75799779 Mathematical Fallacies

8/12/2019 75799779 Mathematical Fallacies

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F ALLACIES

A fallacy is an error in reasoning that results in an invalid argument

Three common fallacies:

Vague or ambiguous premises

Begging the question (assuming what is to be proved)

Jumping to conclusions without adequate grounds

It is possible for a valid argument to have false conclusion and for an

invalid argument to have a true conclusion.



The Mistake, the How ler and th e Fallacy

Perhaps the fi rst really prom inent modern treatment of the sub ject

was Fallacies in mathematic s (1963) by E. A. Maxwell. He drew a dist inc tion

between a simp le mistake, a howler and a m athematical fallacy. He dismiss ed

mere mistakes as being "o f l i t t le interest and, one hopes, of even less

consequence" .

.

Maxwel l had mo re to say on what he cal led "fal lacies" and "h owlers" :

"The how ler in mathematics is not easy to descr ibe, but the term may be used

to denote an error wh ich leads innoc ently to a correct resul t . By c ontrast,

the fal lacy leads by g ui le to a wro ng bu t plausible conclu sion.



A correct result obtained by an incorrect line of reasoning is an

example of a mathematical argument that 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 the middle step. Bogus proofs

constructed to produce a correct result in spite of incorrect

logic are described as howlers by Maxwell.[5]

Howlers

http://en.wikipedia.org/wiki/Truth

http://en.wikipedia.org/wiki/Validity

http://en.wikipedia.org/wiki/Mathematical_fallacies

http://en.wikipedia.org/wiki/Mathematical_fallacies

http://en.wikipedia.org/wiki/Validity

http://en.wikipedia.org/wiki/Truth



HOW AND WHERE IS FALLACY FOUND ?

The tradit ional w ay of pr esenting a mathematical fal lacy is to give an

inval id step of d educt ion m ixed in w ith val id steps. Pseudaria, an

ancient lost boo k of false proofs, is attr ibuted to Eucl id .

Mathematical fal lacies exist in many branches o f mathematics.

In elementary algebra , typical examples may involve a step where

div is ion by zero is performed, where a root is incorr ect ly extracted or,

more generally, wh ere different values o f a mu lt ip le valued

funct ion are equated. Well-know n fallacies also exist in elementary

Eucl idean geometry and calculus .

http://en.wikipedia.org/wiki/Euclid

http://en.wikipedia.org/wiki/Elementary_algebra

http://en.wikipedia.org/wiki/Root_of_a_function

http://en.wikipedia.org/wiki/Multiple_valued_function


http://en.wikipedia.org/wiki/Calculus

http://en.wikipedia.org/wiki/Calculus



http://en.wikipedia.org/wiki/Root_of_a_function

http://en.wikipedia.org/wiki/Elementary_algebra

http://en.wikipedia.org/wiki/Euclid



V ARIABLE AMBIGUITY

The problem here is that x is not a true variable. It is actually aconstant, so that its derivative should also be zero.

Moral: Know your variables from your constants!

http://en.wikipedia.org/wiki/Mathematical_fallacy











NEGATIVE ROOTS

1=11=sqrt(1)

1=sqrt( (-1) * (-1) )1=sqrt(-1) * sqrt(-1)

1=i * i1=-1

What went wrong?



NEGATIVE ROOTS

What went wrong?

This is a tougher one! The problem is that for sqrt(x*y) to be

equal to sqrt(x)*sqrt(y), one or both numbers must bepositive. In this case, that wasn't true.












MULTIVALUED COMPLEX LOGARITHMS

From Euler’s Theorem ,we have that

The mistake is that the rule ln(e x ) = x is in general only valid for real x , not forcomplex x . The complex logarithm is actually multi-valued;

http://en.wikipedia.org/wiki/Complex_logarithm

http://en.wikipedia.org/wiki/Complex_logarithm



PROOF THAT 1=2

a=b=1a2=ab

a2-b2=ab-b2 (a+b)(a-b)=b(a-b)

(a+b)=b2=1

What's wrong with this?



PROOF THAT 1=2

What's wrong with this?

The step where (a-b) was eliminated was not mathematically

logical, since (a-b)=0 and dividing by zero is not generallymathematically defined.

