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!)

**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 dogC) 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 = bsubstituting: 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 beunambiguously 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 beunambiguously described in fourteen words or less". **4.**This is a complete and unambiguous description of n infourteen 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 naturalnumber 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 unambiguouslydescribed 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'. **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,**I'm not crazy or bluffing either.

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.**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! **[ home | meet dan | lessons | matica | ask dan | prob of wk | dvc ]**(top of page)