Mathematics

Arithmetic Division zero – Yes



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How many ways can you divide six identical indivisible objects among six people?
It depends. If you want to do it equally, there is exactly one way:
everyone gets exactly one object. That is easy, but it isn't the
complete answer. What if you want to divide them so that one person
doesn't get an object? Then one of the remaining five would get two
and the other four would get one each. There are 6*5 =30 ways of
accomplishing that, because there are five ways to distribute the six
objects among 5 people and 6 choices of person who does not receive
one. So dividing six objects among six people can be complicated.

Now, how many ways can you divide zero objects among six people?
Intuitively, you would say that there is exactly one way: No one gets
anything. So, can we say that 6 / 0 =1? The answer, oddly enough, is
no. This is because we are dealing with counting and distributing
objects, not actual division. Dividing objects among people lies in
the realm of economics or combinatorics, although it may have some
uses in probability. But we are talking about arithmetic, not higher
math.

Another way to look at the problem is to look at it the way most of us
were taught to do long division in school. We can start simply: how
many times can we take 1 from 6? That's an easy answer. Now, how
many ways can we take nothing from 6? That's a bit more difficult.
The answer is that we could take nothing from 6 as many times as we
want, essentially a huge number of times, without having any effect.
The mathematical approach to this is to say "why bother?" and to
consider division (repeated subtraction) by zero as undefined, at
least for whole numbers.

What about division by zero for other than whole numbers? That is more
complicated because we are not only dealing with whole numbers, but
fractions and irrational numbers like pi or the square root of 2.
The same rules apply: we cannot keep subtracting 0 from 6 and get a
meaningful result. However, since we have the luxury of using
fractions, we can try to get closer to an answer.

Start with 1/2. That's easy. How many times can we subtract 1/2 from
6? The answer should come quickly: 12 times, so we can say that
6 / (1/2) = 12. In this case, 6 is called the dividend, 1/2 is the
divisor, and 12 is the quotient.

Continue on to 1/4. 6 / (1/4) = 24. This shows a pattern: the
smaller the divisor becomes, the more times we can subtract it from
6, and the larger the quotient. In fact, we can make the divisor as
small as we want, and this will make the very large. In fact, if we
pick a number small enough, the quotient will be so large as to be
unimaginable. We can therefore never have a divisor that is exactly 0
(although we can get as close as we want), and so division by 0 is
impossible because the quotient eventually becomes meaninglessly
large.

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