The Instigator
Khaos_Mage
Pro (for)
Losing
4 Points
The Contender
KroneckerDelta
Con (against)
Winning
29 Points

0.999999999 etc. does not equal 1

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Post Voting Period
The voting period for this debate has ended.
after 9 votes the winner is...
KroneckerDelta
Voting Style: Open Point System: 7 Point
Started: 2/2/2013 Category: Science
Updated: 4 years ago Status: Post Voting Period
Viewed: 2,997 times Debate No: 29796
Debate Rounds (2)
Comments (23)
Votes (9)

 

Khaos_Mage

Pro

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.
KroneckerDelta

Con

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...
Debate Round No. 1
Khaos_Mage

Pro

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.

1/√2 = √2/2. We come to this equality by eliminating the radical in the denominator by multiplying the “unit” of 1/√2 by 1 (in this case, √2/√2), to achieve the equivalent figure using the identity property of multiplication. But, what is √2/√2? Sure, it is treated as 1 for these calculations, but they are not the same thing, as multiplying this improper "number" by 5/5 would not aid in ridding the denominator of the radical. Somehow, only a specific equal number eliminates the radical, which shows that all equivalencies are not the created equal.

Therefore, 1 does not equal .99999, even if they are equivalent.

Thank you for the debate.

KroneckerDelta

Con

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 Con--if 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.merriam-webster.com...;
Debate Round No. 2
23 comments have been posted on this debate. Showing 1 through 10 records.
Posted by dylancatlow 4 years ago
dylancatlow
Khaos_Mage, Con had the burden of proof, and he failed because his arguments were invalid. You didn't do a great job refuting his arguments, but they sort of refute themselves if the reader has an understanding in basic mathematics. If one has the burden of proof, and his or her opponent doesn't even respond, I give points to his opposition if his argument is false.
Posted by Khaos_Mage 4 years ago
Khaos_Mage
By refute, I mean challenge the math behind them. I took no issue with the way it was presented or the math in the debate, so neither should you as a judge. If you want to challenge the arguments made by Con, then debate him. Don't use your vote as a proxy to say "you're wrong".

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.
Posted by Khaos_Mage 4 years ago
Khaos_Mage
At Ohio Gary, Con, even though the contender, had the BOP as he was making the affirmative claim (that two things equal each other). To my understanding, this is commonplace in debates.

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.
Posted by KroneckerDelta 4 years ago
KroneckerDelta
btw, if you actually watch the entire video, , dylancatlow will see that he is an "April's Fool".
Posted by KroneckerDelta 4 years ago
KroneckerDelta
Here is another example, which I present merely to convince posters that the "algebraic" method they deem incorrect is correct. I will admit, this is an argument I could have presented, so if they feel I should have, I find it valid to not accept my presented arguments. So this is now more for education rather than purposes for 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)
Posted by KroneckerDelta 4 years ago
KroneckerDelta
obviously in the below post, "I (Pro)", should have been "I (Con)"
Posted by KroneckerDelta 4 years ago
KroneckerDelta
There are so many confused people on here, I feel it necessary to explain my source. Assume you have the following geometric series:

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!!!
Posted by KroneckerDelta 4 years ago
KroneckerDelta
I'm very disappointed that dylancatlow gets caught up in a completely invalid argument that they posted from YoutTube. First the "reasoning" used in the video is blatantly incorrect, second they never attacked my second proof, which is based on an infinite series. Your video does not disprove my second proof and fails to show that my first proof is invalid (if you understood the proof of how to sum a geometric series you would see that he so-called "algebraic" proof is perfectly valid).

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...
Posted by Kinesis 4 years ago
Kinesis
I looked up lysdexia.

-_-
Posted by bladerunner060 4 years ago
bladerunner060
Stupid lysdexia....
9 votes have been placed for this debate. Showing 1 through 9 records.
Vote Placed by dylancatlow 4 years ago
dylancatlow
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Reasons for voting decision: I'm giving Pro the arguments because Con simply didn't prove .99999 = 1 while Con gave a reasonable (and correct) argument for it not being equal to 1, Con gave a lot of algebra mumbo jumbo, and left me unconvinced with his many red herrings. Watch this video: http://www.youtube.com/watch?v=wsOXvQn3JuE to understand Con's fallacy.
Vote Placed by likespeace 4 years ago
likespeace
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Reasons for voting decision: Well done by KroneckederDelta. "The Merchant of Venice" keeps coming to mind. To succeed in a purely semantic argument, one's semantics must be flawless. Kronecker revealed many flaws in Con's semantics and, since special definitions were not introduced in round one, we must accept the standard ones. Con also cited many sources.
Vote Placed by RoyLatham 4 years ago
RoyLatham
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Reasons for voting decision: Con gave correct proofs of .999... = 1. Correctly identified Pro's argument as over the meaning of "equals." The context of the debate made the meaning of "equals" clear at the outset. Pro fails.
Vote Placed by OhioGary 4 years ago
OhioGary
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Reasons for voting decision: I certainly walked into this debate with 0.9999 not equaling 1. I was thinking of a finite 0.9999 and not the irrational repeating 0.9 to infinity, so I thought that Pro would have cinched the debate. Not only did Con happen to provide more sound evidence, but Con took on the BOP as Contender and still won the debate. My. Mind. Is. Blown. :0
Vote Placed by wiploc 4 years ago
wiploc
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Reasons for voting decision: Firstguy's argument depends on two halves not equaling one whole.
Vote Placed by drafterman 4 years ago
drafterman
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Reasons for voting decision: I LOL'd at "0.9999... is irrational."
Vote Placed by AlwaysMoreThanYou 4 years ago
AlwaysMoreThanYou
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Reasons for voting decision: Con wins hands down on arguments and sources. Pro tried some weak semantic argument and failed to properly address Con's arguments. Con had a hell of a lot more sources, so sources to Con as well.
Vote Placed by Jarhyn 4 years ago
Jarhyn
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Reasons for voting decision: Con proved his argument, plain and simple, according to the axioms of mathematics, and his sources support that position. It's pretty open and shut that CON wins the debate.
Vote Placed by Grape 4 years ago
Grape
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Reasons for voting decision: Neither Pro nor Con gives any convincing arguments. Pro really fundamentally misunderstands mathematics and does not give any arguments that make sense at all. Con gives two proofs that are frequently accepted as correct, but they are not. Both proofs presuppose that 0.999... = 1. In order to prove this, you need to specify that you are working with real numbers and prove the completeness of the reals. This equality does not hold for all sets of numbers! You could give this to Con if you don't feel like being this demanding because his arguments made much more sense.