Invalid proof

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In mathematics, there are a variety of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle. These fallacies are normally regarded as mere curiosities, but can be used to show the importance of rigor in mathematics.

Most of these proofs depend on some variation of the same error. The error is to take a function f that is not one-to-one, to observe that f(x) = f(y) for some x and y, and to (erroneously) conclude that therefore x = y. Division by zero is a special case of this; the function f is x → x × 0, and the erroneous step is to start with x × 0 = y × 0 and to conclude that therefore x = y.

Contents 1 Examples 1.1 Proof that 1 equals −1 1.2 Proof that 1 is less than 0 1.3 Proof that 2 equals 1 1.4 Proof that a equals b 1.5 Proof that 0 equals 1 1.5.1 Alternative proof 1.6 Another proof that 2 equals 1 1.7 Proof that 4 equals 5 1.8 Proof that any angle is zero 2 Conclusion 3 See also 4 External links

Examples

Proof that 1 equals −1

• We start with

  <math>-1 = -1</math>

• Then we convert these into vulgar fractions

  <math>\frac{1}{-1} = \frac{-1}{1}</math>

• Applying square roots on both sides gives

  <math>\sqrt{\frac{1}{-1}} = \sqrt{\frac{-1}{1}}</math>

• Which is equal to

  <math>\frac{\sqrt{1}}{\sqrt{-1}} = \frac{\sqrt{-1}}{\sqrt{1}}</math>

• If we now clear fractions by multiplying both sides by <math>\sqrt{-1}</math> and then <math>\sqrt{1}</math>, we have

  <math>\sqrt{1}\sqrt{1} = \sqrt{-1}\sqrt{-1}</math>

• But any number's square root squared gives the original number, so

  <math>1 = -1</math>

Q.E.D.

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

<math>\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}</math>

This principle is only correct when y is a positive number. In the "proof" above, this is not the case. Thus the proof is invalid.

Proof that 1 is less than 0

• Let us first suppose that

  <math>0 < x < 1</math>

• Now we will take the logarithm on both sides. As long as x > 0, we can do this because logarithms are monotonically increasing. Observing that the logarithm of 1 is 0, we get

  <math>\ln (x) < 0</math>

• Dividing by ln (x) gives

  <math>1 < 0</math>

Q.E.D.

The violation is found in the last step, the division. This step is wrong because the number we are dividing by is negative as can clearly be seen in the previous line. A multiplication with or division by a negative number flips the inequality sign; in other words, we should obtain 1 > 0, which is indeed correct.

Proof that 2 equals 1

• Let a and b be equal non-zero quantities

  <math>a = b</math>

• Multiply through by a

  <math>a^2 = ab</math>

• Subtract <math>b^2</math>

  <math>a^2 - b^2 = ab - b^2</math>

• Factor both sides

  <math>(a - b)(a + b) = b(a - b)</math>

• Divide out <math>(a - b)</math>

  <math>a + b = b</math>

• Observing that <math>a = b</math>

  <math>b + b = b</math>

• Combine like terms on the left

  <math>2b = b</math>

• Divide by the non-zero b

  <math>2 = 1</math>

Q.E.D.

The fallacy is in line 5: the progression from line 4 to line 5 involves division by <math>(a - b)</math>, which is zero since a equals b. Since division by zero is undefined, the argument is invalid.

A variation:

• Let x and y be equal, non-zero quantities

  <math>x = y</math>

• Add x to both sides

  <math>2x = x + y</math>

• Take 2y from both sides

  <math>2x - 2y = x - y</math>

• Factorise

  <math>2(x - y) = x - y</math>

• Divide out <math>(x - y)</math>

  <math>2 = 1</math>

Q.E.D.

The fallacy here is the same as above in that by dividing by <math>(x - y)</math>, you are dividing by zero and as such, this argument is invalid.

Proof that a equals b

• We start with:

  <math>a - b = c</math>

• Now, square both sides:

  <math>a^2 - 2ab + b^2 = c^2</math>

• Since <math>a - b = c</math>, substitute:

  <math>a^2 - 2ab + b^2 = (a - b)c</math>

• Write out the multiplication:

  <math>a^2 - 2ab + b^2 = ac - bc</math>

• Rearranging all, we get:

  <math>a^2 - ab - ac = ab - b^2 - bc</math>

• Factorize both members:

  <math>a(a - b - c) = b(a - b - c)</math>

• Cancel the common factor:

  <math>a = b</math>

Q.E.D.

The catch is that since a − b = c, a − b − c = 0, and as a result we have performed an illegal division by zero.

Proof that 0 equals 1

• Start with the addition of an infinite succession of zeros

  <math>0 = 0 + 0 + 0 + \cdots</math>

• Then recognize that <math>0 = 1 - 1</math>

  <math>0 = (1 - 1) + (1 - 1) + (1 - 1) + \cdots</math>

• Applying the associative law of addition results in

  <math>0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \cdots</math>

• Of course <math>-1 + 1 = 0</math>

  <math>0 = 1 + 0 + 0 + 0 + \cdots</math>

• And the addition of an infinite string of zeros can be discarded leaving

  <math>0 = 1</math>

Q.E.D.

The error here is that the associative law cannot be applied freely to an infinite sum unless the sum would converge without any parentheses. In this particular argument, the second line gives the sequence of partial sums 0, 0, 0, ... (which converges to 0) while the third line gives the sequence of partial sums 1, 1, 1, ... (which converges to 1), so it is unclear in what sense these expressions can be considered equal.

