No

by Marie B

Created on: August 29, 2007

When we use math, we usually use it in one of two ways: tangible and intangible methods. This kind of question asks us to negate everything we have been taught, ever proved to be logically true, and ask that big "What if" question. Since math does explain that which cannot be seen, we can actually look at this from a different perspective.

Math allows us to identify quantities with labels. This is shown with the letter "i", which is the square root of -1. Now "i" is an intangible entity, but it can be used tangibly in problems. Can we actually take the square root of -1?

It's the same type of question! But since it comes up so much, mathematicians realized it can't be zero, it can't be infinity, and who knows what it really is. Rafaello Bombelli called it an imaginary number, and named it i. So why can't we do the same for division by zero?

Suppose we give division by zero, ie 1/0, an appropriate label, Ze. Further, lets pick an arbitrary problem, something relatively simple, A series: 1/0, 2/0, 3/0....infinity/0.

Now lets represent that series with the new term, Ze:

nZe where n=1 to infinity

So - what does that tell us? Well, it tells me that I either earned my Applied Math and Statistics Degree or I'm channeling Leibniz (and not by choice, mind you). Because it's just a polite way of representing division with zero! It doesn't say whether I can do it or not, but it doesn't just shut the problem down either. It leaves open the possibility of division by zero.

Now - lets get off the paper reality for a moment and think about the logic of division by zero. Division by zero means dividing something by essentially an entity that we consider to be, nothing. Not 1, which results in the same entity, but NOTHING! Multiplication by nothing results in nothing by our current rules. And why is that? Lets take a look:

If we were to do a simple multiplication like 2 x 3, we get this:

2 x 3 = 2 + 2 + 2 = 6 add 2 three times: this is simple math

But what if we had 2 x 0?

2 + ? well add 2 zero times. huh?

What really happens here? How do we explain that 2 x 0 = 0? Can we say, 2 collapses into itself because you can't multiply something by 'nothing' times? In the other example you simply add the first number by itself, the amount of times being determined by the second term. That's easy enough. But if the second number is zero? Houston, we have a minor problem!

Never fear - we can use one of our trusty rules (Associative property for multiplication) and turn it around:

2 x 0 = 0 x 2 Aha! now that we can work with!

0 + 0 = 0

Logically, that says "Nothing multiplied by more nothing, is pretty much nothing". Makes sense! I'll buy that because science tells me matter is neither created nor destroyed. At least we still think that's true... right?

So with that said, we can say

n x 0 = 0

But what if we divided both sides of that equation by 0?

(n x 0)/0 = 0/0

Ut Oh. Now, if we believe that anything divided by itself is 1, we have a dilemma. Because if was assume that 0/0 = 1, the following situation happens:

n x (0/0) = 0/0 or

n x 1 = 1 or n = 1?

That simply cannot be! And it's why 0/0 is undefined! That says any number you choose for n will always be 1? Logically, that's poop!

So...can it be assumed that not every zero is the same?

Hmmm...they all look the same to me! Lets bring back the Ze for a minute. Lets see what that does for this problem:

Let Ze = 1/0:
(n x 0) Ze = 0 X Ze
0 x Ze = 0 x Ze Eureka! Now that's tangible - both sides are the same!

So what I have shown? I have shown that division by zero doesn't follow convention, but that it may have some kind of "tangibleness" to it. Because when represented by Ze, it can be manipulated in the same way any other number can. And I think we should give division by zero its proper due place in mathematics, and designate it with a term rather than just shrugging our shoulders and saying "infinity"! But then again...wouldn't collapsing nothing onto nothing be considered a black hole of sorts? Whoa...maybe it really is infinity in a big way!

Perhaps the reason why its such an ordeal is because it is dealing with something we haven't completely dealt with tangibly. And that is the concept of nothing being a tangible! Which again, only brings into mind black holes and other scientific entities. But I've shown it has significant meaning, and deserves to be investigated further.

Therefore, I conclude, that division by zero is not impossible. We just haven't completely figured out its significance or how to represent it.

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