Mathematical Fallacy - Wikipedia the Free Encyclopedia

Mathematical fallacy

From Wikipedia, the free encyclopedia

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, asillustrations of a concept of mathematical fallacy. There is a distinction between a simple mistakeand a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the sameway, but in the best-known examples of mathematical fallacies, there is some concealment in thepresentation of the proof. For example the reason validity fails may be a division by zero that ishidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically

presented, it leads not only to an absurd result, but does so in a crafty or clever way.[1]

Thereforethese fallacies, for pedagogic reasons, usually take the form of spurious proofs of obviouscontradictions. Although the proofs are flawed, the errors, usually by design, are comparativelysubtle, or designed to show that certain steps are conditional, and should not be applied in the casesthat are the exceptions to the rules.

The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixedin with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy.The latter applies normally to a form of argument that is not a genuine rule of logic, where theproblematic mathematical step is typically a correct rule applied with a tacit wrong assumption.Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (such as the

introduction of Pasch's axiom of Euclidean geometry).[2]

Pseudaria, an ancient lost book of false

proofs, is attributed to Euclid.[3]

Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typicalexamples may involve a step where division by zero is performed, where a root is incorrectlyextracted or, more generally, where different values of a multiple valued function are equated.Well-known fallacies also exist in elementary Euclidean geometry and calculus.

Contents

1 Howlers

2 Division by zero

2.1 All numbers equal zero

2.2 All numbers equal all other numbers

2.3 Determinants equal to zero

3 Multivalued functions

3.1 Multivalued sinusoidal functions

3.2 Multivalued complex logarithms

4 Calculus

4.1 Indefinite integrals

4.2 Variable ambiguity

5 Infinite series

5.1 Associative law

5.2 Divergent series

6 Power and root

6.1 Positive and negative roots

6.2 Extraneous solutions

6.3 Complex roots

6.4 Fundamental theorem of algebra

Mathematical fallacy - Wikipedia, the free encyclo... http://en.wikipedia.org/wiki/Mathematical_fallacy

1 of 14 Wednesday 08 September 2010 05:34 PM

7 Inequalities

8 Geometry

8.1 Any angle is zero

8.2 Fallacy of the isosceles triangle

9 See also

10 Notes

11 References

12 External links

Howlers

A correct result obtained by an incorrect line of reasoning is an example of a mathematical argumentthat is true but invalid. This is the case, for instance, in the calculation

Although the conclusion 16/64 = 1/4 is correct, there is a fallacious invalid cancellation in themiddle step. Bogus proofs constructed to produce a correct result in spite of incorrect logic are

known as howlers.[4]

Division by zero

The division-by-zero fallacy has many variants.

All numbers equal zero

The following example[5]

uses division by zero to "prove" that 1 = 0, but can be modified to "prove"that any real number equals zero.

The error here is in going from the second to the third line. The reasoning is started by assuming xto be equal to 0; therefore, on the second to the third line: if x = 0, then division by x isarithmetically meaningless.

All numbers equal all other numbers

The following example uses division by zero to "prove" that 2 = 1, but can be modified to prove thatany number equals any other number.

1. Let a and b be equal non-zero quantities

2. Multiply through by a



3. Subtract

4. Factor both sides

5. Divide out

6. Observing that

7. Combine like terms on the left

8. Divide by the non-zero b

Q.E.D.[6]

The fallacy is in line 5: the progression from line 4 to line 5 involves division by a − b, which is zerosince a equals b. Since division by zero is undefined, the argument is invalid. Deriving that the onlypossible solution for lines 5, 6, and 7, namely that a = b = 0, this flaw is evident again in line 7,where one must divide by b (0) in order to produce the fallacy (not to mention that the only possiblesolution denies the original premise that a and b are nonzero). A similar invalid proof would be to saythat since 2 × 0 = 1 × 0 (which is true), one can divide by zero to obtain 2 = 1. An obviousmodification "proves" that any two real numbers are equal.

Many variants of this fallacy exist. For instance, it is possible to attempt to "repair" the proof bysupposing that a and b have a definite nonzero value to begin with, for instance, at the outset onecan suppose that a and b are both equal to one:

However, as already noted the step in line 5, when the equation is divided by a - b, is still division byzero. As division by zero is undefined, the argument is invalid.

Determinants equal to zero

[citation needed]

A "proof" that all numbers are equal to zero follows. Suppose we have the following system of linearequations:

Dividing the first equation by c1, we get x1 + x2 + · · · + xn = 1. Let us now try to solve the system

via Cramer's rule:



Since each column of the coefficient matrix is equal to the resultant column vector, we have

for all i. Substituting this back into x1 + x2 + · · · + xn = 1, we get

.

