Question

Prove that 2=1 ::::::::::::::::::::::::::

Answer 1

Let's begin our journey into the bizarre world of apparently correct, yet obviously absurd, mathematical proofs by convincing ourselves that 1 + 1 = 1. And therefore that 2 = 1. I know this sounds crazy, but if you follow the logic (and don't already know the trick), I think you'll find that the "proof" is pretty convincing.

1. Here's how it works:
2. Assume that we have two variables a and b, and that: a = b
3. Multiply both sides by a to get: a2 = ab
4. Subtract b2 from both sides to get: a2 - b2 = ab - b2
5. This is the tricky part: Factor the left side (using FOIL from algebra) to get (a + b)(a - b) and factor out b from the right side to get b(a - b).
6. If you're not sure how FOIL or factoring works, don't worry—you can check that this all works by multiplying everything out to see that it matches.
7. The end result is that our equation has become: (a + b)(a - b) = b(a - b)
8. Since (a - b) appears on both sides, we can cancel it to get: a + b = b
9. Since a = b (that's the assumption we started with), we can substitute b in for a to get: b + b = b
10. Combining the two terms on the left gives us: 2b = b
11. Since b appears on both sides, we can divide through by b to get: 2 = 1
12. Wait, what?! Everything we did there looked totally reasonable. How in the world did we end up proving that 2 = 1?

What Are Mathematical Fallacies?

The truth is we didn't actually prove that 2 = 1. Which, good news, means you can relax—we haven't shattered all that you know and love about math. Somewhere buried in that "proof" is a mistake. Actually, "mistake" isn't the right word because it wasn't an error in how we did the arithmetic manipulations, it was a much more subtle kind of whoopsie-daisy known as a "mathematical fallacy."

Answer 2

We can substitution b in for a to get : b+b=b.

combining the two terms on the left gives us 2b=b.

since b appeared on both sides, we can divided through by b to get :2=1.......