Alternative proof

• Begin with the evaluation of the integral

  <math>\int \frac{1}{x} dx</math>

• Through integration by parts, let

  <math>u=\frac{1}{x}</math>

  <math>dv = dx</math>

• Thus,

  <math>du=-\frac{1}{x^2}dx</math>

  <math>v = x</math>

• Hence, by integration by parts

  <math>\int \frac{1}{x} dx=\frac{x}{x} - \int \left ( - \frac{1}{x^2} \right ) x dx</math>

  <math>\int \frac{1}{x} dx=1 + \int \frac{1}{x} dx</math>

  <math>0 = 1</math>

Q.E.D.

The error in this proof lies in an improper use of the integration by parts technique. Upon use of the formula, a constant, C, must be added to the right-hand side of the equation. This is due to the derivation of the integration by parts formula; the derivation involves the integration of an equation and so a constant must be added. In most uses of the integration by parts technique, this initial addition of C is ignored until the end when C is added a second time. However, in this case, the constant must be added immediately because the remaining two integrals cancel each other out.

Another proof that 2 equals 1

• By the common intuitive meaning of multiplication we can see that

  <math>4 \times 3 = 3 + 3 + 3 + 3</math>

• It can also be seen that for a non-zero x

  <math>x = 1 + 1 + \cdots + 1</math> (x terms)

• Now we multiply through by x

  <math>x^2 = x + x + \cdots + x</math> (x terms)

• Then we take the derivative with respect to x

  <math>2x = 1 + 1 + \cdots + 1</math> (x terms)

• Now we see that the right hand side is x which gives us

  <math>2x = x</math>

• Finally, dividing by our non-zero x we have

  <math>2 = 1</math>

Q.E.D.

The error: In line two our definition of x assumed that x was an integer; this equation is not meaningful for non-integer real numbers. Functions are only differentiable on a continuous space such as the reals, not on integers. For any particular integer x, you get a true equation. But to differentiate both sides you need an equation of functions, not an equation of integers. The right-hand function <math>x + x + \cdots + x</math> "with x terms" is not a meaningful function on the reals (how can you have x terms?) and thus not differentiable.

Proof that 4 equals 5

• Start with the identity

  <math>-20 = -20</math>

• Express both sides in slightly different, yet equivalent ways

  <math>25 - 45 = 16 - 36</math>

• Factor both sides

  <math>5^2 - 5 \times 9 = 4^2 - 4 \times 9</math>

• Add the same thing to both sides

  <math>5^2 - 5 \times 9 + \frac{81}{4} = 4^2 - 4 \times 9 + \frac{81}{4}</math>

• Now factor both sides again

  <math>\left(5 - \frac{9}{2}\right)^2 = \left(4 - \frac{9}{2}\right)^2</math>

• Square root both sides

  <math>5 - \frac{9}{2} = 4 - \frac{9}{2}</math>

• Cancel the common factor

  <math>5 = 4</math>

Q.E.D.

The error in the proof comes from the fact that <math>x^2 = y^2</math> does not imply that <math>x = y</math>. The arithmetic up until this point is correct, and in fact

<math>-\left(5 - \frac{9}{2}\right) = 4 - \frac{9}{2}.</math> It is also important to note that if we subract the term 9/2 from 4, we end up with -1/2. If we then square the term, we arrive at a positive 1/4th. The next logical mathematical step is to take the square root of both sides. If we do this, we can see that 1/2 is equal to 1/2. The original problem of -20=-20 does in fact result in a correct identity if the problem is worked out in the proper way.

Proof that any angle is zero

Image:Angle fallacy.PNG Construct a rectangle ABCD. Now identify a point E such that CD=CE and the angle DCE is a non-zero angle. Take the perpendicular bisector of AD, crossing at F, and the perpendicular bisector of AE, crossing at G. Label where the two perpendicular bisectors intersect as H and join this to A, B, C, D, and E.

Now, AH=DH because FH is a perpendicular bisector; similarly BH=CH. AH=EH because FH is a perpendicular bisector, so DH=EH. And by construction BA=CD=CE. So the triangles ABH, DCH and ECH are congruent, and so the angles ABH, DCH and ECH are equal.

But if the angles DCH and ECH are equal then the angle DCE must be zero.

Q.E.D.

The error in the proof comes in the diagram and the final point. An accurate diagram would show that the triangle ECH is a reflection of the triangle DCH in the line CH rather than being on the same side, and so while the angles DCH and ECH are equal in magnitude, there is no justification for subtracting one from the other; to find the angle DCE you need to subtract the angles DCH and ECH from the angle of a full circle (2π or 360°).

Conclusion

These arguments do constitute valid proofs, but not of the claimed assertions. For example, there is no a priori reason why division by zero should be defined (it's not a field axiom, for example, though 1 ≠ 0, from which 2 ≠ 1 follows, is an axiom), and the "proof" that 2 = 1 is, in fact, simply a demonstration that division by zero cannot be defined in general. A proof that division by zero could be defined would demonstrate a contradiction and show that the axiomatic system we are working under is logically inconsistent!

On the other hand it is possible to construct useful mathematical systems where 1 is equivalent to 2. Mathematics in domains modulo 1 are one example. In such domains <math>0.5 + 0.5 = 0 = 1 = 2\dots .</math>

See also

• Paradox

External links

• Audience of Two's proof that 28 is the largest number
• More invalid proofs from AhaJokes.comes:Demostración inválida