Q.E.D.

This proof is fallacious because Cramer's rule can only be applied to systems with a unique solution;however, all the equations in the system are obviously equivalent, and thus are insufficient toprovide a unique solution. The fallacy occurs when we try to divide |Ai| by |A|, as both are equal to 0.

Multivalued functions

Functions that are multivalued have poorly defined inverse functions over their entire range.

Multivalued sinusoidal functions

[citation needed]

x = 2π

sin(x) = 0

x = arcsin(0)

x = 0

2π = 0

Q.E.D.

The problem is in the third step, where we take the arcsin of each side. Since the arcsin is an

infinitely multivalued function, x = arcsin(0) is not necessarily true.

Multivalued complex logarithms

[citation needed]

From Euler's formula we see that

and



so we have

Taking logarithms gives

and hence

Dividing by πi gives

Q.E.D.

The mistake is that the rule ln(ex) = x is in general only valid for real x, not for complex x. The

complex logarithm is actually multi-valued; and ln( − 1) = (2k + 1)πi for any integer k, so we

see that πi and 3πi are two among the infinite possible values for ln(-1).

Calculus

Calculus as the mathematical study of infinitesimal change and limits can lead to mathematicalfallacies if the properties of integrals and differentials are ignored.

Indefinite integrals

[citation needed]

The following "proof" that 0 = 1 can be modified to "prove" that any number equals any othernumber. Begin with the evaluation of the indefinite integral

Through integration by parts, let

and dv = dx

Thus,

and v = x

Hence, by integration by parts

Q.E.D.



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

In other words, the second to last line is correct (1 added to any antiderivative of 1/x is still an

antiderivative of 1/x); but the last line is not. You cannot cancel because they are not

necessarily equal. There are infinitely many antiderivatives of a function, all differing by a constant.In this case, the antiderivatives on both sides differ by 1.

This problem can be avoided if we use definite integrals (i.e. use bounds). Then in the second to lastline, 1 would be evaluated between some bounds, which would always evaluate to 1 - 1 = 0. Theremaining definite integrals on both sides would indeed be equal.

Variable ambiguity

[citation needed]

Take the statement

x = 1

Taking the derivative of each side,

The derivative of x is 1, and the derivative of 1 is 0. Therefore,

Q.E.D.

The error in this proof is it treats x as a variable, and not as a constant as stated with x = 1 in theproof, when taking its derivative. Taking the proper derivative of x yields the correct result, 0 = 0.

Infinite series

As in the case of surreal and aleph numbers, mathematical situations which involve manipulation ofinfinite series can lead to logical contradictions if care is not taken to remember the properties ofsuch series.

Associative law

[citation needed]

Start with the addition of an infinite succession of zeros

Then recognize that 0 = 1 − 1

Applying the associative law of addition results in



Of course − 1 + 1 = 0

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

Q.E.D.

The error here is that the associative law cannot be applied freely to an infinite sum unless the sumis absolutely convergent (see also conditionally convergent). Here that sum is 1 − 1 + 1 − 1 + · · ·, aclassic divergent series. In this particular argument, the second line gives the sequence of partialsums 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 these expressions need not be equal. This can be seen as acounterexample to generalizing Fubini's theorem and Tonelli's theorem to infinite integrals (sums)over measurable functions taking negative values.

In fact the associative law for addition just states something about three-term sums:

(a + b) + c = a + (b + c). It can easily be shown to imply that for any finite sequence of terms

separated by "+" signs, and any two ways to insert parentheses so as to completely determine whichare the operands of each "+", the sums have the same value; the proof is by induction on the numberof additions involved. In the given "proof" it is in fact not so easy to see how to start applying thebasic associative law, but with some effort one can arrange larger and larger initial parts of the firstsummation to look like the second. However this would take an infinite number of steps to "reach"the second summation completely. So the real error is that the proof compresses infinitely manysteps into one, while a mathematical proof must consist of only finitely many steps. To illustrate this,

consider the following "proof" of 1 = 0 that only uses convergent infinite sums, and only the law

allowing to interchange two consecutive terms in such a sum, which is definitely valid:

Divergent series

[citation needed]

Define the constants S and A by

.

Therefore



Adding these two equations gives

Therefore, the sum of all positive integers is negative.

A variant of mathematical fallacies of this form involves the p-adic numbers[citation needed]

Let .

Then, .

Subtracting these two equations produces.

Therefore, .

The error in such "proofs" is the implicit assumption that divergent series obey the ordinary laws ofarithmetic.

