Every PROOF you've seen that .999... = 1 is WRONG

It's true, so why so many wrong proofs?

There are many incorrect proofs that .999... = 1 on YouTube and elsewhere. In this video we see why all those proofs are wrong, and then demystify what is actually meant when a mathematician says .999... = 1 and I explain why it is actually true. Did your favorite math creator make one of these mistakes?

*Title only applies before the first viewing of this video and only to people who have not studied real analysis.

References on the standard definition of decimals and repeating decimals (please read if you are confused or unsure about the standard notation):

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[nameu]

The proofs are all correct, the justifications are simply left as an exercise to the reader

[johnmeo]

My very first day of college the math department head advisor was showing us this proof. Me, having seen the algebraic proofs and trying to make a good impression, went up and showed him a more “elegant” version after the lecture. When he pointed out the fatal flaws, I learned that I didn’t actually know what constituted a rigorous proof.

[RandallStephens]

"A proof of something is not simply a list of true statements that ends in the one you're looking for."
Too many people need to hear and understand this.

[marcrindermann]

7:30 to be totally honest, those algebraic proofs never gave me an 'irky feeling' in my stomach. What did, however, was the constant 'dot 99 repeating' because a simple 'dot 9 repeating' would have been sufficient.

[andyl.]

2:10 For anyone wondering, ...999 = -1 is "true" in the "10-adic" number system, specifically as a ten's complement. The analogy is the two's complement (...1111 = -1), the way computers store negative integers.

EDIT: I have edited to put "true" and "10-adic" in hand-waving quotations, because people correctly pointed out that p-adic numbers are only well-defined when p is prime.

[naptastic]

"A proof is not just a series of true statements that ends in the one you're looking for"

here's somebody who gets it

[marcopollom]

My multivariable professor called me a "math magician" when I showed him the initial 'proof.'

[KingBobXVI]

Not a formal proof, but when confronted with this in high school, I figured out this fun fact: you can actually construct any repeating decimal as a fraction by dividing the repeating portion by that quantity of 9's. The obvious being 1/3 = 3/9 = 0.33... but also things like 123/999 = 0.123123...

And with that established, it then follows that you can construct 0.99... with the fraction 9/9, which of course is equal to 1.

[VivekYadav-dsoz]

Video: "Why every proof you've seen is WRONG"
Me, watching this the 2nd time: *VISIBLE CONFUSION*

[anonymousjohnson]

The real difficult thing to understand in these proofs is the idea of a value “approaching” another number meaning that that value is equivalent to that number. Like in an infinite sum that converges to a value, we say that the infinite sum is equivalent to that value, but it almost feels like we’ve just redefined what it means for numbers to be equivalent.

[robinofficial]

"Every proof you've seen is wrong", debunks one proof. (The 1/3 never felt like a proof to me, TBH.)

My teacher once told me "If 0.p9 does not equal 1, there has to be a third number between them." That convinced me the most.
I think that's pretty close to the limit interpretation.

[UltimateDurzan]

The thing is... repeating is really just the result of the limitations of dividing a number a base-ten system by something other than 2 or 5. 3/3 is another way of saying 1, but you can't conveniently write 1/3 in base 10 without getting an endless number.

[johnsavard]

You are correct in saying that the common proofs lack mathematical rigor, by failing to say that .999... is a limit. However, people unfamiliar with calculus can do long division, and they can see that dividing one by 3 yields 3 over and over again. So by viewing decimal numbers as common constructed objects, the common proofs are satisfying, and most people don't go beyond that.

[yazanalj]

My math teacher who used these demonstrations:
"Oh? you're approaching me?"

[BurgerFred]

Algebra teacher: "Prove that .99...=1"
The kid who knows about geometric series: *pushes up glasses* "My time has come."

[justsomeboyprobablydressed]

Good video. But I would say that, although the real numbers are generally presented axiomatically (as a complete, ordered field), there are some treatments of the real numbers that _define_ a real number to be any infinite decimal. So, if you are following that treatment, there is nothing wrong with taking for granted that 0.999... exists.

[amirh]

Man i went to shower and you had 14.7K subs
I'm back and now you have 15K subs
Your channel is grown really fast
Keep up the good work!

[Daniel_VolumeDown]

4:41 "Why are you so okay with one-third being exactly equal to 0.33 repeating?"
It's not just because of what I think is called "long division" in English? (I'm not a native speaker)

[julianfogel]

Something worth adding is that the reals are usually defined axiomatically, using formal logic and set theory, rather than as an infinite sequence of decimal digits as taught in high school.
The decimal representation of a real can be shown as valid because every axiomaticallly defined real has such a representation and likewise every decimal representation refers to a unique real number.
There is the minor caveat that some reals have two decimal representations, one ending in repeating 9s and the other in repeating 0s. For example, is the same real number as This is a consequence of the axioms for reals and the way decimal notation is defined.

[jhawley]

My favorite "proof" was from one of my professors
He pulled down the chalk board and got a new stick of chalk like he was about to write out a long proof then simply wrote: "It is trivial to see that 0.9̅ = 1 ▪︎" i should mention that this was just a joke after class