Dividing by zero is impossible. It is one of the cornerstones of mathematics that has been learned since elementary school. But Why can’t it be done? Who forbids us? And what happens if we do? In reality it is not a ban, but an operation that it doesn’t have a resultand therefore it has no meaningin the arithmetic field, contrary to what happens in the field of calculating limits, and with certain calculators, where this operation can be done in some way.
Why it is impossible to divide by zero: the proof
In arithmetic, division is the inverse operation of multiplication, so calculate 12 ÷ 4 means find that number that multiplied by 4 equals 12.
From a practical point of view, if we want to divide 12 sandwiches equally between 4 people, we can distribute a sandwich each in turn and in the end each of the 4 people will have 3 sandwiches, and we can say that 4 × 3 = 12. In other words when we write
12 ÷ 4 = 3
We are also saying that
4 × 3 = 12
and vice versa.
But then what does dividing by zero mean? From a practical point of view for split the 12 sandwiches between 0 (zero) peoplewe should give 1 sandwich at a time to each of the 0 people involved until the 12 sandwiches are used up, but since there is no one there we are unable to distribute any sandwiches, It’s something we just can’t do.
What happens instead if we try to calculate 12 : 0 from an arithmetic point of view? Let’s try, there is no law that prohibits us.
As we have seen calculate 12 ÷ 0 yes it means find a number which multiplied by zero gives 12, a result we can call x since we don’t know how much it’s worth. In other words if
12 ÷ 0 = x
then it must count
0 × x = 12
but, as we all know, zero multiplied by any number results in zero: we will never be able to find a number x that when multiplied by 0 gives the result 12. Therefore the result of division 12 ÷ 0 it doesn’t exist, but it’s not forbidden to look for it, we just tried and nothing serious happened. Basically when you say that
it is impossible to divide by zero
it means that
if you try to do it you can’t find a result because it doesn’t exist.
It is a kind of mathematical dead end, not a crime punishable by prison, however this is not always the case because there are branches of mathematics, such as the calculation of limits, in which dividing by zero acquires meaning, as some calculators well know.
Dividing by zero is possible when calculating limits
If we try to calculate 12 ÷ 0 with the calculator we can get, depending on the calculator we use, various types of error messages. However some calculatorsincluding Google’s online calculator, instead of an error message they write the result “∞”. In mathematics the symbol ∞ represents infinity and is not an error message, but Not And not even a numberso what do these calculators intend to tell us when they write 12 ÷ 0=∞?
These calculators refer to the calculation of limits, an area of mathematics in which we do not simply consider fixed numbers, but quantities in progress. In this context when calculating 12 ÷ 0 you don’t look at 0 as a numberbut rather as an infinitely small quantity that we imagine can become increasingly smaller.
Basically calculate 12 ÷ 0 in this context it means asking yourself
what happens if you divide 12 by sequences of smaller and smaller numbers?
For example we can divide first by 0.1, then by 0.01, then by 0.001, by 0.0001, and so on. Let’s try to do it:
- 12 ÷ 0.1 = 120
- 12 ÷ 0.01 = 1200
- 12 ÷ 0.001 = 12000
- 12 ÷ 0.0001 = 120000
- …
- 12 ÷ 0.00000000000000000001 = 1200000000000000000000
As we can see the smaller it is the dividerthe number by which we divide, e the greater the result. When the divisor approaches zero it is said to become infinitesimal and the result grows indefinitely: it can be seen as an infinitely large quantity. Using the language of limits we say that if the divisor tends to 0 then the result tends to ∞. In a certain sense, therefore, we can say that ∞ it is the result of division 12 ÷ 0. But then can it be divided by 0?
From the point of view of calculating the limits the calculators that write “∞” they are not wrong, but be careful, infinity in mathematics is not a number so in arithmetic, where we are talking about numbers, 12 ÷ 0 still does not have a result because ∞ it’s not a number.
