Division of numbers
Dividing numbers is the opposite process to multiplying numbers. At the same time, division is a good source of humour because, as everyone knows, we can't divide by zero because it would burn our notebook.
What is division of numbers
We can think of division as an example of dividing a large whole into equally sized smaller parts. We might have a dad who needs to cut a 100 cm long board into five equal sized pieces. How big will each piece be?
We can do the reverse - if we add up the lengths of all five cut parts, we must get back the number 100. What number does this satisfy? It is the number 20, because 20 + 20 + 20 + 20 + 20 = 100. We can also use multiplication instead of addition. We can ask what number multiplied by five gives 100? Again, the number 20, because 5 · 20 = 100.
Graphically, we could express this as dividing a line segment of length 100 into five equal lengths:
If we want to divide the number 100 by 5, we find "how many times does the number five fit into the number one hundred". We write the division either with a slash: 100 / 5 or with a colon: 100 : 5. The result of dividing two numbers is a quotient. In our examples, the number 20 is the quotient of the numbers 100 and 5. The number on the left is called the divisor, the number on the right is called the quotient. We can also write the division using a fraction, instead of 100 / 5 we can write $\frac{100}{5}$. Summary of nomenclature:
$$ a : b = \frac{a}{b} = c $$
- a is called a divisor,
- b is called the divisor,
- c is called a quotient.
Definition by multiplication
The simplest way to define a quotient is by multiplication. For an example, let's take the quotient of the following numbers: 35 / 7. What will be the result? We are looking for a number that, when multiplied by 7, would give the number 35. If we were dividing 48 / 8, we would be looking for a number that, when multiplied by 8, would give the number 48.
So the result of 35 / 7 is the number 5, because 7 · 5 = 35. The result of 48 / 8 is the number 6, because 8 · 6 = 48.
If we generally divide a / b, then the result is the number c, for which b · c = a holds.
Although we have been showing all the examples using natural numbers, we can divide any other numbers because of the definition using multiplication. For example, we can find the quotient −85,76 / 6,7 by finding the number x, for which 6,7 · x = −85,76 holds. This is true for x = −12,8.
We can divide zero by another number
We can divide zero by some other number, this is a valid expression: 0 / 15. Basically what we are saying is that we want to divide zero into 15 equal parts. We can imagine that we have no pie and we want to divide this pie that we don't have into 15 parts - how big will the parts be? They will be zero, because we simply don't have any pie. Alternatively, imagine that you have zero crowns in your wallet and you want to divide these zero crowns between three children. How much does each child get? They get nothing, because you have nothing. That's why 0 / 15 = 0.
We can divide zero by another number, but we can't divide another number by zero. This expression is invalid: 8 / 0.
But why can't we divide by zero?
We can't divide by zero because the Eleventh Commandment forbids it.
But there are more compelling reasons. For now, we'll look at the case where the quotient is different from zero - division is of the form a / 0, where a ≠ 0. We'll list two reasons:
- Let's try to stick with the analogy of dividing a whole into smaller, equally sized, parts. We have 15, the cruelest, Pokémon. Now we want to divide them among zero children. How many Pokémon does each child get?
Eh????
Yes, you read the assignment correctly, and yes, it doesn't make sense. We can't ask how many Pokémon each child got if we didn't have any children to give the Pokémon to. That's why dividing by zero doesn't make sense either, and why we say the term x / 0 is undefined.
- We introduced division by multiplication. When we try to divide by zero 15 / 0, we are looking for some number x, for which x · 0 = 15 holds. Only whatever times zero is zero, we will never find x, for which this equation makes sense.
What about the fraction 0/0?
That is undefined just like x / 0. Reasons:
This time, we have zero of the cruelest Pokémon to distribute among zero children. Again, this is a nonsensical request.
According to the definition of division, when we divide 0 / 0, we are looking for a number x for which x · 0 = 0 applies. We would find a solution, or more precisely: any real number is a solution to this equation. Whether we substitute 4, 1 or π after x, the equation will be satisfied. Of course, it is inadmissible for the result of division to be the whole set of real numbers; that makes no sense.
We would have to pick one particular number from the whole set and say that this particular number is the result of dividing 0 / 0. But which one? Should it be equal to one? Eight? Zero? Minus one? And why is that? Let's try to make arguments for zero and for one:
- We know that if we have a fraction of the form x / x, then this fraction is equal to one. For example, 7 / 7 = 1 or 3 / 3 = 1. From here we could deduce that 0 / 0 should be defined so that 0 / 0 = 1 holds.
- Let's look at this sequence of surely valid fractions. In each of these, we will divide zero by a number that gets closer and closer to zero:
$$\begin{eqnarray} \frac{0}{1} &=& 0\\ \frac{0}{0{,}1} &=& 0\\ \frac{0}{0{,}01} &=& 0\\ \frac{0}{0{,}001} &=& 0\\ \frac{0}{0{,}0001} &=& 0\\ &…&\\ \frac{0}{0{,}0000000001} &=& 0\\ &…&\\ \frac{0}{0} &=& 0?\\ \end{eqnarray}$$
We see that no matter how close we divide by a number close to zero, we still get zero as the result of the fraction. From this we might deduce that 0 / 0 = 0.
These are both valid arguments for how we might define the proportion 0 / 0. Just which one is better? In the end, even if we chose one of them and declared it the only valid one, we would still end up with a conflict with other parts of mathematics.
That's why division by zero is better left undefined.
The final argument against division by zero
If the preceding arguments were not enough for you, perhaps the following photo documentation of a case where someone actually tried to divide by zero will convince you:
