0.999999999 etc. does not equal 1
Voting Style:  Open  Point System:  7 Point  
Started:  2/2/2013  Category:  Science  
Updated:  5 years ago  Status:  Post Voting Period  
Viewed:  3,472 times  Debate No:  29796 
This debate is pretty straight forward and has been done before; however, I believe I am correct in asserting that 0.999 repeating does not equal 1. As this is a two round debate, I have an odd rule that must be honored, otherwise it results in automatic loss for Con. In Con's first round, they must state their proof as to why they believe that .99999 equals 1. I am sure we both know what that will look like... Con has the Burden of Proof, as they will be making the positive assertion, and must do so in round 1. I will refute them in Round 2. Con then may attempt to refute my refutation in Round 2. Good luck to whoever accepts this debate, and, please, no trolling. I will prove (in this round) beyond a shadow of doubt that Pro's resolution is false. Specifically, I will prove that .9999... etc is indeed equal to one.
I will prove this in two different ways which Pro will have to logically show is incorrect (i.e that BOTH are incorrect): 1) Let's assume that x = .999... From this, we can see that 10x = 9.999... = 9 + x This yields the equation: 10x = 9 + x Solving this equations yields: 10x  x = 9 > 9x = 9 > x = 9/9 > x = 1 q.e.d. 2) .999... can be written as an infinite sum: .999... = 9/10 + 9/100 + 9/1000 + ... > .999... = 9 * (1/10¹ + 1/10² + 1/10³ + ...) > .999... = 9 * ∑(1/10)^i [ from i = 1 to i = ∞] From Wolfram[1], when r < 1 (which r = 1/10 is indeed < 1) ∑r^i [from i = 1 to i = ∞] = r / (1  r) This gives: .999... = 9 * (1/10) / (1  1/10) = 9 * (1/10) / (10/10  1/10) = 9 * (1/10) / ( (10  1)/10 ) = 9 * (1/10) / (9/10) = 9 * 1/10 * 10/9 = 1 q.e.d. It might be easier for Pro to understand the 2nd bit of Algebra from a nicely formatted PDF file as given by [2] [1]http://mathworld.wolfram.com... [2]http://my.cs.utsa.edu... 

Numbers represent abstract ideas, and are connected by logic. However, the physical representation of these numbers is hardly the same as their abstract functions imply. Let us, for example, take the equality of 2/2 = 1. We both know these are equivalent functions, but two halves are not the same as one whole. Furthermore, 2 + 2 = 4. However, one side of this equation denotes two distinct sets, while the other includes them as one set. 0.99999 is irrational, while 1 is a whole number. Something can be 1, a whole entity of itself. However, the only way an irrational number can exist, is from a calculation. For example, pi is the ratio of circumference to diameter. The relevant irrational cannot even be expressed as a fraction or ratio, only as a sum. For example, 1/3 = .3333 and if we have three of them, that “equals” 1, but you have three separate entities, not 1.
Therefore, 1 does not equal .99999, even if they are equivalent. Thank you for the debate. Con agrees that I have the burden of proof in this debate but Con believes that the instigator of a debate has the responsibility of properly framing the resolution. Based on Pro's round 2 rebuttal, Pro had no intention of debating whether or not .999... = 1, rather to debate the meaning of the word equal. Nearly all statements in Pro's round 2 rebuttal show a complete lack of understanding of the definition of common terms.
