What is Division?
Arguably one of the most fascinating and widely misunderstood problems in math is the problem of division by zero. To appreciate the problem and to understand the answer we must ask ourselves: what does the division operation mean? To answer this question consider a problem of dividing 8 by 2. When we divide 8 by 2 we want to know how many times do we need to take the ‘2’ in order to make up the ‘8’. So the real question that we are asking when dividing is ‘how many times’. Non-integer division (e.g. dividing 8 by 1.7) can be understood in the same terms, except that rather than dealing with ‘wholes’ we are now dealing with ‘fractions’, but the question we are trying to answer remains to be ‘how many times’.
Dividing by Zero
Now let us apply the same logic to dividing by zero. What do we get when we divide 8 by 0? Well, we do not get anything. Literally there is no answer. Because logically it does not matter how many times we compound zeros we will never get the eight. Even if we spend a billion years taking a zero, after zero, after zero we will not get the eight, or four, or one, or any other number for that matter (except of perhaps for zero itself, but more on it later). We can draw another common sense parallel: if you have absolutely no money, regardless of how long it takes to search your pockets and regardless of how hard you try to grow your non-existent capital you will end up exactly with nothing. If anything our intuition is telling us about the Nature it is that one cannot get something out nothing. Therefore the answer to the question ‘what do we get when we divide 8 by zero’ is ‘nothing’ which has a mathematical and logical meaning of ‘undefined’ because this ‘nothing’ is not zero, or one, or infinity. In fact this ‘nothing’ it is not a number at all. We do not know what it is, it does not make sense and therefore we treat it as undefined.
Division by Zero as a Limiting Case
Of course we can try to define the outcome of division by zero on the grounds of continuity and argue that the result is infinity. E.g. if we divide 8 by 1 we get 8. If we divide 8 by 0.1 we get 80. If we divide 8 by 0.01 we get 800, etc. Clearly the limiting case of this progression is infinity, and perhaps this is the reason when we divide a number by zero on a digital computer we get an infinity, which is defined as a special number with a special digital representation. Perhaps in certain algebras this outcome makes sense. After all, when working with specialized algebras we can define any operation however we want. But in the case of ‘common sense’ approach to division the outcome remains unchanged and quite clear: when we divide any number by zero the result is undefined because such operation does not make (common) sense.
Dividing Zero by Zero
What about dividing a zero by zero? Shouldn’t the outcome be clearly defined in this case as one? Not really. And here is why. On one hand we can easily agree that the answer to the question ‘how many times do we need to take zero to make up zero’ is of course one. We can take zero one time to get zero (e.g. it is sufficient to take nothing once to end up with nothing). On the other hand this answer is not unique because we can take zero twice and still get zero. We can take zero a million times and we still get zero. E.g. if you ask a million of absolutely broke people that do not have a single penny for money you still get no money, regardless of how many such such people you attempt to shake down. Therefore we do not really know the exact answer to the question ‘how many times do we need to take zero to make up zero’ because there are many answers to this question ranging from one to infinity, and every one of them seems logically and factually correct.
Multiplying by Zero
The situation becomes trickier if we think about what happens when we take zero zero times, i.e. is zero also a valid outcome of dividing zero by zero? It does not appear to be so because it is not easy to define what happens when you attempt to take nothing zero time. This question can be rephrased like this: is the absence of nothing amount to something? It would seem so. But what does this ‘something’ amount to exactly? How much of ‘something’ is the opposite of ‘nothing”? Any amount it would seem. I.e. if we take zero zero times we ought to get any number except for zero, thus making 0 x 0 logically complimentary to 0 ÷ 0. Unfortunately, this is not how multiplication by zero is defined. According to textbooks 0 x 0 = 0. Literally this means that taking nothing zero times is still nothing. Or by owing nothing to nobody you still owe nothing (which makes sense of course). One can argue that this definition (0 x 0 = 0) was adopted because it has a clear computational utility although it fails the logical consistency (taking nothing zero times amounts to the absence of nothing, which should amount to something rather than be nothing).
Conclusion
Thus, division by zero makes no sense logically because such operation fails to provide us with a unique and meaningful answer in our everyday experience, it fails the common sense. In the same time, we can define the outcome however we want when we are designing our own algebra (which we often do). After all, algebras exist to facilitate calculations and we can calculate however we want as long as we are happy with the outcome and the outcome of such calculation has a practical utility. In any case we make up the rules as we are the gods of our mathematical Universe. Therefore the result of division by zero can be defined as something in a specialized algebra even though for the most of us it would remain logically ‘undefined’.