The product of two negative numbers produces a positive number. Did it ever make you wonder why? The sum of two negative numbers is still negative, but why positive when multiplied? Come to think of it, how is it even possible to multiply a negative number by a negative number?
One of the easiest ways to understand the concept of negative number is to think of gain as positive, and loss as negative. This way may explain addition and subtraction, but when it comes to multiplication and division, it gets confusing. Stendhal, a French writer famous for his novel The Red and the Black, says in his autobiography, “suppose the negative quantities are a man’s debts, how by multiplying 10,000 francs of debt by 500 francs, will he or can he manage to acquire a fortune of 5,000,000, five million?” He puzzled over the improbable conversion or “changing sides” across a horizontal boundary in the image below.
Stendhal, quite the intellect of the age himself, shows that there were souls who agonized over this conceptual problem. As a matter of fact, the concept of a negative number was formally introduced much later than the concept of an irrational number in Europe, which might mean that people were more comfortable with operations on the latter than on the former. Math historian Ronald Calinger writes that even Blaise Pascal “derided those who thought of taking four away from nothing and getting minus four.”
In the 7th century, Indian mathematician Brahmagupta first discovered that a minus times a minus equals a plus, but even after a thousand years there were dissenting voices. In 18th century Europe many scholars overtly argued against it on the basis of reasoning that’s like “-3 is smaller than 2, then how can square of -3 be greater than square of 2?” So we can imagine the resistance that the concept of negative number created among people.
Model of positive and negative: gain and loss
Like Stendhal did, there is a model of gain and loss to represent the concept of signs. Adding a positive number would be gain, or profit, and adding a negative number (subtracting positive number) is loss. When gain becomes less than zero, it is interpreted as debt. We human beings, by our very nature, tend to apply the same thinking mechanism that was once successful again and again even in different circumstances, so it is natural for us to try to understand the multiplication and division under the same logic.
Then why is it inappropriate when operating on multiplication?
Do not multiply debt by debt
We realize that explaining multiplication of positive numbers by using the model does not make sense even before we get to negative numbers. Does multiplying gain by gain produce gain? When we have 10 of 100 dollar (bills), it means we have 1,000 dollars. (10 x 100 = 1,000.) It sounds perfectly reasonable according to the example. However, units are different; 100 is in dollars while 10 is a number of bills. We could interpret each 100 dollars as gain, but 10 is in a different unit of gain. It may be a contextually valid statement, but in mathematical sense, it is flawed because the unit of 10 simply disappears in the result.
The statement that gain times gain is gain may be invalid, but the model itself has no problem explaining that a plus times a plus equals a plus if we ignore the measurement issue and only consider the contextual interpretation. In addition, it can also explain a minus times a plus equals a minus. For example, when there are 10 people who owe you 100 dollars, you know they owe you 1,000 dollars as a whole. -100 x 10 = -1,000
But the model’s biggest drawback is that it cannot explain a minus times a minus even in a contextual sense because there is no measurement to quantify a negative number of, well, people, and things.
Then, how can we understand that a minus times a minus equals a plus?
1. Finding a pattern
The rows tell us that each product changes by the same amount when the number on the right gets smaller. The columns also show the same pattern: the first column decreases by 2, the second by 1, the third by 0, the fourth column increases by 1, and the fifth by 2. Accordingly, the next row would be,
From this, we learn that a minus times a minus produces a plus.
2. A horizontal line
One of the basic approach to negative number is giving the number a direction. It is to associate positive and negative number with positive direction and negative direction, so they move in the opposite way on a horizontal line. 3 and -3 are apart from the origin by the same distance, but their opposite directions make them different numbers. In fact, people finally stopped resisting the concept of signs when the horizontal line model was introduced. For example, multiplying 2 by 3 is adding three 2’s, so it is like moving by the amount of 2 three times from zero to the right on the line, which equals to 6. In a similar sense, multiplying 2 by -3 is moving by 2 from zero to the left (opposite direction of 2) three times.
It is the same as thinking of moving by -2 three times (the direction of -2), which tells that 2 x (-3) = (-2) x 3.
Finally (-2) x (-3) is moving by -2, yet in the opposite direction of (-2) three times, which basically cancels out and happens to move to the right. It leads to our desirable conclusion (-2) x (-3) = 6.
3. Gain and loss model revisited
We can explain multiplication of negative numbers by expanding the rules of the model. We include negative number operations beside the positive ones. Adding a negative number means decreasing gain, or increasing loss, or debt. Subtracting a negative number means decreasing loss, or debt, in other words, increasing gain. So for example, when a is a positive number, subtracting (-a) means that debt is decreased by the amount of a, so gain is increased by a. In short, -(-a) equals a, and from the model, we can logically justify that (-a) x b = -(a x b) = a x (-b) when a and b are positive. Then let’s say b is a negative number. We can express b equals -c for some positive number c. By substitution, (-a) x (-c) = a x (-(-c)) = a x c.
Thus, a negative number times a negative number is a positive number.
4. Mathematical proof
I explained that a minus times a minus equals a plus, but technically it wasn’t a proof. To prove (-a) x (-b) = ab, I will use the associative law of addition, the distributive law of addition and multiplication, and the fact that zero is the identity element under addition. We know 0 = 1+ (-1). Multiply both sides by -1, and apply the distributive law.
0 = (1+ (-1)) x (-1)
= 1 x (-1) + (-1) x (-1)
Since 1 x (-1) = -1, the equation above become 0 = -1 + (-1) x (-1). Add 1 to both sides, and we get 1= (-1) x (-1).
The formal proof will follow this format. But before I do that, I will show 0 x b =0.
0 = 0 x b – (0 x b)
= ((0+0) x b) – (0 x b)
= (0 x b + 0 x b) – (0 x b) = 0 x b
Without the loss of generality, a x 0 = 0 is true.
Now I want to show (-a) x (-b) = ab
(-a) x (-b) – a x b = (-a) x (-b) + a x (-b) – a x (-b) – a x b
= ((-a) + a) x (-b) – (a x (-b) + a x b)
= 0 x (-b) – (a x (-b + b))
= 0 – a x 0 = 0 – 0 = 0
Stendhal, The Life of Henry Brulard, trans. John Sturrock (New York: NYRB, 2002), 364-66.
Calinger, Ronald. Vita Mathematica: Historical Research and Integration with Teaching (New York: Cambridge University Press, 1996), 36.