It is Con's opinion that as the resolution was stated in round 1, that Con has undoubtedly won this debate. Without any definitions given in round 1, Pro, Con, and voters must accept commonly accepted definitions of terms. As such, Con has met their BoP (it remains unchallenged) and has won the debate. Con will now go through Pro's rebuttal, first showing that Pro does not correctly use terms as they are commonly accepted (rendering Pro's arguments invalid) and finally, Con will show that Pro has merely changed the definition of the word equal such that it no longer has any discernable meaning. If voters are satisfied with the above, it is unnecessary to read the rest. function: In the statement "... take the equality of 2/2 = 1. We both know these are equivalent functions...", Pro misuses the term function when in fact, they mean to use the term expression. expression  a group of symbols that make a mathematical statement. [1] Indeed this is what 2/2 and 1 are, they are groups of symbols (2, /, and 1) that form mathematical statements. It is correct to say that these two expressions are mathematically equal (or equivalent). function  A relation that uniquely associates members of one set with members of another set. The term "function" is sometimes implicitly understood to mean continuous function, linear function, or function into the complex numbers. [2] For the less mathematically inclined, a function basically means: give me a value and I will give you another value (based on your input). So for instance it would have been correct to say, given the two functions f(x) = 2/2 and g(x) = 1, that f and g are equivalent functions, meaning that for every value of x in the domain of f and g, f(x) = g(x). In this equation, f(x) and g(x) represent expressions that are the statement of the value resulting from plugging in x to the two functions. Con admits that this is a fairly trivial distinction, as it is clear to the reader what Pro was attempting to say. However, Pro has decided to make this a semantics argument and as such Con feels it necessary to point out the many places where Pro incorrectly uses terms. Set: Con states "...2 + 2 = 4. However, one side of this equation denotes two distinct sets, while the other includes them as one set." Again, the two sides of this equation are in fact expressions not sets. set  A finite or infinite collection of objects in which order has no significance and multiplicity is generally also ignored. [3] 2 + 2 is not a set, it is an expression! For the purposes of this immediate discussion, Con does not distinguish the terms equivalent or equal. In fact, it would be correct to state that the following two sets are equivalent: A = {1, 1, 2} and B = {1, 2}. The definition of equivalent sets merely means that each set is a subset of the other [4]. This is a technical way of saying all of the objects in set 1 appear in set 2 and, conversely all objects in set 2 appear in set 1. Similar to the above discussion on functions vs. expressions, Con agrees that most readers will understand what Pro is attempting to say (although this mistake is a little more egregious than the first). Once again, Pro is attempting to make a semantic argument, yet in their own rebuttal, they are extremely lose with the terms they choose to use. Irrational Numbers: Con states "0.99999 is irrational, while 1 is a whole number." At this point, I expect many readers would realize this is a blatant misuse of the term irrational number. rational number  A real number that can be written as a quotient of two integers. [5] An irrational number is simply the opposite of a rational number, i.e. a number that cannot be written as the quotient of two integers. Furthermore, Pro is confused about the different ways to represent numbers. Fractions are one way to represent numbers and decimals are another. In fact, both of these representations can only be used to represent rational numbers! Furthermore, decimals are unable to fully represent some rational numbers. A generally accepted rule of thumb is that rational numbers can be represented by either finite or repeating decimals while irrational numbers have decimal representations that never end and never repeat (which can be proved, but Con doesn't see the need to present such a proof). Unlike the two previous misuses of terms, Con does not see this misuse as trivial. Rather it shows a lack of understanding of mathematical concepts on the part of Pro. Lastly, Pro's final analysis of the procedure for rationalizing denominators is very misleading. Pro states "But, what is √2/√2...but [it is] not the same thing...[as] 5/5". Pro attempts to justify this by merely making the misguided statement that multiplying by 5/5 will not yield the final result √2/2. Again, this displays a gross misunderstanding of the difference between the value of an expression and how that expression is represented. In fact, multiplying by 5/5 yields the exact same number as multiplying by √2/√2 because both expressions are equal to 1! 1/√2 = 5/(5√2) = √2/2. Con encourages Pro and readers to copy and paste into google the following expressions (just the left side...you should get the result on the right): 1 / sqrt(2) ≈ 0.70710678118 5 / (5 * sqrt(2)) ≈ 0.70710678118 sqrt(2) / 2 ≈ 0.70710678118 Again, Pro is confusing the value of an expression with the representation of that expression. Finally, Con, unlike Pro offers an actual definition of equal: equal  a (1) : of the same measure, quantity, amount, or number as another (2) : identical in mathematical value or logical denotation : b : like in quality, nature, or status c : like for each member of a group, class, or society<provide equal employment opportunities> [6] Clearly both defintions a(1) and a(2) are the way Con has interpreted the meaning of the word equal and, as such would mean Pro has lost the debate since they did not refute Con's proof in round 1. So what about the other two definitions? a(2b)Con argues that two numbers that have the same value are like in nature. Con argues that definition a(2c) does not readily apply to two numbers and thus cannot be used as a definition in this context. Finally, as Pro has defined the word equal, it has no meaning. In fact, by Pro's definition you could not say that 2 = 2. Think about it. Write it down on a piece of paper. Are the two things you wrote identical? No, they will likely be ever so slightly different and thus, by Pro's definition are not equal (even on the computer screen, they would not be the same because of how your monitor is is using light and physical media to display the two numbers). In fact, this would render any two quantities in the real world not equal. In conclusion, as Pro failed to provide a definition in round 1, the assumed definition must be used. Con requests that voters look at round 1 and decide whether they interpreted the meaning of the resolution the same as Conif so, voters must vote Con. Furthermore, Con has presented a case that despite Pro wanting this to be a semantics debate, they have failed to live up to their own standards by repeatedly being very lose with the definition of the terms they chose to use. The only logical vote here is for Con. Sources: [1] www.wolframalpha.com/input/?i=expression [2] www.wolframalpha.com/input/?i=function&a=*C.function_*MathWorld [3] www.wolframalpha.com/input/?i=set&a=*C.set_*MathWorld [4] http://www.proofwiki.org...; [5] www.wolframalpha.com/input/?i=rational+number&a=*C.rational+number_*MathWorld [6] http://www.merriamwebster.com...; 
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To be clear, I know these terms are equal. I wanted to try a semantic argument, so I tried something that I knew would get a response and felt I could argue. Clearly, I miscalculated.
