Prove that one equals zero

Answer 1

"Proof" that 1 = 0.

Postulates:

We know that one third = 1/3 which in decimals is written 0.3333...

("..." will represent recurring decimals)

We also know that three thirds = 3/3 which is a whole, i.e. 1.

Corollary:

So 1 - 3/3 = 0 by definition. This implies that 1 - (3 x 0.3333...) = 0

Tricky:

But multiply 0.3333... by 3 and you get 0.9999... (You can check this by multiplying each decimal by 3. Since none of them will exceed 10 {as 3x3=9}, each can be calculated separately and then strung back together)

Following the reasoning above, 1 - 0.9999... = 0

Which you can rearrange to get 1 = 0.9999...

So subtract 1 from both sides, you get that 0.0000...0001 = 0

Divide both sides by 10 infinitely* repeatedly until all the decimal zeroes are removed and you get 1 = 0.

NOTE: This is a common [[fallacy]] in the transformation between fractions and decimals which often misguides students into making false assumptions (similar to saying that 1 divided by 0 is infinity "because 0 goes into 1 infinite times").

*The fact that we had to use infinity at all to prove this (divide by 10 infinitely) should alert the student that even the most basic presumptions have to be reviewed (see note).