Why is negative times negative, a positive?

By Chandrahas M. Halai

The most common answer to this question is that this is a rule. But in mathematics every rule has a logical reason.

Let us examine or have a fresh look at what is multiplication, what are negative numbers and how multiplication between a positive and a negative numbers is defined and lastly how multiplication of a negative number with a negative number is defined.

Counting

In the very beginning, humans invented natural numbers (or is it that they discovered them, or is it both) for the purpose of counting and measurement. One can imagine a scenario where early human needed numbers to keep a tab on his flock of cattle or horses. Cave paintings by early humans show us that they used to keep a score of their kills during their various hunting expeditions by painting lines on the walls of the caves they used to dwell in or by etching notches/lines on dead animal bones ^[1].

When humans started living in groups or tribes and there was a conflict between two tribes/groups the strategist/chieftain needed to compare the number of fight worthy men in both the tribes to decide whether it is wise to fight the battle or make a truce or make good a retreat.

Addition

As the human population grew and it started living in communities and settling in villages or small towns, so did the number of live stock and other things in its possession. People started exchanging or bartering things. Let us say a tribe chieftain wants to know the total of all livestock/cattle or horses they have at all the stables in the village. Or a General wants to know the total number of guards present at all the gates or watch towers of a fortification or a chief wanting to know the population of his tribe after merger with various small groups or after a split. Simple counting of things became tedious and time consuming, this scenario warranted the development of a method/algorithm to aid counting. These led to the development of Addition as an aid or extension to counting. The development of addition also led to the development of Subtraction as an inverse operation.

Multiplication

Soon people found that many times they were adding the same/equal numbers repeated times. You can imagine a scenario that a father, Shanti Prasad wants to buy 6 apples for each of his five children. He must add 6, five times to know the total number of apples he needs to buy from the fruit vendor. Hence Multiplication was developed as an extension of addition. Division was developed as an inverse operation to multiplication. You can visualize the multiplication process in the present example as shown in figure 1.

6 apples / child x 5 children = 30 apples

Figure 1

Distributive Rule of Multiplication

It was soon found that multiplication obeys the rule of distribution. i.e.

6 apples / child x 5 children = 6 x (3 girls + 2 boys) = 18 apples for the girls + 12 apples for the boys = 30 apples

We know that 6 x 5 = 30

Figure 2

Negative Numbers

As human civilizations evolved, they started living in cities; trade and commerce grew. Arithmetic was used to keep accounts, calculate taxes, measure areas and volumes. Algebra evolved in ancient and medieval India to deal with problems involving unknown quantities. In algebra situations arose when early mathematicians got perplexed or hit a stone wall. For e.g.

X + 5 = 2

Hence, X = 2 – 5

Early mathematicians used to wonder, how can a larger quantity be subtracted from a smaller quantity? Indian mathematicians found a way to solve such equations by inventing negative numbers.

Brahmagupta and operations with Negative Numbers

Though earlier Indian mathematicians may have worked with negative numbers, it was Brahmagupta who first wrote about the rules for doing arithmetic with negative numbers in his book Brahmasphutasiddhanta (The Opening of the Universe), in 628.

He also gives arithmetical rules in terms of fortunes (positive numbers) and debts (negative numbers):

A debt subtracted from zero is a fortune.
A fortune subtracted from zero is a debt.
The product or quotient of two fortunes is one fortune.
The product or quotient of two debts is one fortune.
The product or quotient of a debt and a fortune is a debt.
The product or quotient of a fortune and a debt is a debt.

We can imagine a scenario wherein a man returns home and tells his wife that he lost 5 silver coins at the gambling den. His wife is puzzled, how can he lose 5 coins when he had only 2. The husband clarifies that not only did he lose the 2 silver coins he had but also the 3 silver coins that he had borrowed from his friend. Hence he is now in debt of 3 silver coins.

i.e. 2 – 5 = -3

Twelfth century A.D. Indian mathematician, Bhaskaracharya also used to consider negative quantities as debt (ऋण) or losses ^[2]. Till the seventeenth century A.D. some European mathematicians used to consider negative numbers as absurd and resisted to accept their existence ^[3].

