Sina Math Blog

Dividing by Zero

Here's a question that my students never ask, but really should: Why can't we divide by zero?

That's a good question. We learn in school that dividing by zero is 'undefined'. We even often have to write restrictions on rational expressions to make sure that the denominator of a fraction is not zero, because when we have a fraction we are actually dividing by the denominator. So we can't have the denominator being zero because that would mean we were dividing by zero. Now why is it that we can't divide by zero? It's simple...

Suppose, in some world, we could divide by zero. Now we know that:

0 = 0 (obviously)

We can rewrite this as:

0x1 = 0x2

If we could divide by zero, then in this expression we could divide both sides of the equation by zero and we'd get:

1=2

Which obviously makes no sense (1 does not, will never, ever equal 2).

So there you have it. We can't divide by zero simply because, if you could, 1 would equal 2 and that would just be crazy. The whole universe would collapse or something.

If you're a math teacher, I suggest you share this information with your students. It's surprising: I've never had a student who knew why we can't divide by zero. They would know how to write restrictions, they would know that dividing by zero is not allowed, but they had no idea why. They just simply accept it as fact, for some reason.

Note: The choice of using 1 and 2 were arbitrary. I could have picked any other two numbers, say x and y, and I would have been able to show that x=y (if dividing by zero were allowed).

The Da Vinci Code and Fibonacci

So I saw The Da Vinci Code a few days ago. If you saw it, you know that they talk about Fibonacci sequences and stuff. And if you read the book, because they don't mention in it in the movie, it also talks about the "Golden Ratio"

, and goes on to claim that the Fibonacci sequence and

are related in a subtle way.

[Side Note: The book The Da Vinci Code actually says that

exactly, which isn't true. You can see that phi is actually an irrational number.]

First off, let me tell you what the Fibonacci sequence is in case you don't know. The Fibonacci sequence is defined as:

,

, and

for integers

. That is, to get the next number in the sequence, you simply add the two previous numbers. The sequence looks something like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Now according to The Da Vinci Code, the ratio

of the Fibonacci sequence approaches

as n gets large, which is true.

But is there a proof? When I got home from the movies I searched the web for a proof that this ratio converges to

. I found a few, but they were messy and I couldn't follow the logic. The proofs seemed flawed. I even looked at Wikipedia, and their proof just doesn't make sense to me.

So here's an easy proof:

Proof:

We want to show that

,

where

is defined above.

Let:

Then:

(since

for

and n is large)

Now notice that:

(think about this and convince yourself that it is true)

So we get:

(since

)

This now a simple quadratic that we can solve.

You might want to note the quadratic formula:
If

then

In our case we get:

We have now two solutions because of the plus-or-minus. You can see that the minus version yields a negative number. But we know thatx must be positive. So we must take the positive solution:

Which is exactly

, so we are done. ■

So what's the big deal around

? Good question. I have no clue. Well actually, I kinda sorta know, but not really. First of all,

comes up in a lot in geometry. The diagonal of a regular pentagon, for example, is

times its side length. There are also some suggestions that the human body has proportions equal to

. And also

comes up a lot patterns in nature (the shape of flowers for example). There are probably countless other places where

comes up. But who knows. At least for sure,

is a beautiful number.

P.S. No one has solved the problem in the "Euler's Formula" post. It's not that hard.

Euler's Formula

Okay, so we all know Euler’s Identity that says

for any Real number

, and where

(if you don't know about Euler's Identity, here is a quck explanation from MathWorld. The article also contains two seperate proofs). Let me just say that Euler's Identity is absolutely remarkable, probably because it seems so unlikely. But I was playing around with this identity one night, and I came across something strange mathematics…

First of all, we can see that