Power and root

Fallacies involving disregarding the rules of elementary arithmetic through an incorrectmanipulation of the radical. For complex numbers the failure of power and logarithm identities hasled to many fallacies.

Positive and negative roots

Invalid proofs utilizing powers and roots are often of the following kind:[7]

The fallacy is that the rule is generally valid only if at least one of the two numbers x

or y is positive, which is not the case here.

Although the fallacy is easily detected here, sometimes it is concealed more effectively in notation.

For instance,[8]

consider the equation

cos2x = 1 − sin2x

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

cosx = (1 − sin2x)1 / 2

so that

1 + cosx = 1 + (1 − sin2x)1 / 2.

Squaring both sides gives



But evaluating this when x = π implies

or

0 = 4

which is absurd.

The error in each of these examples fundamentally lies in the fact that any equation of the form

x2 = a2

has two solutions, provided a ≠ 0,

and it is essential to check which of these solutions is relevant to the problem at hand.[9]

In theabove fallacy, the square root that allowed the second equation to be deduced from the first is validonly when cos x is positive. In particular, when x is set to π, the second equation is rendered invalid.

Another example of this kind of fallacy, where the error is immediately detectable, is the followinginvalid proof that −2 = 2. Letting x = −2, and then squaring gives

whereupon taking a square root implies

so that x = −2 = 2, which is absurd. Clearly when the square root was extracted, it was the negativeroot −2, rather than the positive root, that was relevant for the particular solution in the problem.

Alternatively, imaginary roots are obfuscated in the following:

The error here lies in the equality, where we are ignoring the other fourth roots of 1,[10]

which are−1, i and −i (where i is the imaginary unit). Seeing as we have squared our figure and then takenroots, we cannot always assume that all the roots will be correct. So the correct fourth are i and −i,which are the imaginary numbers defined to be .

Extraneous solutions

[citation needed]

Start by attempting to solve the equation

Taking the cube of both sides yields



Replacing the expression within parenthesis by the initial equation and canceling common termsyields

Taking the cube again produces

Which produces the solution x = 2. Substituting this value into the original equation, one obtains

So therefore

Q.E.D.

In the forward direction, the argument merely shows that no x exists satisfying the given equation. Ifyou work backward from x=2, taking the cube root of both sides ignores the possible factors of

which are non-principal cube roots of one. An equation altered by raising both sides to a

power is a consequence, but not necessarily equivalent to, the original equation, so it may producemore solutions. This is indeed the case in this example, where the solution x = 2 is arrived at while itis clear that this is not a solution to the original equation. Also, every number has 3 cube roots, 2complex and one either real or complex. Also the substitution of the first equation into the second toget the third would be begging the question when working backwards.

Complex roots

[citation needed]

Assume the following equation for a complex x :

Then:

Divide by x (assume x is not 0)

Substituting the last expression for x in the original equation we get:



Substituting x=1 in the original equation yields:

Q.E.D.

The fallacy here is in assuming that x3 = 1 implies x = 1. There are in fact three cubed roots of

unity. Two of these roots, which are complex, are the solutions of the original equation. Thesubstitution has introduced the third one, which is real, as an extraneous solution. The equationafter the substitution step is implied by the equation before the substitution, but not the other wayaround, which means that the substitution step could and did introduce new solutions.

Note that if we restrict our attention to the real numbers, so that x3 = 1 implies x = 1 is in fact

true, the proof will still fail. In fact, what it is saying is that if there is some real number solution tothe equation, then 0=3. This is true, but it is a big "if". The quadratic formula erases any lingeringfears that 0=3 by showing that no real number solution exists.

Fundamental theorem of algebra

The fundamental theorem of algebra applies for any polynomial defined for complex numbers.

Ignoring this can lead to mathematical fallacies.[citation needed]

Let x and y be any two numbers Thenlet Let Let's compute:

Replacing , we get:

Let's compute Replacing :

So:

Replacing :



Q.E.D.

The mistake here is that from z³ = w³ one may not in general deduce z = w (unless z and w are bothreal, which they are not in our case).

Inequalities

Inequalities can lead to mathematical fallacies when operations that fail to preserve the inequality

are not recognized.[citation needed]

Let us first suppose that

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

Dividing by ln (x) gives

Q.E.D.

The violation is found in the last step, the division. This step is invalid because ln(x) is negative for 0< x < 1. While multiplication or division by a positive number preserves the inequality,multiplication or division by a negative number reverses the inequality, resulting in the correctexpression 1 > 0.

Geometry

Many mathematical fallacies based on geometry come from assuming a geometrically impossiblesituation. Often the fallacy is easy to expose through simple visualizations.