At dylancatlow, that is a bad vote, even if it is in my favor. Even if Con made up these formulas and was spouting lies, I did not refute them. Therefore, Con's arguments should have been considered true for the purposes of this debate.
What is the fraction that represents the following repeating decimal: 0.98989898...?
So first, let's see why multiplying by 10 DOESN'T WORK!
10 * .989898... = 9.898989... < we CANNOT SAY this is 9 + x
Instead you need to multiply by 100:
100 * .989898... = 98.989898... = 98 + x
>
100x = 98 + x > x = 98/99
What about the following value: 1.123888...?
you STILL just need to multiply by 10!
10 * 1.123888... = 11.23888...
> well that's just
10x = 11.238 + .000888.... = 11.238 + (x  1.123)
> (notice to get .000888, we need to subtract 1.123 from the original value)
10x = 11.238  1.123 + x = 10.115 + x
>
9x = 10115/1000
>
x = 10115/9000
(again, check in google or your favorite calculator to see this is correct)
S = sum(r^i) [i = 0, i = n]
Can we find this sum? Well yes, it's actually quite simple, just multiply the series by the common ratios:
S = 1 + r + r^2 + r^3 + ... + r^(n  1) + r^n
r * S = r + r * r + r * r^3 + ... + r * r^(n  1) + r * r^n, distribution principle of multiplication
Simplifying, we get r * r = r^2, r * r^2 = r^3, r * r^(n  1) = r^n, r * r^n = r^(n + 1)
writing it out again, we get:
S = 1 + r + r^2 + ... + r^n
rS = r + r^2 + ... + r^n + r^(n + 1)
When we subtract those two EVERY TERM IN THE MIDDLE CANCELS!!! Leaving only:
S  rS = 1  r^(n + 1)
> factor out S, on the left
S(1  r) = 1  r^(n + 1)
>
S = (1  r^(n + 1)) / (1  r) or = (r^(n + 1)  1) / (r  1)
Now, since clearly .999... is the infinite geometric sum where r = .1:
.9 * sum(.1^i) (i = 1 to i = infinity)
> we can write this as first a finite sum:
.9 * (1  .1^(n + 1)) / (1  .1)
> now take the limit as n tends to infinity
lim(n > infinity) { .1^n } = 0 because.1 < 1 (basically you have 1 / 10^infinity, which is 1 / infinity which is 0)
This leaves:
.9 * 1 / (1  .1) = .9 * 1 / .9 = 1, qed
Now one might ask, is this comment just something that I (Pro) should have presented in my original case? ABSOLUTELY NOT and in fact, I DID PRESENT THIS EXACT ARGUMENT BY CITING THE FORMULA FOR AN INFINITE GEOMETRIC SERIES FROM WOLFRAM: http://mathworld.wolfram.com...
Had the voters bothered to look at my source, they would see that Wolfram PROVES THIS FORMULA. Like I said in a previous response, I need not prove that, for the particular case of .999.... = 1, the proof of a general geometric series when this has already been done and is WELL KNOWN FORMULA! THIS IS THE PURPOSE OF SOURCES PEOPLE!!!
The video you posted incorrectly uses this example which they use to reach a completely bull$h!t claim:
They assert you can do the following, take 0.222... and attempt to use the same reasoning (except they don't):
They make a mistake very quickly that if you failed elementary school algebra, you wouldn't catch. Assume x = 0.222..., so what does 2x equal? They say it's 2.222... TOTALLY WRONG!!! Surely you can see that:
2 * .222... = .444..., NOT 2.222...
In fact to get 2.222... you need to multiply by TEN:
x = .222... > 10x = 10 * .222... = 2.222... = 2 + .222... = 2 + x
> yields
10x = 2 + x > 9x = 2 ? x = 2/9, the absolutely correct value for .222...
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