Today,
we’ll look at the vexing question – what does it really mean to divide by zero?
Why does it result in concepts like “undefined”
or “infinity”? To answer this, we’ll first need to understand what “0” really is. So let’s go
find out the history of the concept “zero”.
At
first glance, “zero” represents the
concept of nothingness – indeed the name “zero”
itself came from the Arabic word “SIFR”,
which in turn came from the Sanskrit word “Shunya”, meaning nothingness. But
Zero is much more than just
nothingness. The concept of zero is
one of the linchpins on which all modern mathematics, and by consequence, our
understanding of the universe, rests.
Let’s
take a brief look at history. You’ve studied the Roman number system – it’s functional, but a bit clumsy. Addition
and multiplication are just plain hard in that system, not to mention that
writing down large numbers could be even harder. And there was nothing like “zero” in there. The good part was
that it was a decimal system. The Sumerians and the Mesopotamians had more
sophisticated number systems, but their system was a base-60 one (the decimal
system is a base-10 system). The interesting part about their system was that
it was a positional one, the value of a “digit”
depended on its position in the number (same as both 5000 and 50 have the
same digit “5”, but it’s value depends on the position of the digit in the
number – the same 5 has a much larger value in the first number than in the
second one). And in this system, they also had a symbol that indicated the
positional zero – a symbol that indicated that the number had no value at that
position. So they did have a notional representation of zero.
So let us return to the question we
started out with. Given that “0” signifies nothingness, what does the concept
of dividing something by 0 mean? Unlike what some of us were taught,
dividing a number by zero (for example 15/0) does not give the result
of “infinity”. 15/0 (or 1/0) or any number
divided by zero) is actually undefined. The exception is
which is indeterminate.
What do we mean by this?
Let us go back to the basics of
division, and look at it in terms of the physical meaning. Taking a problem that
you all had encountered in elementary school – If you have 10 pencils, and had
to distribute it among 5 friends, how many pencils will you give to each
friend?
The answer is simple – 10 pencils to
be distributed (or “divided”) among 5 people. Divide (or “distribute equally”)
10/5 to get 2. You give 2 pencils to each friend.
The meaning of divide here is to “distribute equally”, but it is more
than that. It is to distribute equally into separate groups. You should be able
to distribute these into a different groups so that each group has equal number
of the item (be it a pencil, or just an abstract number) at the end of the
operation. And the number of groups distributed into is defined by the
denominator in the division operation.
So, in this example above 10/5 meant,
you distribute the 10 pencils (specified in the numerator) objects equally to
the group of 5 friends (specified in the denominator), we get the answer that
each gets 2 pencils. The correctness of the operation can be verified by two
things
- Each
of the five now has 2 pencils
- The
number of pencils in total is 10
Everything is wrapped up and
consistent.
Now, take another problem. What is
the value of 10/0?
Let’s consider it in terms of our
previous problem on division. You have 10 pencils, and have to distribute it
among “0” friends. Just imagine the scene in your head.
There you are standing with 10
pencils, ready to distribute it, but there’s a problem. There’s no one, or “0”
friends, you can distribute it to.
Now you see that you cannot be solved,
because the operation required of you just cannot be completed. This is also
the reason we say that dividing a number by 0 is undefined.
The other aspect is 0/0. We can’t
even define this in physical terms (It would be rather strange to say that you have
no pencils and no one to distribute it to. One would be tempted to ask why you
are standing there at all with nothing to do and no one to do it with). So
let’s look at it from a logical sense
Let’s say 0/0 = a
Then 0 = 0 x a
But we know that’s true of any
number a, whether a is 1 or 100 or -74 or 22/7.
This is why we say 0/0 is indeterminate –
it cannot be uniquely determined.
Now that that part is settled, things do get a little more
complex. You see, what we stated above for division by 0 is true only for
elementary mathematics (which is the stuff you’ve been learning in primary and
middle school). In more advanced subjects like calculus, the concept of limits
comes in, and we can then start to treat division by 0 differently. Then
come more advanced mathematical areas like Abstract Algebra and Riemannian
Geometry where there are other interpretations of division by zero.
