# 0=-1 (Calculus Puzzle)

Has this turned into a thread of trying to find a math trick that Ether doesn’t know how to work it? If so, i want a chance to disprove it before he dose .

I have an article here one of my math professors gave me that has a dozen “proofs” that 1=2. (She gave me this after I showed her my “proof,” which happened to be in the article… I guess I didn’t come up with it first…)

One that’s not so hard to disprove is as follows:

0 = (1 - 1) + (1 - 1) + (1 - 1)…

Rearrange parenthesis:
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1)…
0 = 1 + 0 + 0 + 0…
0 = 1

(Ether’s going to get this oh-so-fast…)

//Andrew

Not if the other Andrew gets there first!

after you rearrange parenthesis, you should have the one at the end that was freed up, therefore
0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1)… …-1
0 = 1 + 0 + 0 + 0… …-1
0 = 0

But can’t the one at the end just be moved into the next (1 - 1) term?

Sorry, I was offline all day.

The problem with the above “proof” is that the associative law of arithmetic is not universally valid for infinite sums. So you aren’t allowed to re-arrange (or remove) the parentheses.

There is no 1 “at the end”. It’s an infinite sum.

Ah, ok. Not entirely how I was trying to communicate it, but I guess being an infinite sum invalidates that anyways.

Hmmm… that’s different than what I was going to say… (the article I have doesn’t list the answers…)

I thought that because the series doesn’t converge, it doesn’t equal zero to begin with.

//Andrew

The original infinite sum you wrote:

(1 - 1) + (1 - 1) + (1 - 1)…

does converge. It converges because of the parentheses. The term that is being repeatedly added is (1-1). It is equal to zero.

If you remove all the parentheses, so that you have

1 - 1 + 1 - 1 + 1 - 1 …

then you are alternately adding plus or minus 1, so the sum never converges: it oscillates between 1 and 0.

The error was re-arranging the parentheses. The associative law does not always hold for an infinite sum: You cannot re-arrange the parentheses in an infinite sum unless certain criteria are satisfied.

The lack of a “p” was a typo, also, I don’t like using alt to insert symbols either, but if you use google translate to enter the text the symbol keys (;’,./]=-) are used to input special letters like vowels with accents

Also, nobody has attempted to figure out this one yet

Back in college years, my two buddies and I decided to go to Las Vegas for spring break. We piled into my Chevy Cavalier station wagon (jealous?) and headed West.
About 4am, we got to the point where we were all falling asleep, none of us were able to drive, and the car was too small to comfortably fit three guys. We found a small motel in Kansas, paid \$30 for a room (\$10 each), and retired for the night.
The clerk was filling out his books when he realized the motel had a policy of \$25 per night after 1am. He took five \$1 bills and headed to our room. Along the way, he realized we were not going to be able to split five bucks equally among the three of us. He shoved two of the singles in his pocket, gave us each a dollar, and that was that.
Here’s the issue:
We were initially charged ten dollars a piece - total of \$30.
After the clerk realized his error, the room charge was \$25. We paid \$9 apiece (\$10 - the \$1 that was returned) for a total of \$27; he put two dollars in his pocket for a total of \$29.

29 = 30 ?

Guy 1: \$10
Guy 2: \$10
Guy 3: \$10
Clerk: \$0
Hotel: \$0

Guy 1: \$0
Guy 2: \$0
Guy 3: \$0
Clerk: \$0
Hotel: \$30

Guy 1: \$0
Guy 2: \$0
Guy 3: \$0
Clerk: \$5
Hotel: \$25

Guy 1: \$1
Guy 2: \$1
Guy 3: \$1
Clerk: \$2
Hotel: \$25

Total: \$30. Anywho the error comes into play with the 27. You dont add the 2 to the 27, instead subtract from it to get 25.