Instigator / Pro

0.999 repeating equals 1


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Round 1
RESOLVED: 0.999 repeating equals 1

  • equal: "In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object." 
    • Denoted with "=" sign. 
  • 0.999 repeating: 0.999… represents a sequence of terminating decimals where each number in the sequence is a string of 9's after the decimal point. The first number in the sequence is 0.9, the second number is 0.99, the third number is 0.999, the fourth number is 0.9999, and so on. 

NOTE: "..." indicates repeating.
  • 1 ÷ 3 = 0.333... in decimal form.
  • 1 ÷ 3 = 1/3 in fraction form. 
    • Applying this 1/3 * 3 = 1 
    • Therefore; 0.333...* 3 should also be equal to 1

  • Having established this, 3 * 0.333... = 0.999...
    • Therefore, 0.999 must be equal to one. 

  • We can further illustrate this quite easily with this proof:
    • If x = 0.999...
    • Then 10x = 9.999...
        • 10x - x = 9x
        • 9.999... - 0.999... = 9.000...
        • so 9x = 9.000...
          • When we solve this equation, x = 1. 
            • Therefore 0.999... = 1. 


A few thoughts and counterthoughts to begin with, but, asan introduction:
The opponent’s own sources  disprovetheir own arguments, and one sources seems like a poor authority onthe subject.

Business Insider is not a mathematics journal, forum oreducational resource; the author, Andy Kiersz, has given no indication if they've been awarded degrees, or what they are; and their understanding of congruence and mathematical equality contradict Wikipedia’s.

However, given the pro side has restrictedthe character count to a tenth of their other debates, I have to be brief, then expand in later rounds. This means I can only provide a skeleton of a proper argument, due to the opponent's restrictions, and hope the opponent can providebetter sources and proofs, while also disproving universallyaccepted mathematical definitions and concepts, while also providing betterarguments than my own.
First, the Business Insider source and the definition of MathematicalEquality.
1)Their definition of convergence is applied wrong in theirargument. Convergence does not imply equality.
2)Within their own article, the author contradictsthemselves in their closing lines.
"..non-terminatingdecimal expansion actually representsa sequence of terminatingdecimals that gets arbitrarily close to some number."
Then, in another line:
“all getting really close to 1
A does not equal ~A.
Definitions of Mathematical Equality:
1)Theopponent’s given definition of equality neither satisfy their implicitdefinition of equality (the definition required to satisfy their argument withtheir mathematical proofs)
2)Nor does itrepresent the full scope or definition of what is meant or understood bymathematical equality.
The incorrectly apply their own definition, and they don’tunderstand the full definition of mathematical equality.

So, to address their proofs.
1/3 does not exactly equal .333…
1/3 is understood to represent a more hypothetical value which is represented in decimal form by the approximate value of .333…

Where theopponent states, “.333…*3 should also be equal to 1 …. Therefore, .999 must beequal to one,”
“The binary relation ‘is approximately equal’ (denoted bythe symbol [~]) between real numbers or other things, even if more preciselydefined, is not transitive (since many small differences can add up tosomething big).” - Wikipedia
This definition of approximate equality--the formof mathematical equality one would use when discussing congruence--becomesespecially true using multiples of x = .999…

For every multiple of x, the approximate equality become greater and greater.

In addition:

10x – x does equal 9x, but 9x does not equal 9.000…

So there isa flaw in the math presented here.
In addition to expanding on these, I would also like to discuss operations and functions inmathematics necessitated by approximate values, but, with limited space, this is what I have. The opponent now has a vast gulf of discrepancies and 3000 characters.
Round 2
  • Everything CON says about the Business Insider source is irrelevant. It was only used in the framework for the "definition"/explanation of .999 repeating so unless CON disagrees with what .999 repeating means, there is nothing to address. 
  • CON accepts the definition of equality and therefore our framework is agreed upon. 

  • My C1 can be restated in the following syllogism. As you will see CON fails to rebut any of the premises 
p1) .333... = 1/3
p2) 1/3 * 3 = 1
p3) .333... * 3 = .999...
c) Therefore .999... = 1

CON argues that I "incorrectly apply [my] own definition," and that [I] "don’t understand the full definition of mathematical equality" 
    • I don't know how CON got this but looking at their arguments they all ultimately fail to refute anything I said in round one. 

CON first says that "1/3 does not exactly equal .333…"
    • Totally false 1/3 is exactly equal to 0.333...
    • We can use any basic mathematical proof to demonstrate this

  • Let's say 0.333... is equal to x 
  • x = 0.333...
  • Therefore 10x = 3.333... (multiplying both sides by 10)
  • Therefore 9x = 3 (subtracting x from both sides)
  • Therefore x = 3/9
  • Therefore x = 1/3

    • CON in your response please tell me which one of these steps you disagree with. 

"1/3 is understood to represent a more hypothetical value which is represented in decimal form by the approximate value of .333…"
    • Source needed? 
    • Also, the value isn't approximate to 1/3. That would imply it was rounded but as I have just shown, .333... is exactly equal to 1/3.

CON says "This definition of approximate equality--the form of mathematical equality one would use when discussing congruence--becomes especially true using multiples of x = .999…"
  • CON's entire case can be summarized on the pillar that .999 repeating is approximately equal or rounded to 1
  • This is not the case because I did not use rounding in any of the formulae used to prove that .999 repeating is equal to one. The argument is therefore moot. 
    • I have proven that .333... = 1/3 exactly and CON drops all other steps of my equations, so they have not refuted my C1.
    • Frankly, anything CON posted related to approximate equality is irrelevant 

CON says "In addition: 10x – x does equal 9x, but 9x does not equal 9.000…"

  • But in denying the truth of basic math, CON does not say how or why 9x does not equal 9.000... in my equation, nor does CON state specifically what is wrong with the step. 
  • I can even restate the equation and justify each step 

    1. If x = 0.999..
    2. 10x - x = 9x (multiplying bothnsides by 10 and 10 - 1 = 9)
    3. 9.999... - 0.999... = 9.000... (subtracting .999... from 9.999... produces an infinite string of 0s = 0)
    4. Therefore 9x = 9.000... 

  • So we have the equation here. Let's see how CON will deny or insist that 9x does not equal 9.000... in the equation
  • As of now, CON has failed to disprove any of my arguments 

Back to CON and vote PRO

To reiterate a few things I said.
Business Insider is not an academic or scholastic source on mathematical theory. There are contradictions to what my opponent said, as well as within the Business Insider article itself, and my opponent’s other source, the Wikipedia article, even contradicts what the opponent says on mathematical equality.

The opponent quoted one line from the Wikipedia article about mathematical equality, and seems not to have read anything on congruence and approximate values.
.333… is only approximately equal to 1/3, because there will always be an infinitesimally small amount that will be needed to make.333… equal to 1/3, just as there will always be an infinitesimally small amount that will need to be added to .999… in order for it to be equal to 1.
This is why you get strange properties with numbers, such as 1/3 * 3 = 1, but .333…*3 = .999…
They are not the same value. But, for the sake of simplicity, we say that 1/3 * 3 = .333… *3, even though it does do not.
As far as the steps you take for your proof, you found (you didn’t, someone on Business Insider did) one of the many minor absurdities of mathematics, and you’re abusing the fact that we simplify repeating decimals when converting them into fractions.
The opponent has clearly not read the Wikipedia article still, or else they would understand what an approximate value is. Thus, they have also proven they still incorrectly apply their own definitions, and that they still don’t have a full understanding of mathematical equality.
The next part with x = .999… further proves this. I didn’t say anything about rounding. They still obviously don’t understand what approximate equality is.
An approximate equality is something which almost equals something, but doesn’t quite equal something. Because of this, in mathematics, there’s an idea of errors in approximate values, because you are getting impossibly closer to a hypothetical value (one third being hypothetical because you can never divide a single, real-world object into perfect thirds, only approximate thirds), and a compounding of errors if you were to add approximate values together.
In short, .333 will never quite equal 1/3, and .999… is even further from 1 than .333… is from 1/3
 As far as C2, 9 * .999… = 8.999…, and we already went over approximate values, so I don’t need to explain this all over again.
Here’s a source on Xeno’s paradox.
It’s another example of an absurdity/paradox in math. There’s a number of them.
Here’s also this quick little thing:
0 x 1 = 0
0 x 2 = 0
0 x 1 = 0 x 2
(0 x 1) / 0 = (0 x 2) / 0
Which simplifies to
0/0 x 1 = 0/0 x 2
And then
1 = 2
Math is weird, yo.

Round 3

  • CON, unfortunately, continues to argue that not equal to 1/3.
  • Last round I illustrated a mathematical proof showing they are exactly equal
  • CON is unable to show how the math is flawed or wrong. 
CON says In my proof I am "abusing the fact that we simplify repeating decimals when converting them into fractions"
  • I did not simplify anything im my proof. If you believe I did, show the specific step.

"The next part with x = .999… further proves this. I didn’t say anything about rounding. They still obviously don’t understand what approximate equality is."
  • I am not saying approximate equality in of itself means rounding. 
  • I'm saying when a quantity is deemed approximately equal it must be rounded up to the quantity of approximation in order to be stated as equal to the given quantity. 
  • If .333... is approximately equal to 1/3 I must have rounded somewhere in my proof to determine their exact equality, and CON should be able to point that out easily. 
  • This is not the case as .333... or .999.... as they are exactly equal. CON did now show anywhere I have rounded, therefore, the argument is moot.
  • CON is disagreeing with a fact here. 

"As far as C2, 9 * .999… = 8.999…, and we already went over approximate values, so I don’t need to explain this all over again."
  • CON just says this without showing anything as to how they got the answer? Regardless, the answer is incorrect. 
      • 9 * .999 = 9.000... (repeating into an infinite string of 0s).
  • CON is still unable to pinpoint any step in the mathematics that is flawed, illogical, or faulty. Remember this was my C2 proof.:

      • x = 0.999...
      • 10x = 9.999... (multiplys both sides by 10)
          • 10x - x = 9x (subtracts x from left side)
          • 9.999... - 0.999... = 9.000... (subtracts x from right side 9)
      • 9x = 9.000... (infinite string of 0s = 0)
      • x = 1 
      • Thereofre 0.999... = 1. 

  • Both paradoxes CON brings up are completely irrelevant, but I will address the one part because CON's own source refutes their argument. Let's see what it says about the 1=2 "paradox." 
      • "The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number. However, it is possible to disguise a division by zero in an algebraic argument,[3] leading to invalid proofs that, for instance, 1 = 2"
      • CON doesn't point out that his/her source says that the conclusion is both invalid and fallacious because it uses the divide by zero operation in the wrong way (anything divided by zero is undefined). 
  • CON paints this as a form of inconsistency or gap in mathematics, however, the math is just incorrect. CON's own source literally uses it as an example of fallacious and invalid math. 

  • My main argument is simple
p1) .333... = 1/3
p2) 1/3 * 3 = 1
p3) .333... * 3 = .999...
c) Therefore .999... = 1
  • CON has failed to disporve any of the premises 
  • Therefore .999... = 1

-I'm arguing a truism
-You didn't
-Opponent should why 1 doesn't equal 2

Asking about where you simplified the number is like asking, “Show me in the cooking instructions where it says I should wash my hands.” Cooking instructions usually don't have that step.

It's not about rounding. An approximately equal value is almost equal to something, so we say, “It’s basically equal, for the sake of simplicity, since we don't want a debate on infinity every time we try to divide shit," but they’re not actually equal.


9 * .9 = 8.1
9 * .99 = 8.91
9 * .999 = 8.991

Let’s say I want to find three equally-sized numbers that, when added together, equal ten.
3+3+3=9. Nope.

4+4+4=12. Nope.

3.3+3.3+3.3=9.9. Nope.

3.4+3.4+3.4=10.2. Nope.

I continue doing this for 3.33 & 3.34, 3.333 & 3.334, and so on, until I find the three numbers I can add together to equal ten.

And I get an infinite series of "Nooope".

I never will find three equal numbers that add up to 10 [or 1], however, because I will always have to add an infinitely smaller 3 at the end of the decimal.

There is no A that can satisfy 1 / 3 = A, because there is no A that can satisfy A+A+A = 1.

It’s not about rounding.

It’s that there are no three equal numbers numbers you can add together to get 1, so we just say, "1/3 = .333...".


As far as the 0’s.
It’s a fallacy because they simply decided to call it a fallacy because they don’t know what to do with 0’s--not because there’s some inherent attribute a zero has that makes it a fallacy, or some other logical inconsistency to the otherwise infallible and satisfactory operation. They decided it was a fallacy, because they don’t know what happens when you decide by zero.

And that was the whole point of using that in this argument.

I used that as an example of why,  sometimes, mathematicians make decisions on weird, fringe cases, such as 1=2 when you divide by 0, or 1=.999,

To reiterate, math is weird.
Finally, if you read the page on Zeno’s Paradox and understood it as well as you understood the part about 0’s, then I’m sure you have a solid grasp on calculus now. So, you ought to understand what a limit and an asymptote is. The easiest one is the inverse function, f(x) = 1/x or y  = 1 / x.

So, 1/x = y

1 / 0 = error
1/1 = 1
1/10 = .1
1/100 = .01
1/1000 = .001
1/10000 = .0001
1/100000 = .00001

As X gets bigger, Y gets smaller.
As X gets closer to infinity, Y gets closer to zero.
However, Y will never =0. Y will only approach 0.
Similarly, .333... approaches 1/3, but isn't actually 1/3; and .999... approaches 1, but isn't actually 1.