Talk:Mathematical fallacy/Archive 1

readability

I had a readability issue when I was looking at this page. In some of the proofs, the explanations of the infallacies refer to line numbers, but none (or at least few) of the proofs are numbered. Also, there are various formats for the proofs. Some of them have each line bulleted with an explanation as well as mathmatical statement, while others are just simply one line. I suggest coming up with the same format, layout, and numbering for each proof, to make the page easier to read. One way might be a 3 column table, with the line number in the first column, the mathematical statement in the 2nd column, and the explanation in the 3rd column. But not being an expert mathematician, before I or someone else went through and changed everything, I wanted to see if that format presents any issues. --JerryFu 23:04, 2 August 2006 (UTC)

Is there a word for a statement like "1 = 3", besides "wrong"? I'm trying to concisely express the concept of connecting an ideal voltage source (across which is a finite, non-zero voltage) to an ideal short circuit (across which is 0 volts). Is there a mathematical term for this? - Omegatron 22:25, August 6, 2005 (UTC)

"On the other hand it is possible to construct useful mathematical systems where 1 is technically equal to 2. Mathematics in domains modulo 1 are one example. In such domains 0.5 + 0.5 = 0 = 1 = 2..." WolfKeeper

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Could someone create an entry and "solution" for this equation: http://bash.org/?522860 I know it's wrong but I'm not sure about the proof. Jdm 20:53, 25 August 2005 (UTC)

Can't be bothered, but the main error is when they take square roots of both sides near the end. If you consider that the square root can be negative then there's no problem.WolfKeeper

Just as I thought, thanks! Jdm

Theorem: All numbers are equal.

Proof: Choose arbitrary a and b, and let t = a + b. Then

1. a + b = t

2. (a + b)(a - b) = t(a - b)

3. a^2 - b^2 = ta - tb

4. a^2 - ta = b^2 - tb

5. a^2 - ta + (t^2)/4 = b^2 - tb + (t^2)/4

6. (a - t/2)^2 = (b - t/2)^2

7. a - t/2 = b - t/2

8. a = b

So all numbers are the same, and math is pointless.

line 6 with a + b = t substitution:

(a - a/2 - b/2)^2 = (b - b/2 - a/2)^2

(a/2 - b/2)^2 = (b/2 - a/2)^2

remember the definition abs(x) = sqrt(x^2)?

abs(a/2 - b/2) = abs(b/2 - a/2)

abs(-1(b/2 - a/2)) = abs(b/2 - a/2)

abs(b/2 - a/2) = abs(b/2 - a/2)

divide by abs(b/2 - a/2) (a constant) and you get 1 = 1

.9999bar = 1

.9999bar = 1

.9999bar = .3333bar + .3333bar + .3333bar

.3333bar = 1/3

1/3 + 1/3 + 1/3 = 1

therefore...

.9999bar = 1

HAHA!

Seems to be a valid proof. Mathematically, they are the same number expressed differently (if you take the sum of 9 x 10^-n as n tends to infinity the difference between it and 1 goes as 10^-n, which tends to zero.) So they are equal.WolfKeeper

you're wrong though. .3333bar does not = 1/3 - but it is infinitely close. I'd like someone to prove that 1/3 = .3333... because they are just as much the same thing as pi = 22/7 (ie they're not equal) 68.6.112.70 03:05, 11 January 2006 (UTC)

You are incorrect, a repeating .9 does indeed equal 1. But the above proof is not mathematically solid. The proof I see most often is:

x = .9bar

10x= 9.9bar

9x = 9

x = 1

But even this proof is not completely rigorous. It assumes that the series .9 + .09 + .009 + .0009 ... converges. That's not difficult to prove though. --Starx 03:27, 11 January 2006 (UTC)

Well, this proof gets into the way you handle infinities and infintesimals. For example, such a proof would not work for a finite series of 9's such as .999 . x=.999, 10x = 9.99, 9x = 8.991. My argument is that by saying 10 = 9.9bar, you are "extending" the infinite series of 9's by one 9. Another way to put it is that .9999.... approaches 1 as the number of 9's approach infinity, but this is not to say that it is 1 when the number of 9s is infinity. 68.6.112.70 20:11, 17 March 2006 (UTC)

The bar means there is no last 9. Its been pretty well established it is in fact 1, not approaching 1. This is something people tend to resist, but it follows from the definition of limits and recuring decimals. See the .999... article Brentt 21:20, 17 March 2007 (UTC)

woman = evil

anyone know the proof of woman=evil? I know it involves time=money and $money={\sqrt {evil}}$ --Taejo | Talk 09:21, 24 October 2005 (UTC)

I love that one. First we state the women require time and money.

