1/2 why

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. 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.

Everything we did there looked totally reasonable. What Are Mathematical Fallacies? 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. What was the fallacy in the famous faux proof we looked at? Like many other mathematical fallacies, our proof relies upon the subtle trick of dividing by zero. And I say subtle because this proof is structured in such a way that you might never even notice that division by zero is happening.

Where does it occur? Take a minute and see if you can figure it out… OK, got it? It happened when we divided both sides by a - b in the fifth step. But, you say, that's not dividing by zero—it's dividing by a - b. That's true, but we started with the assumption that a is equal to b , which means that a - b is the same thing as zero!

And while it's perfectly fine to divide both sides of an equation by the same expression, it's not fine to do that if the expression is zero. Because, as we've been taught forever, it's never OK to divide by zero! Which might get you wondering: Why exactly is it that we can't divide by zero? We've all been warned about such things since we were little lads and ladies, but have you ever stopped to think about why division by zero is such an offensive thing to do? There are many ways to think about this.

We'll talk about two reasons today. The first has to do with how division is related to multiplication. Let's imagine for a second that division by zero is fine and dandy.

We don't know what it is, but we'll just assume that x is some number. We can also look at this division problem as a multiplication problem asking what number, x , do we have to multiply by 0 to get 10?

Of course, there's no answer to this question since every number multiplied by zero is zero. Which means the operation of dividing by zero is what's dubbed "undefined.

In other words, as we divide 1 by increasingly small numbers—which are closer and closer to zero—we get a larger and larger result. In the limit where the denominator of this fraction actually becomes zero, the result would be infinitely large. Which is yet another very good reason that we can't divide by zero. A proof is a finite sequence of formulas see here , where each formula is either an axiom or follows from the previous ones by some inference rule.

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Gerenuk Gerenuk 1, 2 2 gold badges 14 14 silver badges 21 21 bronze badges. The point was to put the foundation of mathematics on firmly rigorous ground in a highly axiomatized system. Show 11 more comments. Active Oldest Votes. I have to check that. Add a comment. Dejan Govc Dejan Govc Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post.