Any angle is zero

[citation needed]

Construct a rectangle ABCD. Now identify a point E suchthat CD = CE and the angle DCE is a non-zero angle. Takethe perpendicular bisector of AD, crossing at F, and theperpendicular bisector of AE, crossing at G. Label wherethe two perpendicular bisectors intersect as H and join thispoint to A, B, C, D, and E.

Now, AH=DH because FH is a perpendicular bisector;similarly BH = CH. AH=EH because GH is a perpendicularbisector, so DH = EH. And by construction BA = CD = CE.So the triangles ABH, DCH and ECH are congruent, and sothe 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 showthat 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 forsubtracting one from the other; to find the angle DCE you need to subtract the angles DCH and ECHfrom the angle of a full circle (2π or 360°).

Fallacy of the isosceles triangle

The fallacy of the isosceles triangle, from (Maxwell 1959, Chapter II, §1), purports to show that every triangle is isosceles, meaning that twosides of the triangle are congruent.

Given a triangle △ABC, proof that AB = AC:

Draw a line bisecting ∠A1.

Call the midpoint of line segment BC, D2.

Draw the perpendicular bisector of segment BC, which contains

D

3.

If these two lines are parallel, AB = AC; otherwise they intersect

at point O

4.

Draw line OR perpendicular to AB, line OQ perpendicular to AC5.

Draw lines OB and OC6.

By AAS, △RAO ≅ △QAO (AO = AO; ∠OAQ ≅ ∠OAR since AO bisects ∠A; ∠ARO ≅ ∠AQO are both

right angles)

7.

By SAS, △ODB ≅ △ODC (∠ODB,∠ODC are right angles; OD = OD; BD = CD because OD bisects

BC)

8.

By SAS, △ROB ≅ △QOC (RO = QO since △RAO ≅ △QAO; BO = CO since △ODB ≅ △ODC; ∠ROB ≅

∠QOC since they are vertical angles)

9.

Thus, AR ≅ AQ, RB ≅ QC, and AB = AR + RB = AQ + QC = AC10.

Q.E.D.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BCin the same way.

All but the last step of the proof is indeed correct (those three triangles are indeed congruent). Theerror in the proof is the assumption in the diagram that the point O is inside the triangle. In fact,whenever AB ≠ AC, O lies outside the triangle. Furthermore, it can be further shown that, if AB islonger than AC, then R will lie within AB, while Q will lie outside of AC (and vice versa). (Anydiagram drawn with sufficiently accurate instruments will verify the above two facts.) Because ofthis, AB is still AR + RB, but AC is actually AQ - QC; and thus the lengths are not necessarily thesame.

See also

Paradox

Fallacy

Proof by intimidation

2 + 2 = 5

List of fallacious proofs

Notes

^ Maxwell 1959, p. 91.

^ Maxwell 19592.

^ Heath & Helberg 1908, Chapter II, §I3.

^ Maxwell 19594.



^ Maxwell 1959, p. 75.

^ Harro Heuser: Lehrbuch der Analysis - Teil 1, 6th edition, Teubner 1989, ISBN

978-3835101319, page 51 (German).

6.

^ Maxwell 1959, Chapter VI, §I.27.

^ Maxwell 1959, Chapter VI, §I.18.

^ Maxwell 1959, Chapter VI, §II9.

^ In general, the expression evaluates to n complex numbers, called the nth roots of unity.10.

References

Barbeau, Edward J. (2000), Mathematical fallacies, flaws, and flimflam, MAA Spectrum,

Mathematical Association of America, MR1725831 (http://www.ams.org/mathscinet-

getitem?mr=1725831) , ISBN 978-0-88385-529-4.

Bunch, Bryan (1997), Mathematical fallacies and paradoxes, New York: Dover Publications,

MR1461270 (http://www.ams.org/mathscinet-getitem?mr=1461270) , ISBN 978-0-486-29664-7.

Heath, Sir Thomas Little; Heiberg, Johan Ludvig (1908), The thirteen books of Euclid's

Elements, Volume 1, The University Press.

Maxwell, E. A. (1959), Fallacies in mathematics, Cambridge University Press, MR0099907

(http://www.ams.org/mathscinet-getitem?mr=0099907) .

External links

Invalid proofs (http://www.cut-the-knot.org/proofs/index.shtml) at Cut-the-knot (including

literature references)

More invalid proofs from AhaJokes.com (http://www.ahajokes.com/math_jokes.html)

More invalid proofs also on this page (http://www.jokes-funblog.com/categories/49-Math-Jokes)

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Categories: Proof theory | Proofs | Logical fallacies

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