• Women = Time * Money

And everyone knows that "time is money".

• Time = Money

So line one can be restated as:

• Women = Money * Money
• Women = Money^2

We also know that money is "the root of all evil"

• $Money={\sqrt {Evil}}$

So line 4 means

• $Women=({\sqrt {Evil}})^{2}$
• Women = Evil

Q.E.D. Mr. Quertee 01:25, 7 April 2006 (UTC)

1 = 2

On the other hand it is possible to construct useful mathematical systems where 1 is technically equal to 2. Mathematics in domains modulo 1 are one example. In such domains 0.5 + 0.5 = 0 = 1 = 2...

Umm... does this even have anything to do with division by zero? I think this doesn't really belong in the article. - furrykef (Talk at me) 08:53, 9 December 2005 (UTC)

I think it's important to note that equivilance classes are different than te normal number system. And that different rules apply to invalid proofs regarding them. Full Decent 04:44, 10 January 2006 (UTC)

Integration by parts

I added an invald proof involving integration by parts but I haven't been able to identify the invalid step yet. If anyone can find it, please add that. The proof for women = evil goes as follows:

• Women = Time * Money
  • Time = Money
• Women = Money^2
  • Money = Sqrt(Evil)
• Women = Evil

Splat 02:06, 24 December 2005 (UTC)

-

I think this proof shows that the slope of 1 equals the slope of zero, not that 1=0. Finding a derivative is really finding the slope. 1 and zero are both constants, so they have the same slope. When finding an anti-derivative, aren't you supposed to assume that the answer is always "+ C", where C is a constant in the Real Number System? So having the "1 + integral" really doesn't change a thing. Sorry if my Calculus is off. --Jguy100 01:57, 26 December 2005 (UTC)

I just figured it out today. You are essentially correct. In most presentations of integration by parts, the addition of C is saved until the end. But in deriving the formula for integration by parts, one must go through the process of integration, thus resulting in the addition of C. Normally, this can be ignored since the first C adds to the second C (when you integrate the final integral), resulting in another C. In the case of this manipulation of integrals, however, this cannot be done. Because the integrals are eliminated and the second C is never added, the first C must nonetheless be included. Splat 06:14, 26 December 2005 (UTC)

1 is the greatest real number

How about :

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Proof that the largest real number is 1 :

Let x be the largest real number. Therefore x^2 = x, which rearranges to x(x-1) = 0. This has solutions x=0 and x=1. We know 1>0, therefore 1 is the largest real number.

The fallacy is that we've assumed that there exists a largest real number, which is not true. There is no largest real number, because you can always "add one" to any suggested upper bound.

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Someone might want to explain it a bit more elegantly than me, but you get the basic premise of it.

You have the reason wrong. A "real number" as currently definined has no largest real number. However, if you took as a *given* that x is the largest real number, then you would assume *not* that x^2 = x, but that the set of all real numbers is not closed over addition or multiplication. This is to say that x^2 might not be a member of the set. For example, say you have a set {1,2,3,4} and you postulate that 4 is the largest of the set, 4^2 is obviously not equal to 4. 4^2 is still 16, it just isn't part of that set. Fresheneesz 20:23, 17 March 2006 (UTC)

Also, even if there were a largest real number, would there be any reason to assume it has the property that x^2==x? (I know it seems obvious, but...you know)Brentt 22:17, 16 March 2007 (UTC)

Editing "Proof that 0 equals 1"

I wanted to edit this section. Please let me know if this is accurate. Editing the part after QED:

The error here is that the associative law cannot be applied to an infinite sum unless the sum would converge without any parentheses. In this particular argument, the second line gives the series of partial sums 0 + 0 + 0 + ... (which converges to 0). However, without parenthesis, the series would be 1 - 1 + 1 - 1 + ..., which is divergent. Therefore, the third line employs an illegal usage of the associative law.

