Paradox - 2=1

[_echo]

This is an old one Im sure most all of you have seen. It is a paradox that trys to show how 2=1, however I have some issues with it.

First, here is how it is explained.

a = b

Multiplying both sides by a,

a² = ab

Subtracting b² from both sides,

a² - b² = ab - b²

Factorizing both sides,

(a + b)(a - b) = b(a - b)

Dividing both sides by (a - b),

a + b = b

This proves to be true, that a+b=b. Even if we added in numbers.

2 = 2

Multiplying both sides by 2,

2² = 2x2

Subtracting 2² from both sides,

2² - 2² = (2x2) - 2²

Factorizing both sides,

(2 + 2)(2 - 2) = 2(2 - 2)

Dividing both sides by (2 - 2),

2 + 2 = 2

So in the end, 2+2=2, or 4=2, IE: 2=1 However, My problem with this comes from the Order of Operations. PEMDAS, or, Parentheses, Exponents, Multiplication/Divition, Addition/Subtraction. In order for the above equation to be done properly, you must simplify before moving on to the next step. So right from Step 2 we have the problem. Step 2 says that 2² = 2x2. Which, when simplified becomes 4=4. Now, Step 3 wants us to subtract b² from both sides, and as we know, b=2, and b²=4. As a result, we must subtract 4 from both sides of 4=4, leaving us with 0=0, making the rest of the problem pointless as we are now working with 0.

2 = 2

Multiplying both sides by 2,

2² = 2x2 >>> 4=4

Subtracting 4 from both sides,

4-4=4-4 >>> 0=0

Factorizing both sides,

(0+0)(0-0)=0(0-0)

Dividing both sides by (0 - 0),

0+0=0

So what do you all think?

[nottlong]

Order of operations is important and you have proved that the equation is balanced.

[Yevic]

*ahem*

ahhhh, math!!

[Hanaisse]

This made my brain hurt.

[athray]

That is the whole point. To slip past the reader that you are dividing by 0 in that step, which, of course, is a big no-no. It is less of a proof than it is a trick on the reader.

http://en.wikipedia.org/wiki/Division_by_zero#Fallacies_based_on_division_by_zero

Edited August 26, 2010 by athray