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For example: (a) 3 × (–7) = –(3 × 7) = –21

(b) –5 × 4 = – (5 × 4) = –20
In general, (+ve) integer × (–ve) integer = (–ve) integer

(–ve) integer × (+ve) integer = (–ve) integer

In chapter 18 of Brahma-sphuṭa-siddhanta, Brahmagupta describes rules of multiplication on
negative numbers.
• The product of a negative and a positive is negative.
• The product of two negatives is positive.
• The product of two positives is positive.

Example 2: Find the product of the following.
(a) 12 × ( –8) (b) (–13) × 5 (c) (+7) × (–9)

Solution: (a) 12 × ( –8) = – (12 × 8) = –96 (b) (–13) × 5 = – (13 × 5) = –65
(c) (+7) × (–9) = – (7 × 9) = –63
Example 3: Multiply the following.

(a) 17 by 5 (b) 17 by (–5) (c) –17 by 5 (d) –17 by –5.
Solution: (a) 17 × 5 = 85 [Q (+ve) integer × (+ve) integer = (+ve) integer]

(b) 17 × (–5) = – (17 × 5) = –85 [Q (+ve) integer × (–ve) integer = (–ve) integer]

(c) (–17) × 5 = – (17 × 5) = –85 [Q (–ve) integer × (+ve) integer = (–ve) integer]
(d) (–17) × (–5) = (17 × 5) = 85 [Q (–ve) integer × (–ve) integer = (+ve) integer]

Multiplication of More Than Two Negative Integers

Let us observe the following products.
(a) (–2) × (–3) × (–4) = +(2 × 3) × (–4) = + 6 × (–4) = – (6 × 4) = –24
(b) (–2) × (–3) × (–4) × (–5) = + (2 × 3) × (–4) × (–5)
= + 6 × (–4) × (–5) Knowledge Desk
= – (6 × 4) × (–5) Euler was one of the
= (–24) × (–5) first mathematicians who
= + (24 × 5) = 120 attempted to prove
It is clear from the above products, that: (–1) × (–1) = 1.
• The product of even numbers of negative integers is positive.
• The product of odd numbers of negative integers is negative. Quick Check
Let us observe the multiplication pattern of (–1) by itself. What is the product of:
(–1) × (–1) = +1 (a) (–1) × (–1) × (–1) ×

(–1) × (–1) × (–1) = [(–1) × (–1)] × (–1) = 1 × (–1) = –1 ... 197 times?
(–1) × (–1) × (–1) × (–1) = [(–1) × (–1) × (–1)] × (–1) = (–1) × (–1) = + 1 (b) (–1) × (–1) × (–1) ×
Thus, we observe: ... 216 times?
• If the integer (–1) is multiplied even number of times to itself the product is +1. That is (–1) ×
(–1) × (–1) × ... even number of times = 1

13 Integers