dansmath > Paradoxes and Bogus Proofs
 
 
 

 < Paradoxes and Bogus Proofs >
 
 
 
All math teachers are required by law to know a few paradoxes,
such as Xeno's Paradox, and some Bogus Proofs, like why 2 = 1.
(You might also want to check out dan's jokes page!)
 
 
Page Contents at the moment:
 
Xeno's Paradox - Tortoise and Hare
 
Whoever Knows the Least Earns the Most
 
Alexander the Great had an Infinite Number of Limbs
 
Perfect Squares Don´t Exist
 
A Freaky Infinite Series
 
An Elephant Weighs the Same as a Mosquito
 
Proof that My Dog Has Three Tails
 
Every Natural Number can be Described in 14 Words or Less
 
A Ham Sandwich is Better than Eternal Happiness
 
Ray Charles is God
 
Heaven is Hotter than Hell!
 
Proof that Girls are Evil
 
2 = 1 (bad algebra)
2 = 1 (bad calculus)
-1 = 1 (just plain bad)
1 = 0 (blame it on Euclid)
 
Proof that 4 = 5 !!!!
 
Joe's suggestions
 
Two Opposing Questions
 
Click here to suggest a paradox or bad proof!
 
 
Xeno's Paradox - Tortoise and Hare - One of the most famous mathematical paradoxes! (top)
 
A hare (rabbit) can run ten times as fast as a tortoise (turtle);
the tortoise gets a ten-yard headstart. By the time the hare
runs the 10 yards to catch up, the tortoise has crawled a yard.
When the hare makes up this yard, the tortoise has gone a tenth
of a yard, and so on. The hare can never catch the tortoise!
Variation: The phone rings across the room. Before you can walk to the phone, you have to walk
halfway there. But before you can walk 1/2 the distance you need to walk 1/4 of the distance,
and before that you have to go 1/8 the distance, and so on. So you can never answer the phone!
 
 
Whoever Knows the Least Earns the Most (the Dilbert Equation) (top)
 
Engineers and scientists can never earn as much as business execs and sales reps.
This can be supported by a mathematical theorem based on two postulates:
P1: Knowledge is Power, and P2: Time is Money.
Basic physics definition: Power = Work / Time
Substituting the postulates, Knowledge = Work / Money
A little algebra gives us: Money = Work / Knowledge
Now as Knowledge --> 0 , Money --> oo (infinity).
So for any amount of work, :Whoever knows the least earns the most!
 
Alexander the Great had an Infinite Number of Limbs - from Joel Rubin - grad school days! (top)
 
Alexander the Great was forwarned of his fate by an oracle.
Forewarned is fore-armed.
Four arms + 2 legs = 6 limbs.
Six is an odd number of limbs for a man!
Six is even.
No finite number is both odd and even
Therefore Alexander had an infinite number of limbs!
 
 
Theorem : Perfect Squares Don´t Exist - by Prof Hendrik Lenstra (top)
 
Proof : Suppose that n is a perfect square. Look at the odd divisors of n.
They all divide the largest of them, which is itself a square, say d^2.
This shows that the odd divisors of n come in pairs a, b, where ab = d^2.
Only d is paired to itself.Therefore the number of odd divisors of n is odd.
This implies that the sum of all divisors of n is also odd. In particular, it is not 2n.
Hence n is not perfect, a contradiction : Perfect squares don´t exist. QED.
 
 
A Freaky Infinite Series - from an AOL member! (top)
Let T = 1 + 2 + 4 + 8 + 16 + 32 + . . .
2T - T should equal T, but:
2T = 2+4+8+16+32+64...
- T = -(1+2+4+8+16+32+64...)
T = -1+0+0+0+00+00+00...
Therfore : -1 = 1 + 2 + 4 + 8 + 16 + 32 + . . .
 
 
An Elephant Weighs the Same as a Mosquito (top)
 
Let x = weight of an elephant, and y = weight of a mosquito.
Call the sum of the two weights 2v ; we have x + y = 2v
From this equation we obtain x - 2v = -y and x = -y + 2v
Multiplying these together gives x^2 - 2vx = y^2 - 2vy
Adding v^2 to each side: x^2 - 2vx + v^2 = y^2 - 2vy + v^2
Factor: (x - v)^2 = (y - v)^2 , take square root: x - v = y - v
Adding v to both sides gives x = y. The weights are the same!
 
