Integers

Multiplication of integers

Multiplication of a Positive and a Negative Integer

While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. The result is a negative integer.

In general, for any two positive integers a and b we can say a × (– b) = (– a) × b = – (a × b).

Look at the following table.

Expression	Next step	Value
(–33) × 5	= –(33 × 5)	= –165
33 × (–5)	= –(33 × 5)	= –165
–55 × 15	= – (55 × 15)	= – 825
15 × –16	= – (15 × 16)	= – 240

Multiplication of two Negative Integers

While multiplying two negative integers, we multiply them as whole numbers and put the positive sign before the product. The result is a positive integer.

In general, for any two positive integers a and b, we have (– a) × (– b) = a × b.

Look at the following table.

Expression	Next step	Value
(–10) × (–12)	= + (10 × 12)	= 120
(–15) × (– 6)	= + (15 × 6)	= 90
(–31) × (–100)	= + (31 × 100)	= 3100

Example. Find the value of (– 4) × (–3).

Solution. It is the product of two negative integers. Therefore, the result will be a positive integer.

We have (– 4) × (–3) = + (4 × 3) = 12.

Product of three or more Negative Integers

If the number of negative integers in a product is even, then the product is a positive integer.

If the number of negative integers in a product is odd, then the product is a negative integer.

Example. Find the value of (– 4) × (–3) × (–2).

Solution. It is the product of three negative integers. So, we need to apply the above result for number of times. Using associativity, we have

(– 4) × (–3) × (–2)

= [(– 4) × (–3)] × (–2)          

= + (4 × 3) × (–2)

= 12 × (–2)

= – (12 × 2)

= – 24.

Example. Find the value of (– 5) × (– 4) × (–3) × (–2).

Solution. It is the product of four negative integers. Now

(– 5) × (– 4) × (–3) × (–2)

= [(– 5) × (– 4)] × (–3) × (–2)

= 20 × (–3) × (–2)

= [20 × (–3)] × (–2)

= (–60) × (–2)

= + (60 × 2)

= 120.

A Special Case

Euler in his book Ankitung zur Algebra(1770), was one of the first mathematicians to attempt to prove (–1) × (–1) = 1. Consider the following simialr type of statements and the resultant products:

1. (–1) × (–1) = +1
2. (–1) × (–1) × (–1) = –1
3. (–1) × (–1) × (–1) × (–1) = +1
4. (–1) × (–1) × (–1) × (–1) × (–1) = –1
5. (–1) × (–1) × (–1) × (–1) × (–1) × (–1) = +1
6. (–1) × (–1) × (–1) × (–1) × (–1) × (–1) × (–1) = –1

This means that if the integer (–1) is multiplied an even number of times, the product is +1 and if the integer (–1) is multiplied an odd number of times, the product is –1. You can check this by making pairs of (–1) in the statement. This is useful in working out products of integers.