: Woopsy. Fixed.

Math you say?

@blacklegraph: What is the area of the region enclosed by the graphs of y = 2x and y = x^2?

You have 24 hours to respond, and time is running out!

Okay....

@pipxeroth: Isn't that.. taught in high school?

Math is flawed.

@princearagorn1: I was never taught that in high school. >.>

really? Maybe some class for extra competitive exams then..

The third 1/3 should be 1.000000000 instead of 0.999999999.

17/3 should be 5.666666666 or 5.666666667

5.666666661 = 0.333333333 x 17 though

@princearagorn1: Technically I wasn't taught it, but I did coach myself for Math League at the end of my senior year for exactly that (competitive exams) >.>

@talic: The third 1/3 or the 3/3 = 0.99r was used to show that 0.33r x 3 = 0.99r because the 3's will infinitely multiply out to nine, and that there won't be a 1 leftover to make it actually equal to 1 if you think about it that way. If you just think about it fraction-wise then you'd get the full 1 and because of this, 3/3 = 1 and 3/3 = 0.99r, then 0.99r = 1.

@dr_harlequin:

@blacklegraph: That's it...

Pipxeroth...Deploy the anti-life equation!

@dr_harlequin: Roger!

loneliness + alienation + fear + despair + self-worth ÷ mockery ÷ condemnation ÷ misunderstanding × guilt × shame × failure × judgment n=y where y=hope and n=folly, love=lies, life=death, self=dark side

@pipxeroth:

It's called rounding!

That's pretty cool. I never thought of it like that before.

Math is easy.

@nefarious: Not if it's AP Calculus...ugh.

No.