Moral: Be careful with your algebra!















COMPLEX ROOTS

x2 + x + 1 = 0x2 = -x -1x=-1 - 1/xSubstitute x into the initial equation

x2

+ (-1 - 1/x) + 1 = 0x2 - 1/x = 0x2=1/xx3 =1x=1

Substitute this x into the initial equation12 + 1 + 1 = 03 = 0

What went wrong?



COMPLEX ROOTS

The problem is that x3=1 really has three roots, and the one chosen,x=1, was an extraneous solution given the previous mathematicalcontext.

Moral: Make sure your solution is a real mathematical and physicalsolution.





http://en.wikipedia.org/wiki/Extraneous_solution

http://en.wikipedia.org/wiki/Extraneous_solution








INFINITE SERIES ASSOCIATIVE LAW



What is wrong ?

The error here is that the associative law cannot be applied freely to aninfinite sum unless the sum is absolutely convergent (see also conditionallyconvergent). Here that sum is 1 − 1 + 1 − 1 + · · ·, a classic divergent series.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 ofpartial sums 1, 1, 1, ... (which converges to 1), so these expressions need notbe equal. This can be seen as a counterexample to generalizing Fubini'stheorem and Tonelli's theorem to infinite integrals (sums) over measurablefunctions taking negative values.

http://en.wikipedia.org/wiki/Associative_law

http://en.wikipedia.org/wiki/Infinite_sum

http://en.wikipedia.org/wiki/Absolutely_convergent

http://en.wikipedia.org/wiki/Conditionally_convergent


http://en.wikipedia.org/wiki/Grandi's_series

http://en.wikipedia.org/wiki/Equality_(mathematics)

http://en.wikipedia.org/wiki/Fubini's_theorem








http://en.wikipedia.org/wiki/Equality_(mathematics)

http://en.wikipedia.org/wiki/Grandi's_series



http://en.wikipedia.org/wiki/Absolutely_convergent

http://en.wikipedia.org/wiki/Infinite_sum

http://en.wikipedia.org/wiki/Associative_law



DIVERGENT SERIES



ANOTHER FORM OF THE PREVIOUS FALLACY



INDETERMINATE INTEGRALS



WHAT’S WRONG WITH IT ?

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.

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

still an antiderivative of ); 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 last line, 1 would be evaluated between some bounds, which would

always evaluate to 1 − 1 = 0. The remaining definite integrals on both sides would

indeed be equal.

http://en.wikipedia.org/wiki/Arbitrary_constant_of_integration

http://en.wikipedia.org/wiki/1_(number)

http://en.wikipedia.org/wiki/Definite_integral



http://en.wikipedia.org/wiki/Definite_integral


http://en.wikipedia.org/wiki/Arbitrary_constant_of_integration



POSITIVE AND NEGATIVE ROOTS

Invalid proofs utilizing powers and roots are often of the following kind

The fallacy is that the rule is generally valid only if at least one of the twonumbers x or y is positive, which is not the case here.



WHAT’S WRONG WITH IT ??

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

equation of the form

x 2 = a

2 has two solutions, provided a ≠ 0,

x=+- a.

and it is essential to check which of these solutions is relevant to the problem

at hand. In the above fallacy, the square root that allowed the second

equation to be deduced from the first is valid only when cos x is positive. In

particular, when x is set to π, the second equation is rendered invalid.



EXTRANEOUS SOLUTIONS Let’s attempt to solve the equation



WHAT’S WRONG WITH IT ?

In the forward direction, the argument merely shows that no x exists satisfying the

given equation. If you work backward from x = 2, taking the cube root of both sides

ignores the possible factors of which are non-principal cube roots of negative 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 produce more solutions. This

is indeed the case in this example, where the solution x = 2 is arrived at while it is

clear that this is not a solution to the original equation. Also, every number has 3 cube

roots, 2 complex and one either real or complex. Also the substitution of the firstequation into the second to get the third would be begging the question when working

backwards.

http://en.wikipedia.org/wiki/Begging_the_question

http://en.wikipedia.org/wiki/Begging_the_question



Thank You

(P.S. Wake ur neighbor up)

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75799779 Mathematical Fallacies