Division by zero – No

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Division 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.

More about this author: Jess Boling

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