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   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   Proof that Girls are Evil   2 = 1 (bad algebra) 2 = 1 (bad calculus) -1 = 1 (just plain bad)   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!

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!

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.

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.

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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