Division by zero is arguably one of the biggest taboos in mathematics. Legend has it that doing so will break the law of nature and mess up the whole universe.
But seriously, why? What’s the mathematical reason behind this forbidden operation? Let’s figure it out… by actually dividing by zero.
We are going to discuss this topic in the language of fractions, or more specifically, rational numbers. To understand what’s so bad about things like 3/0 and 0/0, let’s look at two principles that “good” rational numbers abide by.
Principle 1
We say that a/b = c/d precisely when ad = bc.
For instance, 2/7 = 6/21 because 2 × 21 = 7 × 6. On the other hand, 2/3 ≠ 4/5 because 2 × 5 ≠ 3 × 4.
Principle 2
The fraction a/b is that unique number such that when we multiply it by b, we get a, i.e.
Equivalently, a/b is the unique solution to the equation bx = a.
For example, 2/7 is the one and only solution to 7x = 2, i.e. 7 × (2/7) = 2.
Note that although 7 × (6/21) = 2 as well, by Principle 1, both 2/7 and 6/21 are considered as the same object in the world of fractions, and so there’s really just one solution.
Now that we understand how “good” rational numbers behave, we are ready to see why it’s a bad idea to define fractions with zero denominators. There are two possible scenarios, however, one for nonzero numerators (e.g. 3/0), and one for zero numerator (i.e. 0/0), so let’s analyse them separately.
Why isn’t 3/0 defined?
If we were to define 3/0, then according to Principle 2, we would require 3/0 to be the solution to the equation 0x = 3. However, as any number multiplied by 0 is equal to 0, this equation has no solution. In other words, there is no way we could define 3/0 without breaking Principle 2, and hence it is undefined. Ditto for every other “fraction” a/0 with nonzero numerator a.
What about 0/0?
According to Principle 2 again, we need 0/0 to be that one number that satisfies the equation 0x = 0, which is… any number! We have a different issue now, i.e. the problem is not about the existence, but rather the uniqueness of the solution, and because of this, we regard 0/0 as an indeterminate form.
You might say: “Since 0/0 could be anything, why don’t we just artificially assign a convenient value to it?”
While the interpretation of the word “convenient” is highly subjective, there’s another more serious reason as to why we avoid doing so:
Once defined, 0/0 is always equal to every other rational number!
This follows immediately from Principle 1. For example, we necessarily have 0/0 = 2/3 because 0 × 3 = 0 × 2, and similarly 0/0 = 4/5. Consequently,
Such a glaring contradiction!
Therefore, as long as 0/0 is there, any pair of rational numbers a/b and c/d will be equal to each other. Essentially, defining 0/0 turns it into a black hole that contracts the whole universe of rational numbers to a single point, thus breaking Principle 1.
What’s the upshot?
The main idea of division is to undo the process of multiplication. For example, if we allow “division by 8”, then the process of “multiplication by 8” is reversible. Take a look at the following two equations:
If we multiply Equation (1) by 8 on both sides, we get Equation (2). Conversely, if we divide Equation (2) by 8 on both sides, we get equation (1).
We say that Equations (1) and (2) are equivalent in the sense that solving either one is the same as solving the other. Symbolically, we usually use a double-headed arrow to denote the equivalence:
On the other hand, since we do not allow “division by 0”, the process of “multiplication by 0” is not reversible. Consider these two equations:
Multiplying Equation (3) by 0 on both sides does give Equation (4). However, there is no way we can go from Equation (4) back to Equation (3) as division by 0 is forbidden!
We therefore say Equations (3) and (4) are not equivalent, which makes sense as their solutions are completely different, i.e. x = 2 is the only solution to Equation (3), whereas every x is a solution to Equation (4) (which is really just a trivial identity 0 = 0). Hence, the implication is only one-directional as follows:
Note that the above discussion centres around a fundamental and pervasive concept in mathematics, namely the invertibility of an operation. Some quick examples are addition vs. subtraction, multiplication vs. division, and function vs. inverse function. We will definitely explore them in future, but until then, stay acute, stay positive. See you!
Remark 4.1
The fact that the indeterminate form 0/0 has the potential to be any number is, in some sense, the main driving force for differential calculus (more specifically, the derivative of a function). As this intriguing topic is way too advanced to be discussed in a paragraph or two, I will save that for a series of articles in future, so stay tuned!