 
Proof that my dog has three tails (Contributed by a reader!) (top)

A) No dog has two tails
B) One dog has one more tail than no dog
C) Two plus one is three!
 
And the corresponding Cats Have Nine Legs:
a) No cat has five legs; b) One cat has four legs; c) Add the equations!
 
A Ham Sandwich is Better than Eternal Happiness (Contributed by a reader!) (top)
 
A ham sandwich is better than nothing...
Nothing is better than eternal happiness...
Therefore...
A ham sandwich is better than eternal happiness.
 
 
Ray Charles is God (from Patrick in Arizona) (top)

P1: God is Love - G = L
P2: Love is blind - L = b
substituting: G = b
Fact: Ray Charles is blind,
therefore Ray Charles is God.
 
 
Heaven is Hotter than Hell! (from a Colorado physics class) (top)
A reader shows what happens if you mix data from scientific and biblical sources!

The temperature of Heaven can be rather accurately computed. The Bible says,
in Isaiah 30:26, that heaven receives the light of the moon and: "Moreover the
light of the moon should be as the light of the sun and the light of the sun should
be sevenfold, as the light of seven days." Thus Heaven receives from the moon
as much radiation as we do from the sun and in addition, seven times seven (49)
times as much as the Earth does from the sun, or a total of 50 times in all.
Now to compute the temperature H of Heaven: The radiation falling on Heaven
will heat it to the point where the heat lost by radiation is just equal to the heat
received by radiation. In other words, Heaven loses 50 times as much heat as
the Earth by radiation. Using the Stefan-Boltzmann 4th power law for radiation
(H/E)^4 = 50 where E is the absolute temperature of the Earth -300 degrees
Kelvin. This gives H as 798 degrees Kelvin or 525 degrees Celsius.
The exact temperature of Hell can not be computed from existing data, but it
must be less than 444.6 degrees C, the temperature at which brimstone or sulfur
changes from a liquid to a gas. Revelations 21:8 says: "But the fearful, the unbe-
lieving... shall have their part in the lake which burneth with fire and brimstone."
A lake of molten brimstone means its temperature must be below the boiling point,
444.6 C; above this temperature it would be a vapor and - whoops, no more lake.
Summary: Heaven is 525 C and Hell is less than 444.6 C.
Therefore, Heaven is Hotter than Hell. I rest my case.
 
 
Proof that Girls are Evil (Contributed by my student Victoria) (top)
(Dan's note: I don't take responsibility for any sexist overtones here)
 
First we state that girls require time and money
Girls = Time * Money
And we all know that "time is money"
Time = Money
Therefore
Girls = Money * Money = (Money)^2
And because "money is the root of all evil"
Money = (Evil)
Therefore
Girls = (Evil)^2
And we are forced to conclude that
Girls = Evil
 
 
Every Natural Number can be Unambiguously
Described in 14 Words or Less (top)
(from U. of Toronto Math Network)
 
1. Suppose there is some natural number which cannot be
unambiguously described in fourteen words or less.
2. Then there must be a smallest such number. Let's call it n.
3. But now n is "the smallest natural number that cannot be
unambiguously described in fourteen words or less".
4. This is a complete and unambiguous description of n in
fourteen words, contradicting the fact that n was
supposed not to have such a description!
5. Since the assumption (step 1) of the existence of a natural
number that can't be unambiguously described in fourteen words
or less led to a contradiction, it must be an incorrect assumption.
6. Therefore, all natural numbers can be unambiguously
described in fourteen words or less!
 
