Why Minus times Minus equals Plus?
1. Intuitive Understanding via Physics Model
One of the first stumbling blocks in mathematics is the multiplication of negative numbers. We can approach this intuitively using the engineering models of "Velocity" and "Time".
Imagine a straight road stretching East and West. Let "East" be the positive direction and "West" be the negative direction.
Scenario Setup
- Current Position: At point $0$.
- Velocity: Driving at $-50\mathrm{km/h}$.
(Negative, so the car is reversing towards the West) - Time: $-2$ hours.
(Negative time, so we consider 2 hours ago in the past)
Now, where was this car "2 hours ago"?
Since it is currently passing point $0$ while reversing West, if we rewind time, the car must have come from the "East (Positive) direction".
Thus, considering a "reverse movement" while "rewinding time" results in a "positive position." This is the explanation through physical intuition.
2. Algebraic Proof
However, the example above is merely an "interpretation." In mathematics, what matters more is the logical necessity of "why it must be so by definition."
This fact is automatically derived from the following three axioms included in the definition of real numbers:
- 1. Distributive Property:
$a(b + c) = ab + ac$ - 2. Definition of Additive Inverse:
For any real number $x$, $x + (-x) = 0$ holds. - 3. Definition of Multiplicative Identity:
For any real number $x$, $x \times 1 = x$ holds.
Steps of the Proof
As the most basic example, we will prove that $(-1) \times (-1) = 1$.
First, start with the fact that multiplying any number by $0$ yields $0$.
\[ (-1) \times 0 = 0 \]Here, rewrite $0$ as $(1 + (-1))$ (Definition of Additive Inverse).
\[ (-1) \times \{ 1 + (-1) \} = 0 \]Now, apply the Distributive Property to expand. This is the core of the proof.
\[ \underbrace{(-1) \times 1}_{-1} + \underbrace{(-1) \times (-1)}_{?} = 0 \]The first term $(-1) \times 1$ becomes $-1$ by the Definition of Multiplicative Identity. Thus, the equation becomes:
\[ -1 + \{ (-1) \times (-1) \} = 0 \]Add $1$ to both sides of this equation to eliminate the $-1$ on the left side.
\[ 1 + (-1) + \{ (-1) \times (-1) \} = 1 + 0 \]The $1 + (-1)$ on the left side becomes $0$ and disappears. As a result, the remaining equation is:
(Q.E.D.)
3. Dr.WataWata's Insight
This proof demonstrates that the rule "Minus times Minus equals Plus" is not something arbitrarily decided by someone, but an "inevitable consequence of preserving the Distributive Property."
If we were to define $(-1) \times (-1) = -1$, the powerful tool known as the Distributive Property would contradict itself, and the system of algebra (and consequently the circuit theory and control engineering that rely on it) would collapse.
Mathematical rules that appear strange at first glance are actually constructed in a way that they "must be so" in order to maintain the Consistency of the entire system.