Image this: its almost the end of a long week and you are sitting in the middle of math
class, trying desperately to stay awake.
It's at this point you hear the teacher say something that catches your attention:
"... and no matter what you can't have zero divided by zero ..."
You think to yourself, that seems strange.
Of course I can divide zero by zero.
I'll simply write it down... and see, there you go zero divided by zero is equal to zero.
That wasn't so hard.
Or did you?
Afterall just because you can write down an expression, that doesn't make it true.
What gives then?
Why can't we have zero divided by zero, and what is it actually equal to?
To get to the bottom of this answer its important to understand exactly where the problem is.
And, once we do, we might be able to find just enough wiggle room to actually divide by
zero.
By now you've probably seen many different rules and formulas when learning mathematics;
but you may have also noticed that no matter how many new rules you learn, they always
seem to fit perfectly with everything that came before it.
Algebra, Trigonometry, Calculus, Differential Equations,
The list of subjects goes on, but at every stage what you learned previously doesn't
change, and it is in fact neccessary for understanding what you are doing now.
So no matter what new rule or law you decided cook up, its important that it must fit with
all of our other previous rules.
And there in lies the problem.
With our familiar number system, no matter what number we try and assign to dividing
a number by zero, there is always another rule that comes along and conflicts with it.
Let's give it a try.
First we might simplify the problem a bit by only looking at what happens when we divide
non-zero numbers by zero.
This is a good place to start since having zero in the numerator or top is usually not
a problem.
For example we could take 0/5 and the answer is simply zero.
So a huge part of the problem lies with trying to divide by zero.
In this case we might attempt to give it a value of say: zero.
So for 5/0, we'll just call this 0.
But unfortunately this doesn't work.
It doesn't fit with all of our other previous rules.
Multiplication and division are opposites, so in a way this statement is just another
way of saying that 5 = 0*0.
And that is definately not true.
We have a rule that says multiplication by zero should be equal to zero.
In fact no matter what value we give a/0, where a is a non-zero, it will never be consistent
with what we've established in our multiplication rules.
Well that is certainly a problem with something like 5/0, but maybe we can find something that works
for 0/0.
Maybe in this one case we can let it equal 0.
Unfortunately this still doesn't work.
The rule that doesn't fit now is our rule that says any number divided by itself should
be equal to 1.
We might try and fix this by letting it equal something else.
Maybe we'll say that 0/0 = 1. That way it better fits with all of our other rules.
But we just covered that when zero is divided by a number, it should be equal to zero.
So once again we are left in a situation that no matter what value we give it, it simply
will not fit in with our other rules.
Because of this we say that we can not divide 0 by 0, or more mathematical, we say that 0/0
is left undefined.
As you can see the main issue we keep running into is staying consistent with all of the
previous rules we've already agreed on.
But what if we used a different number system and kept our usual operations?
Then could we divide by zero.
The surprising and wonderful answer is yes, absolutely!
Take for example the real number line.
Here we can represent all of the numbers we are familar with.
Now let's make a small change by adding point which we'll represent using the infinity symbol.
This point will exist above all of the real numbers, and also below all of the real numbers.
A good way to visualize where this new point is located is to bend our real line so that
it now connects in a circle at this new point.
(as a fun fact, this this called a one point compactification of the real line)
Under this new structure, we can still add, subtract, or multiply the usual numbers, but
we now must also assign meaning when we using these operations with our brand new point.
So let's do just that.
Let's say that a non-zero number divided by 0 is called inifinity.
So under this new system, 5/0 = infinity.
But what about the corresponding multiplication problem, do we still have issues with that?
In a way we do.
Since 5/0 = infinity and -7/0 = infinity, its unclear what should we call the opposite
problem involving multiplication.
Should it be 5, should it be -7, what do we do 0*infinity.
It seems that any value we try and give it just doesn't seem to fit with our division
rule with infinity.
In this intresting system we can divide by zero with no issues, but when zero is multiplied
by infinity, we must now leave this as undefined.
But there is a system where it does work.
A system where you can divide zero by zero, and more importantly all of the unsual math
rules fit perfectly! but I have to admit, its not a very "big"
space.
In fact, its down right tiny!
For this last number system, imagine a space where we only have a singe point and let's
call this point zero.
We have all of the usual operations, but since we only have one point to deal with, the answer
is always the same.
0 + 0 = 0.
0 * 0 = 0.
0/0 = 0.
Since this is the only point we could ever use, and the only answer we could ever get,
it fits perfectly with all of the usual rules.
0/0 = 0 doesn't cause any problems, this is because 0 = 0*0 is the only multiplication problem
we have to consider.
This system simply doesn't have any non-zero elements to cause any problems.
The unfortunate part about this system is that it is not big enough to do any interesting
problems with.
We can define bizarre operations all day long, but if the only value we can ever get is
zero, then we are not really getting any new information.
This is truly unfortunate since the power behind math is in its ability to give us new
insights, and in this particular space, there is nothing left to see.
Don't worry, all is not lost.
By working really hard to find a space where we can actually divide by zero, we've gained
more understanding about the number system we use every day.
With this in mind, I hope you can now see why it really is best to leave 0/0 undefined.
It leaves us with a number system that is definately large enough for interesting problems,
but also one that has the rules need for modeling the world around us.
Thanks for watching.