Download A Dozen Proofs that 1 = 2

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 3s, 6s and 9s 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 …
But we also know …
Some paint
unit of paint
More paint
unit of paint
So I guess this shows that a half is less than a quarter.
So we have …
???
SHAMELESS
COMMERCIAL BREAK
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:
SHAMELESS
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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… ?
What’s ….99999?
What’s …999.999…?
So
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: