Division by zero – Yes

Dividing by zero is a tricky concept in mathematics and is one that i guess we will never get a definitive answer on as there are good arguments both for and against and this is what makes it such a good topic of debate amongst mathematicians worldwide. I personally believe it is impossible, or at least, a number divided by zero remains undefined (mainly because this is what i have been taught) and i shall argue my reasons why.

The question of the mathematical possibility of division by zero dates back as far as 600AD, where zero was first treated as a number and had mathematical operations performed on it. More recently, in the early 18th century, George Berkeley questioned it’s definition in ‘The Analyst.’ And since then, many a great mathematician has tried to put an end to the debate and still we are no closer to finding a solution.

I think it wise to break it down and think about what we mean by division by any number. In layman’s terms, we mean “share equally what ever quantity you have by the number you wish to divide by” for example, 15 sweets shared between 3 people means that they get 5 sweets each – simple. If we reverse our thinking, we could rephrase it as “how many times does our the number we are dividing by go into our original quantity?” When we do this, in the case of dividing by zero, we come to the conclusion that we can count up an infinite amount of zeros before we get to our quantity, and therefore any number, ‘n,’ divided by zero should be defined as infinity, for any number greater than zero, and n/0 = minus infinity for any number, n<0.

This makes logical sense in a way, although it is still not strictly speaking correct, as zero has to be undefined to satisfy the axioms of a ‘field,’ so how do we show that division by zero is undefined? Well if we introduce a bit of analytical mathematics in the form of limits, we can obtain an explanation. Taking limits basically means that we look what happens to a basic algebraic division, a/b, as b gets closer and closer to zero.

Now by definition, for a limit to exist and be defined, both the left and right limit must be equal (the left and right limit basically means that our number b approaches zero from the left and right respectively on a number line). So the limit of a/b from the right is positive infinity, and the limit of a/b from the left is minus infinity. Therefore, the left and right limits are not equal, and we can conclude that division by zero is indeed ‘impossible’ or more technically, division by zero remains and undefined concept.

Having said all this, mathematics and physics are both ever evolving subjects, with new theories being produced all the time, scientists updating old laws and coming up with their own. I mean, Newton’s Laws of motion remained unchallenged for over 2 centuries until Einstein postulated his theory of relativity, so who’s to say that one day in the future, could be tomorrow, could be in the next millennium, someone may eventually be able to answer this time old question once and for all. Until then, well it’s one word against the other.