It seems to me that this argument is circular. The "invalid proof" is in fact a valid proof of the statement that the associative law does not hold in general for infinite sums. -- Dominus 16:07, 18 March 2006 (UTC)

And I just realized that the "Conclusion" section of the article makes exactly this point. -- Dominus 17:40, 18 March 2006 (UTC)

I got another elegant "2 = 1" proof!

Let x be 0. Therefore, x + x = x or 2x = x. Divide each side by x to get rid of x, and you are left with 2 = 1. Tada! Problem here is that x is defined to be 0 and division by 0 is invalid. —Preceding unsigned comment added by 67.108.48.182 (talk • contribs)

3-=4?

I get Channel 3 and Channel 4 mixed up a lot on TV, and so my family taeases me. Could you "prove" that 3 = 4? SigmaEpsilon → ΣΕ 23:37, 27 April 2006 (UTC)

Made it up based on alot of the examples on this page

when x=1

then x-1 = 0

any number multiplied by zero equals zero (so the following is true when x=1)

3(x-1) = 4(x-1)

divide both sides by (x-1) and you get

3(x-1)/(x-1) = 4(x-1)/(x-1)

cancel (x-1)/(x-1) on each side to give 1, so:

3*1 = 4*1

clean up what remains

3=4

you could reword it better —The preceding unsigned comment was added by 212.50.188.131 (talk • contribs).

Proof that 1=4

• Since an infinitely large plane has the coordinates of (-∞,∞) × (-∞,∞), this means that

  ${\frac {}{}}\infty =[\infty -(-\infty )]^{2}$

• Which can be simplified into

  ${\frac {}{}}\infty =(2\infty )^{2}$

• And finally

  ${\frac {}{}}\infty =4\infty ^{2}$

• So

  ${\frac {}{}}\infty /\infty =4\infty ^{2}/\infty$

• Then

  1=4∞^1

• Therefore

  $1=4$

Beautiful proof that -1=1, uncommon one

I've got a short proof from one of my analysis teachers...

$-1=(-1)^{3}=(-1)^{6/2}=((-1)^{6})^{1/2}=1^{1/2}=1$

—Preceding unsigned comment added by MarvinCZ (talk • contribs)

Interesting proof. Maybe this one could be inserted into the article. x42bn6 Talk 01:37, 22 July 2006 (UTC)

But I am not sure where exactly is the mistake, specifically which step introduces it. I would say it's the third '='. But again, why exactly? --Marvin ^talk 15:24, 22 July 2006 (UTC)

Because there's two solutions to the square root, so the third equality isn't necessarily true.WolfKeeper 17:13, 22 July 2006 (UTC)

You can also argue that the final part where you take the square root of 1 could take either sign. It depends whether your square root operator is positive or not.WolfKeeper 16:41, 23 July 2006 (UTC)

No, that is not the problem here. It's real numbers - it is usually assumed there's but one square root (i.e. the non-negative one).

$y={\sqrt {x}}$ is always non-negative.

The mistake here is indeed the third '=':

$(-1)^{6/2}=((-1)^{6})^{1/2}$ is invalid, according to the rules of Exponentiation. $(-1)^{6/2}=((-1)^{1/2})^{6}$ by definition, and that is just nonsense, thus that move it illegal.83.237.112.97 13:41, 13 November 2006 (UTC)

Not only is it not "just nonsense", but it is actually correct. ((-1)^{1/2})^6 is in fact equal to -1. -- Dominus 14:20, 13 November 2006 (UTC)

I think the problem is in the fact that the equation has (-1)^(1/2). This requires the square root of -1 (which is undefined). —The preceding unsigned comment was added by 212.50.188.131 (talk • contribs).

