Get a Widget for this title
RSS FeedDivision by zero: Is it really impossible?
Results so far:
| No | 46% | 568 votes | Total: 1222 votes | |
| Yes | 54% | 654 votes |
No
Write now
Print
Invite a writer™
Add to FavoritesDivision by 0 is impossible by the simple fact that 0 times anything is 0, this itself being a result of the field axioms of the real numbers. What am I talking about? Let me explain.
Any sort of mathematical structure where one can do addition and multiplication, and for which these operations satisfy the usual ones of the real numbers, is called a field. Everyone has familiarity with one field: the real numbers. You can add reals (in any order), subtract (more on this in a bit), multiply (and the distributive law holds), and divide (so long as the element you are dividing by is not 0, as we will see). Are there any other fields? Yes, indeed there are.
Consider the usual operations of addition and multiplication with integers except take the remainder of everything when it is divided by two. You can see that 0+0 is still 0. As well, 1+0=0+1=1, like usual. We take the step of abstraction when considering that 1+1=0, because the remainder when two is divided by two is 0. Multiplication is exactly as you would think, as well as every other operation commonly done with the reals. For instance, what is -1? Well, what does that even mean? The interpretation of -1 is "the thing you add 1 to in order to get 0". So what do we add to 1 in order to get 0? We've already seen it, its 1!
There are many many examples of other fields: the complex numbers, the rationals, doing arithmetic but take remainders when dividing by a prime, etc.
Now, back to the point of not being able to divide by 0. This operation is in fact impossible in any field. We'll begin by proving that anything times 0, in any field, is 0. Let a be any element of some field.
0*a = (0+0)*a
This follows from 0+0 = 0. We then have:
(0+0)*a = 0*a+0*a
Which comes from the distributive property.
So we've got 0*a = 0*a + 0*a and we know that every element has a negative element, so add -0*a to both sides, cancelling one 0*a from each and giving:
0 = 0*a
So now suppose there exists a multiplicative inverse of 0, that is an element, call it 1/0, such that 0*(1/0) = 1. Then take the fact that for every single field element "a" we have 0*a=0 and multiply both sides by this inverse. You get a=1. That is, if you have a field where 0 has a multiplicative inverse, then every element in the field is 1. This is impossible if the field has more than 1 element. It means that if the real numbers had an element called 1/0 and this element satisfied the usual properties of being a real, then every real number is 1. This is just funky and downright impossible in the real field.
So if you're thinking about dividing by zero, think twice. You sacrifice alot of structure and familiarity with the real numbers if you do. Perhaps someone could define some strange structure where there was a notion of "division by 0", but it wouldn't be important unless its usefulness was well established.
Learn more about this author, Jess Boling.
Click here to send this author comments or questions.
Yes
There is a mathematical concept known as the asymptote. This concept is taught to most students in the second or third year of high school math. An asymptote refers to a straight line that a graph approaches but never touches; later in calculus or at the end of pre-calculus the similar concept of a limit is introduced.
Simply put, the fraction x/0, x being any real number, is an example of an asymptote or limit. However, it is unlike defined limits, such as the limit of the function f(x) = a^-x as x approaches infinity. In the case of f(x) = 2^-x, for instance, the value of the function approaches the definite value of zero as x gets larger. This is not the case with something like x/0, because infinity is not defined. Calculus tells us the limit of x/0 is infinity, but infinity is not a number. It is a mathematical construct used to represent the concept of a number larger than all other numbers.
If you read that last sentence and think about it, you'll see the problem. Let's suppose, for instance, that a famous mathematician tells you, "The largest number in the world is 30,000,000!" You can reasonably ask, what about 30,000,001? The mathematician may then respond, "Yes, 30,000,001 is the largest number in the world, as well as 30,000,000!" Later on in the evening, the famous mathematician will laugh about this with his friends over beer and poker. But that's not the point. The point is, for any number one can declare to be the largest, there will always be a way to obtain a larger number than that; you can simply add one. This is the essence of infinity, and the reason infinity is undefined.
If you divide a number by 0.000001, you will get quite a large number as a result. The closer the denominator gets to zero, the larger the result of the division. This means that the limit diverges; it does not settle near a single point. The closer you get to zero on the bottom, the more you spiral out into infinity.
In short, division by zero yields a result of positive or negative infinity. It is thus impossible to divide by zero and get a defined result. If you can accept infinity as a valid result, then you can reason that dividing by zero is possible. It is unequivocally clear, however, that division by zero is undefined.
Learn more about this author, codehappykid.
Click here to send this author comments or questions.
