Does two plus two equal five?

Answer 1

No!!! 2+2=4, not 5.Here's the so-called "proof".

• A = 1, B = 1
• A = B
• A+A = A+B (adding A to both sides)
• 2A = A+B (simplifying)
• 2A - 2B = A + B - 2B (subtracting 2B from both sides)
• 2A - 2B = A - B (simplifying)
• 2(A - B) = 1(A - B) (factoring)
• 2 = 1 (cancelling)

Of course, as soon as 2 = 1, we can prove that any other pair of numbers are equal. But wait, one problem! On the second last problem, both sides divide by A - B, which is 0, therefore, we have just divided by zero. That is illegal in mathematics, therefore, we get the erroneous answer that 2 + 2 = 5.

Another way, slightly more mathematical, is to note that it is possible to "define" all numbers using Peano's axioms using a certain function. According to these axioms 2 is DEFINED to be equal to 1+1, 4 is DEFINED to be equal to 1+1+1+1, and 5 is DEFINED to be equal to 1+1+1+1+1. So to prove addition all you need to do is regroup the 1s:

2 + 2 = (1+1) + (1+1) = (1+1+1+1) = 4

Anyway, to answer your question, it depends how we define the symbols for the numbers. If we suddenly decided that the symbol 5 would mean "4", then the equation 2+2=5 would be correct, because

2 + 2 = (1+1) + (1+1) = (1+1+1+1) = 5

would be valid, since we defined 5 = 4 in the first place. Under STANDARD mathematical axioms, however, 2+2=5 is FALSE.