Actually, no, there exists a number system where the square root of -1 exists - complex numbers. x42bn6 Talk 20:39, 11 December 2006 (UTC)

It [b]is[/b] nonsense, plain and simple. I've provided the link where exponentiation is defined. Thus, for $(-1)^{6/2}=((-1)^{1/2})^{6}$ to make any sense we should somehow define $(-1)^{1/2}$ which is impossible in real numbers (and we're working with real numbers here, no?). In complex numbers, I concede, it is correct, but imaginary numbers are irrelevant - they spoil all the fun. Regardless, the article definetly contains a mistake in that point.(85.140.3.102 20:23, 15 December 2006 (UTC))

Of course if you are considering only real solutions, this is invalid. However, would it be correct to say that 2+0.5=3 because we are not considering fractions? Complex numbers are far from irrelevant - there is even a section in the exponentiation article about complex powers. We do not say that just because -1 is real, then its square root must also be real because "we can ignore complex roots". x42bn6 Talk 01:54, 16 December 2006 (UTC)

I am sorry, but you are still missing my point. The article mentions that the mistake lies in the fact that there are 'two roots' and the mistake is thus in the last step, while it clearly isn't there. The third '=' is invalid, not fourth (i.e. last). And it remains invalid even with complex numbers, which we may, I surrender, use freely.85.140.1.39 15:59, 16 December 2006 (UTC)

Another variant is:

$\cos ^{2}(x)=1-\sin ^{2}(x)$

$\cos ^{3}(x)=(1-\sin ^{2}(x))^{\frac {3}{2}}$

Let x=π, $LHS=-1,RHS=(1-0)^{\frac {3}{2}}=1$

Therefore $-1=1$

Should this be put in? x42bn6 Talk 12:08, 23 July 2006 (UTC)

Again, the page mentions the wrong reason why this proof is invalid. Two square roots have nothing to do with that, we use only the non-negative root.

The mistake here is ${\sqrt {cos^{2}(x)}}=cos(x)$ which is incorrect. ${\sqrt {cos^{2}(x)}}=|cos(x)|$.83.237.112.97 13:41, 13 November 2006 (UTC)

i=?

There's a section of the article stating that i=the sqrt of -1. The only problem is, there is no sqrt of -1. When it gave an invalid proof, it stated that i squared = 1. If i = the sqrt of -1, then wouldn't i2 be -1? If so, wouldn't this just be another proof that -1 = 1? AstroHurricane001 12:38, 18 October 2006 (UTC)

i is the imaginary square root of -1. See imaginary number for a better explanation than I can ever hope to give. QWERTY | DVORAK 21:36, 24 October 2006 (UTC)

Look at this...

Power = Work / Time Knowledge = Power Time = Money Thus: Knowledge = Work / Money Knowledge × Money = Work Money = Work / Knowledge As Knowledge → 0, Money → infinity. Damn it. The less you know, the more you make. Pity really. Paidgenius 16:37, 17 December 2006 (UTC)

'Nother one

allow x to equal 1.

x-1 = xx0

cancel out the x's

-1 = 0

...thus we prove that ALGEBRA is broken, as real numbers disproves this: x = 1

(1-1) = (1x0)

which leaves us with 0 = 0 —The preceding unsigned comment was added by Tar7arus (talk • contribs) 14:03, 10 February 2007 (UTC).

Yours is an equation and not an identity - you are substituting a value of x. x42bn6 Talk 17:21, 10 February 2007 (UTC)

i, pi or two equals 0

I'm not really sure where the error is in this proof (I think the last step) but if someone knows I think it is a good one to add.

e ^ (i * pi) + 1 = 0

e ^ (i * pi) = -1

(e ^ (i * pi)) ^ 2 = (-1) ^ 2

e ^ (2 * i * pi) = 1

e ^ 0 = 1

e ^ (2 * i * pi) = e ^ 0

ln(e ^ (2 * i * pi)) = ln(e ^ 0)

2 * i * pi = 0

Therefore, either 2, i, or pi must equal 0.