In other words the man will have to earn 5 silver coins to pay off his debt and regain his earlier financial status.

-3 + 5 = -3 + (3 + 2) = -3 + 3 + 2 = 2

Also, x (debt) + 5 = 2

Hence, x = 2 – 5 = -3

Additive Inverse

When a number is subtracted from itself we get zero. i.e.

a – a = 0    (a is a positive integer)

This is the same as a + (-a) = 0, adding a negative of a number is the same as subtracting

Two numbers having the same value but different signs (positive and negative) are called Additive Inverses of each other. In the above examples a and –a are additive inverses of each other.

Multiplication between a positive and a negative

Humans also found that many times they were subtracting the same values repeatedly. Let us again take the help of Shanti Prasad and his children. Shanti Prasad has 50 copper coins and he gives away 3 copper coins to each of his 5 children. To know how many copper coins will remain with him we have to subtract 3, five times from 50. i.e.

50 – 3 – 3 – 3 – 3 – 3 = 35

We can rewrite this repeated subtraction as

50 + (- 3 – 3 – 3 – 3 – 3) = 35

50 + (5 x -3) ?= 35

To balance this equation 5 x -3 has to be -15, hence

50 – 15 = 35

Looking at multiplication between a positive and negative in a different way, we have

a – a = 0    (a is a positive integer)

Let’s multiply both sides by a positive integer b, we get

b x (a – a) ?= b x 0

b (a – a) ?= 0

Now we know that b x a = ab, so multiplying –a (which is the additive inverse of a) by b should give us additive inverse of ab, i.e. –ab, to balance the equation and obey the Distributive rule of multiplication.

Thus, we have

ab – ab = 0

This is how multiplication between a positive and a negative number is defined, such that the multiplication obeys the distribution rule of multiplication. We can now state this as a rule:

“Multiplication of a positive number with a negative number gives us a negative number.”

And now for the final assault.

Multiplication between a negative and a negative

Let us again take the help of Shanti Prasad and his family. He has with him a treasure of 100 gold coins and he instructs his wife to give away 10 gold coins to each of his 5 children. So Shanti Prasad will be left with 100 – 5(10) = 50 gold coins after the distribution. After few days he checks his treasure and finds that he has 65 gold coins left with him. He inquires with his wife and finds that she has distributed 3 less gold coins to each of her 5 children. i.e.

100 – 5(10 – 3) = 100 – 5(7) = 100 – 35 = 65

Also, we have

100 – 5(10 – 3) ?= 65

100 + (-5 x 10) + (-5 x -3) ?= 65

100 – 50 + (-5 X -3) ?= 65

50 + (-5 x -3) ?= 65

To balance this equation -5 x -3 has to be +15, hence

50 + 15 = 65

Looking at multiplication between a negative and a negative in a different way, we have

a – a = 0    (a is a positive integer)

Let’s multiply both sides by –b (b is a positive integer), we get

–b x (a – a) ?= –b x 0

–b (a – a) ?= 0

Now we know that –b x a = –ab, so multiplying –a (which is the additive inverse of a) by –b should give us additive inverse of –ab, i.e. +ab, to balance the equation and obey the Distributive rule of multiplication.

Thus, we have

–ab + ab = 0

This is how multiplication between a negative number and a negative number is defined, such that the multiplication obeys the distribution rule of multiplication. We can now state this as a rule:

“Multiplication of a negative number with a negative number gives us a positive number.”

References:

[1] Calvin C. Clawson, The Mathematical Traveler – Exploring the Grand History of Numbers, Viva Books Pvt. Ltd., India, 2008, pp. 33-35

[2] Calvin C. Clawson, The Mathematical Traveler – Exploring the Grand History of Numbers, Viva Books Pvt. Ltd., India, 2008, pp. 126

[3] Calvin C. Clawson, The Mathematical Traveler – Exploring the Grand History of Numbers, Viva Books Pvt. Ltd., India, 2008, pp. 131-132