Math Blog

Dividing by Zero

2006-06-23T01:08:00.000-04:00

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

2006-05-26T01:02:00.000-04:00

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

2006-04-10T04:00:00.000-04:00

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 by taking (this fact is absolutely amazing by the way).

Now let’s see what happens if . Then:

___ (since )

___ (multiplied by 2)

Now this is obviously not true since . But what is most concerning for me is that this implies that is a REAL number, which is just crazy.

It actually took me a while to figure out what the problem was in my steps, but I found it eventually.

Can you see where the problem is? Let me know what you think.

Functions and Mappings

2006-04-07T03:00:00.000-04:00

So guess what I realized a few days ago.....

It came up when I was introducing the concept of functions to one of my students. When you have f(x): the actual "function" is f, while f(x) is merely what happens to x once you apply f to it. The function f is not a number on its own, it's only the "recipe" of what to do to once you are given x (that's why f is called a function). f(x) however is a number, and it is the number that x becomes under f. Confused yet?

You can sort of picture it like this:

So if you give me x, then I'll apply f to it to get f(x). The function f is represented by the arrow, wherein it is the "process" (or "function") that takes you from x to f(x).

Well actually, I lied. I didn't just realize it. I've known for quite a while now. But I never had a real appreciation for it until recently when I started doing topology and have been using functions that go from "sets to sets" (as opposed to "numbers to numbers"). And when I had to explain to my student for the first time what a function is, I was forced to rigorously understand and define what a function really is.

So that's my story. Meh, I'm probably just rambling again.

Cardinality and Infinities

2006-01-31T00:51:00.000-05:00

Question from Andy Hunter:

Does it make sense to talk about equalities among infinities? What I mean is does the word equal only apply to finite sets?

That's a good question. And it has a tough answer, if I can think of one.

I sort of skipped through all the trouble of infinity (mostly because it's confusing), and assumed most of the ideas of 'sizes' and 'equality' to be true. But maybe we should talk more about it.

Infinity is a very tricky subject. First of all, what is infinity? Most people will say "never ending" or "the largest number", but for me that really means nothing. What is "never ending"? Or more concerning, what is "the largest number"? There's no answer I can give for what infinity is. But as far as natural numbers go (and for even numbers too), we can get around saying that the set is 'infinite' (whatever that means) by instead simply saying that there is no largest natural number. That is, we can get as large as we want, simply by keep adding 1.

Now you're right, how is it possible for infinite things to be equal? I mean, they're infinite. Well, the idea is that many things are infinite, but some things can be more infinite (if that makes sense). And in the same way, some things may be equally infinite. Now this isn't the same as when we say two finite things are equal. I mean: 12 = 7+5 = 4x3 = 12 and that's it. But it's not so easy to think of equality with infinite sets the same way. Simply put, we can't assign the 'size' of these infinite sets an actual number. Now keep in mind that we never really assigned the natural numbers N = {1, 2, 3, . . .} or the even numbers L = {2, 4, 6, 8, . . .} a size. We simply said that their sizes, whatever they are, are equal. And that's really the best we can do. So in short: we can't assign the sizes of infinite sets a number, the best we can do is say that their sizes are equal (or that one is bigger than the other).

In fact, if we were to be rigorous about it, we shouldn't even talk about the 'size' of these sets at all. 'Size' is just a loose term we use. We really should say cardinality. And how we define cardinality is exactly the technique we used: if we can create a one-to-one correspondence between two sets (like we did for the natural numbers and even numbers) then those two sets have equal cardinality. And this is nothing more than a definition.

But if you think about it, you can see and convince yourself why we associate cardinality with 'size'.

I know that's all confusing. But I hope it helped.

Measuring Infinity: Cardinality

2006-01-29T00:30:00.000-05:00

What I'm going to do is show that there are just as many natural numbers N = {1, 2, 3, . . .} as there are even numbers
L = {2, 4, 6, 8, . . .} .

This seems illogical since the natural numbers N contains all even numbers, and since N also includes all odd numbers, common sense would make you expect there to be more natural numbers than even numbers. But this isn't so, and we will show that.

First, let's discuss the logic. Here's the idea:
Suppose you have two sets S = {red, green, blue, yellow} and
T = {apple, lime, blueberry, banana} .

The idea is that if we can create a one-to-one correspondence between the elements in S and T, then the two sets must be the same size. In the case of S and T we can write the one-to-one correspondence:

red ↔ apple
green ↔ lime
blue ↔ blue berry
yellow ↔ banana

You can see there is no element in S or T "excluded", so S and T must be the same size. You can contrast this to the case where you compare S = {red, green, blue, yellow} instead with
T' = {apple, lime, blueberry, banana, kiwi} . (Notice that T' is just T with 'kiwi' added to it)
In the case of T', it is impossible to construct a one-to-one correspondence with S, because an element in T' will always be left out:

red ↔ apple
green ↔ lime
blue ↔ blue berry
yellow ↔ banana
(kiwi is left out)

Thus, S and T' cannot be the same size.

We will use this logic to show that the even numbers
L = {2, 4, 6, 8, . . .} is the same size as the natural numbers
N = {1, 2, 3, . . .}.

It's quite easy actually, all we have to do is create a one-to-one correspondence:

1 ↔ 2
2 ↔ 4
3 ↔ 6
4 ↔ 8
5 ↔ 10
6 ↔ 12
. ↔ .
. ↔ .
. ↔ .
n ↔ 2n

You can see that given any natural number I can associate an even number to it, and given any even number I can associate a natural number to it. Without exclusion. There are no leftovers, no matter how far we extend the correspondence. Thus, the sets
L = {2, 4, 6, 8, . . .} and N = {1, 2, 3, . . .} must be the same size. Meaning there are just as many natural numbers as there are even numbers.

I find this to be quite amazing.

The Monty Hall Problem

2005-12-17T03:38:00.000-05:00

Now this is beautiful mathematics. Here's the problem:

"You are on a game show on television. On this game show the idea is to win a car as a prize. The game show host shows you three doors. He says that there is a car behind one of the doors and that there are goats behind the other two. He asks you to pick a door. You pick a door but the door is not opened. Then the game show host opens one of the doors you didn't pick to show a goat (because he knows what is behind each door). Then he says that you have one final chance to change your mind before the doors are opened, and asks you if you want to change your mind and pick the other unopened door instead. What do you do?

Using your intuition you might think that there is a 50-50 chance that there is a car behind each door. But according to Marilyn vos Savant you should always change and pick the other door because chances are 2 in 3 that there will be a car behind that door. 92% of the readers wrote in to Marilyn and said she was wrong, many of whom have Ph.D's in mathematics."

But Marilyn is right, do you see why? If you're really curious and want to see why, I posted the solution here (look at all comments, but the final solution is in the last comment I posted).

Amazing, huh?

A Magical Beginning

2005-12-15T01:29:00.000-05:00

So what's this blog going to be used for? Well, my friends don't seem to appreciate it when I post anything mathematical on my personal blog. For some reason it upsets them to see mathematics of any kind (some sort of complex developed back in high school). So I decided to start this page which will be solely dedicated to things mathematical that I would regularly post on my personal blog.

So we'll wait and see where this goes. Some of the things I post might be quite basic and easy, while others will be quite complex.

I'm just worried about how I'm gonna be able to write some of the symbols I'll be using (think integrals for example). Yikes, might get messy.