Any ideas on where the error is? 67.167.99.102 04:04, 18 April 2007 (UTC)

The error occurs when you take logs. Logs of complex numbers are multivalued. If you use the principal branch, then $\log(e^{2i\pi })=0=\log(e^{0})$. --Zundark 06:56, 18 April 2007 (UTC)

Problem with first example

I believe that this explanation is not helpful:

• "This proof is also invalid since it applies the following principle for square roots incorrectly::

${\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}$ This is only true when y is a positive number, which is not the case in the "proof" above. Thus, the proof is invalid."

I don't believe that this statement is correct.

Every sqrt operation has two possible solutions, and I don't believe that whether a number is positive or negative has any bearing on the issue cojoco 00:37, 9 July 2007 (UTC)

Actually, the sign of the number in the square root does matter. The invalid proof you're referring to shows what happens when you treat square roots of negative numbers with the same rules as positive numbers. Here is another example if you're still not convinced:

${\sqrt {-4}}\times {\sqrt {-9}}$

Using the correct method, we get the following

${\sqrt {-4}}\times {\sqrt {-9}}=i{\sqrt {4}}\times i{\sqrt {9}}=-6$

But if we assume that we can "combine" the square roots like we can with positive numbers, we get

${\sqrt {-4}}\times {\sqrt {-9}}={\sqrt {-4\times -9}}={\sqrt {36}}=6$

Thus, the rules for manipulating square roots are clearly different when negative numbers are involved.

And while you are correct in saying that there are two solutions for square roots of positive numbers, that fact has nothing to do with the "proof" given since none of the square roots are actually evaluated. b.y.w 09:35, 21 August 2007 (UTC)

I'm with cojoco on this. I would have written something like:

Using one method, we get the following

${\sqrt {-4}}\times {\sqrt {-9}}=i{\sqrt {4}}\times i{\sqrt {9}}=-1\times \pm 2\times \pm 3=\pm 6$

If we assume that we can "combine" the square roots like we can with positive numbers, we get

${\sqrt {-4}}\times {\sqrt {-9}}={\sqrt {-4\times -9}}={\sqrt {36}}=\pm 6$

Thus, the rules for manipulating square roots are clearly the same even when negative numbers are involved. --Twirlip 09:39, 26 August 2007 (UTC)

Are you sure? The square root function is defined to return only the positive square root (it could well have been defined to return the negative square root). True, 36 has two square roots, plus or minus 6, but ${\sqrt {36}}=6$. It's also difficult to do arithmetic with plus/minus signs, because it doesn't always stay a plus/minus. For example, $\cos {A\pm B}=\cos {A}\cos {B}\mp \sin {A}\sin {B}$. If the plus/minus is a plus, the minus/plus is a minus, and vice-versa. And $\pm 6=-1\times \mp 6$ so the two statements above actually give different, not same, results. x42bn6 Talk Mess 16:16, 26 August 2007 (UTC)

Twirlip, you're right in that you can find 2 numbers that, when squared, equal 36, namely 6 and -6. However, as x42bn6 said above, the square root of 36 is just 6. If the square root function returned 2 answers, then it would not be a function, since it is no longer "one to one." (Just visualize the graph of f(x) = sqrt(x)). And just to elaborate on what x42bn6 said, let's assume that the square root function does indeed give 2 answers.

From the first method, we could say

$i{\sqrt {4}}\times i{\sqrt {9}}=-1\times {\sqrt {36}}=-1\times \pm 6=\mp 6$

Since the first and second methods supposedly give the same answers, we can thus conclude

$\pm 6=\mp 6$

So either way, the two methods produce different results. b.y.w 20:29, 27 August 2007 (UTC)

Proof that 2 = 0

How fares this proof that 2 = 0?

2 = 1 + 1

2 = 1 + 1^1/2

2 = 1 + ((−1) × (−1))^1/2

2 = 1 + ((−1)^1/2 × (−1)^1/2)

2 = 1 + ((−1)^1/2)²

2 = 1 + (−1)^1/2×2

2 = 1 + (−1)¹

2 = 1 − 1 = 0

—Preceding unsigned comment added by 84.48.198.46 (talk • contribs) 21:11, 23 October 2007

We already have this one in the article, stated as a proof that 1 = −1. --Zundark 21:22, 23 October 2007 (UTC)