Zero — the most problematic divisor
In division, we can use any numbers other than zero as denominator without any conditions. As you know, when dealing with a fraction, there always is an assumption that the denominator must not be zero. It seems like zero is excluded from division. Why? Why do we have to avoid defining division with zero?
Limiting approach
After trying calculating 1 ÷ 0 with google calculator (see Fig 1), it gives infinity as the answer (note that the infinity is the idea to represent a large, actually very very large, quantity in mathematics). This answer is derived from substituting the denominator with numbers gradually close to zero. For example, we start substituting the denominator with 0.1, 0.01, 0.001, and so on. Then we have
- 1 ÷ 0.1 = 10
- 1 ÷ 0.01 = 100
- 1 ÷ 0.001 = 1000
- …
It is observed that the quotients tends to increase as the decreasing denominator, that is, the closer to the zero of the denominator, the more value of the quotient. Therefore, it can be concluded that the result from 1 ÷ 0 should be infinity.
However, the numbers close to zero do not need to be only positive values. We then consider the numbers adjacent to zero from the negative side instead of positive one.
- 1 ÷ (-0.1) = -10
- 1 ÷ (-0.01) = -100
- 1 ÷ (-0.001) = -1000
- …
As you can see, the quotients now have extra negative signs compared to those from the previous case. Therefore, the best inference that we can make, based on the observations, is that 1 ÷ 0 should be negative infinity, not positive one. Now, we have both positive and negative infinity as the answer although we use the same technique to calculate the same value 1 ÷ 0. How can this even possible?
It seems that the limiting process does not work effectively in this situation. So, back to the meaning of division, the division can be interpreted as splitting an object to several equal segments. For example, we have a group of ten people. In order to make two equal subgroups, how many people in each group are there? ..Well, we just assign five people to each group, and everything sounds perfect because 5 × 2 = 10. Actually, this is the mathematical definition for the division. We can say that for any numbers a, b and c, a ÷ b is equal to c if a = b × c, where b ≠ 0.
How about division by zero?
If we use the concept of the division, we have to partition an object to zero equal group. Wait! ..dividing the object into ZERO group? ..what is it? what does it mean exactly? It unintentionally turns to be a very hard philosophical question if we try to extract the meaning from this statement. Let’s consider in terms of mathematics. What is the value of a ÷ 0 for any numbers a? Although we have not known the value yet, we still can assume that the quotient exists and equals a number, say c. Along this calculation, if we find a contradiction, it means that what we assume is wrong and then it should be concluded that the quotient does not exits. However, if we cannot find any contradictions, it only implies that the assumption just corresponds to the statement, but we cannot use it to infer any other broader conclusions (see more details about the contradiction via this link).
Consequently, from the definition of the division,
for any numbers a, b and c, a ÷ b is equal to c if a = b × c, where b ≠ 0,
we can write a = 0 × c (because the divisor is zero), but the multiplication of zero with any numbers always results in zero (click this link to see the proof). Finally, we get a = 0.
The catastrophe occurs immediately because we did not assign the value of a so a can be any numbers but the calculation will always set a to be zero! From this point, we can divide the catastrophe into two cases depending on a.
Case I: a ≠ 0
For example, we assume a = 2 (this is valid because 2≠ 0). Then the calculation will lead to the contradiction due to the multiplication of zero with other numbers which always give zero. Therefore, we have
but 2 is absolutely not equal to 0. This point is called as a contradiction. As mentioned earlier, if we find a contradiction, it means that the assumption is incorrect. So, the quotient c does not exist, and we can conclude that the answer of a ÷ 0 does not exist.
Case II: a = 0
For this case, we set a = 0, and it is observed that the following equation,
is always true no matter what c is, that is, c can be any numbers. However, c is the quotient! We may be curious that why it possesses several numbers (you can try substituting with any numbers, and this equation is still valid). Due to a lot of numbers that satisfy the equation, we then decide not to define the answer c, that is, the quotient c is undefined.
Summary
In conclusion, we divide into two cases to find the quotient of a ÷ 0.
- If a ≠ 0, there is no c satisfies the definition for the division because of multiplying with zero. Therefore, a ÷ 0 does not exists.
- If a = 0, we can find the value of c but it can be any numbers. Therefore, a ÷ 0 = 0 ÷ 0 is undefined.
(Extra) The well-posed problem
Moreover, there is an idea to describe characteristics of a problem called well-posed problem, proposed by a French mathematician named Jacques Hadamard. We can say a problem is well-posed if
- a solution exists.
- the solution is unique.
- the solution’s behavior changes continuously with the initial conditions.
Therefore, on calculating a a ÷ 0 for the case that a ≠ 0, this problem is not well-posed because a solution does not exists. In case of a = 0, the problem is still not well-posed but the reason is because the solution is not unique.
With the concept of well-posed problem, it will be better if we always ask ourselves before solving a certain problem that
- Is there any solution?
- Is the solution unique?
- If the solution is not unique, what are the possible values for the answer?
Hope you enjoy this article. Happy learning!