Proof that 2 = 1 (bad algebra method) (top)
 a = b  Let's pick two equal numbers, a and b.
 a^2 = a b  Multiply both sides by a.
 a^2 - b^2 = a b - b^2  Subtract b^2 from both sides.
 (a - b)(a + b) = b (a - b)  Factor each side using algebra.
 a + b = b  Cancel the common factor on both sides
 b + b = b ; 2b = b  Substitute a = b (step 1) and simplify
 2 = 1  Divide both sides by b and voila!
Proof that 2 = 1 (bad calculus method) (top)
 x^2 = x + x + . . . + x  With x terms on the right, x^2 = x * x.
 2 x = 1 + 1 + . . . + 1  Take the derivative of each side.
 2 x = x  We added up the x 1's on the right.
 2 = 1  Divide both sides by x. Whoa!
Proof that -1 = 1 (just plain bad) (top)
Hey Dan, Nice bogus proofs on your web page. Have you seen this one! -- Mike
Dan's note -- Yes Mike, in a slightly different form. I supplied the 'reasons'.
 -1 = -1  Reflexive property a = a
 -1 / 1 = 1 / -1  Rewriting the -1 in two ways
 (-1/1) = (1/-1)  Take square root of both sides
 (-1) / (1) = (1)/(-1)  Quotient law of square roots
 i / 1 = 1 / i  Definition of i = (-1)
 i^2 = 1  Cross multiplying
 -1 = 1  Because i^2 = -1 by def'n.
Proof that 1 = 0 (blame Euclid for this) (top)
This one comes from www.cut-the-knot.org
 
 (n+1)^2 = n^2+2n+1  FOIL or expand binomial square
 (n+1)^2 - (2n+1) = n^2 Subtraction Principle of Equality
 (n+1)^2 - (2n+1) - n(2n+1)
= n^2 - n(2n+1)
 Ditto (a.k.a. Euclid's Common Notion)
 (n+1)^2 - (n+1)(2n+1) =
n^2 - n(2n+1)
 Plain old expanding and factoring
 (n+1)2-(n+1)(2n+1)+(2n+1)^2/4 =
n^2-n(2n+1)+(2n+1)^2/4
 Euclid's Common Notion
 [(n+1)-(2n+1)/2]^2 = [n-(2n+1)/2]^2  A little algebra
 (n+1) - (2n+1)/2 = n - (2n+1)/2  Taking square roots of both sides
 n+1 = n  Euclid's Common Notion
 1 = 0  One last time, Euclid!

Proof that 4 = 5 !!!!!!!!!!!!!!!! (top)

I. Rasul wrote (and provided comments and all those !'s):
 16 - 36 = 25 - 45  [ sure, both are -20 ] - Dan
 4^2 - 2*4*9/2 + (9/2)^2
= 5^2 - 2*5*9/2 + (9/2)^2
 adding (9/2)^2 on both sides;
[ rewritie 36 = 2*4*9/2, 45 = ... etc ]
 (4 - 9/2)^2 = (5 - 9/2)^2  [ recognize those perfect squares ]
  4 - 9/2 = 5 - 9/2  [ take square root of both sides ]
  we get : 4 = 5   9/2 gets canceled from both sides
 I know there's an algebraic analog to this but I like the down-to-earth feeling
of actual numbers - you feel more betrayed when things go wrong! -- Dan
 
 
Joe writes in with a general cleanup letter: (top)
 
You can no longer use the walking to the telephone variation to strenthen(?) Xeno's
Paradox,because nowadays we have e-mail and cell phones.When sitting at a computer
terminal,such things as walking and running no longer apply and besides,Xeno was
seemingly using the concept of infinity somewhere in the Paradox.An indicated distance
does not contain an infinite amount of increments,if that were so,any defined numerical
amount of feet,yardage,miles or kilometres would be infinite.
 
Girls=Evil does not work because the operation of multiplication
is assumed.Why not addition? Let's try addition:
Girls=money+time
time=money therefore
girls=time+time=2time...or two timer??
but time is infinite,thus
girls=2(infinity)=infinity
hence girls=infinity Q.E.D!
 
Now the clincher-1+1 does not equal 2 because in the English proof,
it is ASSUMED that 1=1(as in Let 1=1).Peano Postulate assumes
that 1 is in N.Check it out.I also have a longer more beautiful
proof that 1+1 DOES NOT EQUAL 2. I'm not crazy or bluffing either.
 
Two Opposing Questions (top)
 
As a math teacher, especially in algebra, the two most often-asked questions I get are:
1) What are we ever gonna use this (*%@# math) stuff for ?
2) Why do we have to do all these (*%@#) word problems ?
 
 
Click here to suggest a paradox or bad proof!
 
 
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