A DOZEN PROOFS THAT 12 A Misguided Review

A DOZEN PROOFS THAT 1=2 A Misguided Review of Mathematics James Tanton Mathematical Association of America www. jamestanton. com www. gdaymath. com

1. PROOF BY REGROUPING Guidobaldo del Monte (1545 – 1607) Add 1 to both sides to get …

2. PROOF BY SCHOOL ALGEBRA Let Then It is certainly true that Factor Cancel the common term But is one, so …

3. PROOF BY FRACTIONS It is okay to cancel 3 s, 6 s and 9 s in fractions.

So we must have … … just cancel the threes. Multiply … Subtract 26: Divide by 6: Add 1: 0=6 0=1 1=2

ASIDE: Really cool example …

4. PROOF BY COLOURING So … Multiply by 4 … Some paint unit of paint But we also know … More paint unit of paint So I guess this shows that a half is less than a quarter. So we have … ? ? ?

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5. PROOF BY GEOMETRY How long is the diagonal of a square? Pythagoras says:

Alternatively … The diagonal can be approximated arbitrarily close by a “stair case” of segments. Diagonal = and Diagonal = 2

6. PROOF BY EXPERIMENT WATCH!

7. PROOF BY TRAINS

Small Wheel: Radius = 1 Large Wheel: Radius = 2

8. PROOF BY COMPARING LENGTHS A line segment two meters long is twice as long as a line segment one meter long. But we see that there are just as many points on the first line segment as there on the second. Thus 1 meter is just as long as 2 meters: 1 = 2.

9. PROOF BY ROTATION Actually, this is an anti-proof: WATCH! So: Divide by two: Add one:

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BONUS 10. PROOF BY AREA The diagonal line divides in half. Yellow rectangles must have the same area.

11. PROOF BY PURE THOUGHT Let’s ask a strange question: What is the largest counting number? Answer: 1 is the largest counting number. Proof: We show that no other number can be the largest counting number. So that leaves 1 as the only possibility. For any number So , we have . (Multiply by N. ) can’t be the largest. DONE! So … 1 > 2. And clearly 1 < 2. I guess that means 1 = 2.

Believing that answers exist can be delightfully dangerous … What’s 0. 9999… ? So What’s …. 99999? What’s … 999…? So So IS ANY OF THIS TRUE?

12. PROOF BY SHOPPING I was recently at the store and came across a “two for the price of one” sale. I only wanted one item so I asked the store clerk how much a single item would be. “Same as the price for two, ” came the reply. “So one is the same as two? ” I checked. “Yep, sure is!” vouched the clerk.

THANKS!! www. jamestanton. com www. gdaymath. com (WOULD YOU LIKE SOME MORE “PROOFS”? )

13. PROOF BY IMAGINARY NUMBERS Recall that i is the imaginary number whose square is -1: So

14. PROOF BY REARRANGING Let x be the value of the sum:

15. PROOF BY MORE ALGEBRA Let’s solve Since in an unusual way. is clearly not zero, we may divide through by Substituting back into the original equation: Thus: :

BONUS 16: PROOF BY TRIGONOMETRY

BONUS 17. PROOF BY INTEGRATION Integration by Parts formula: