The Singularity of Math:
Why You Must Not Divide by Zero
Where calculators return errors and circuit simulators crash.
Understanding the difference between "Undefined" and "Indeterminate".
1. The Question
"Do not divide by zero." It is a rule everyone learns in elementary school. If you type $1 \div 0$ into a calculator, it displays "E (Error)".
"No, since we are dividing by an infinitely small amount, shouldn't it be infinity ($\infty$)?"
Mathematically speaking, there are two patterns of division by zero: "Undefined" (no answer exists) and "Indeterminate" (any answer is possible). Let's uncover the true nature of these by thinking of division as the "inverse of multiplication".
2. Undefined: A World with No Answer
First, let's consider the case of $1 \div 0$. Let's assume the answer is $X$.
$$ 1 \div 0 = X $$
Converting this back to multiplication:
$$ X \times 0 = 1 $$
Now, does a number $X$ exist in this world that becomes "1" when multiplied by 0?
$1 \times 0 = 0$, and even $100 \times 0 = 0$. No matter how large a number you bring, multiplying by 0 turns it into nothing (zero).
In other words, "There is not a single $X$ in the universe that satisfies this condition." No solution. In mathematical terms, this is called "Undefined". This is the first reason why it is forbidden.
3. Indeterminate: A World where Anything is Possible
Next, let's consider the even stranger case of $0 \div 0$.
$$ 0 \div 0 = X $$
Converting this back to multiplication:
$$ X \times 0 = 0 $$
What is $X$ that satisfies this equation?
- If $X = 1$: $1 \times 0 = 0$ (Correct!)
- If $X = 5$: $5 \times 0 = 0$ (Correct!)
- If $X = 1 \text{ million}$: $1 \text{ million} \times 0 = 0$ (Correct!)
Surprisingly, "any number becomes the correct answer." Since the answer cannot be determined as a single value, it makes no sense as a calculation result. In mathematical terms, this is called "Indeterminate".
It is troublesome when there is "no answer (Undefined)", but when "anything goes (Indeterminate)", the very rules of the game of mathematics break down.
4. Engineering View
For us engineers, division by zero is not just a mathematical play. It implies physical destruction.
In Ohm's law $I = V / R$, what happens if the resistance $R$ is set to $0$ (short circuit)?
$$ I = \lim_{R \to 0} \frac{V}{R} = \infty $$
Mathematically, an "infinite current" flows. In the real world, wires overheat, circuits burn out, and breakers trip.
Even in circuit simulators like SPICE, if the denominator becomes 0 during the matrix calculation process, the infamous "Singular Matrix" error occurs, and calculation becomes impossible. The act of "dividing by zero" is synonymous with creating a "Singularity" that collapses the laws of physics in the virtual world of the simulator.
5. Dr.WataWata's Insight
The Wisdom of "Not Defining"
"Do not divide by zero" is not just a mean school rule. It is a failsafe to protect the system of mathematics, warning that "logic will collapse if you proceed further."
Being "Undefined" or "Indeterminate" is not a failure. The fact that "no answer exists there" is, in itself, an important discovery.
Rather than forcing a calculation, observing what happens at the edge of that singularity (limit) is the true scientific attitude.