2 x 3
To represent this on the number line, we start at 0 and put 2 groups of 3 of the number line.
We end up at 6. So, the product is 6.
Therefore, 2 x 3 = 6 (Positive × Positive = Positive)
2 x (-2)
To represent this on the number line, just start at 0 and put 2 groups of -2 of the number line.
We end up at (-4). So, the product is (-4).
Therefore, 2 x (-2) = 4 [(Positive × Negative) or (Negative x Positive) = Negative]
Steps
1. Multiply them as whole numbers.
2. Put a minus sign (–) before the product.
Example: 12 × (–15)
Solution: First find the product of whole numbers i.e. 12 x 15 = 180
Now, put a minus sign (–) before the product = -180
Steps
1. Multiply the two negative integers as whole numbers.
2. Put the positive sign before the product because product of two negative integers is a positive integer.
In general, for any two positive integers a and b, (– a) × (– b) = a × b
Example: (-9) × (–11)
Solution: First multiply the two negative integers as whole numbers i.e. 9 x 11= 99
Now, put a minus sign (+) before the product = +99
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.
This means,
(a) The product of two negative integers is a positive integer.
(b) The product of three negative integers is a negative integer.
(c) Product of four negative integers is a positive integer.
Let’s understand this with following examples.
Example: (a) (– 4) × (–3)
(b) (– 4) × (–3) × (–2)
(c) (– 4) × (–3) × (–2) × (–1)
Solution: (a) (– 4) × (–3) = 12 (number of negative integers in a product is even, so the product is a positive integer)
(b) (– 4) × (–3) × (–2) = [(– 4) × (–3)] × (–2) = 12 × (–2) = – 24 (the number of negative integers in a product is odd, so the product is a negative integer)
(c) (– 4) × (–3) × (–2) × (–1) = [(– 4) × (–3) × (–2)] × (–1) = (–24) × (–1) (number of negative integers in a product is even, so the product is a positive integer)
Closure property under multiplication states the product of two integers will always be an integer.
Let’s consider the following pairs of integers.
(-5) x (-6) = 30 (Result is an integer)
15 x (-10) = -150 (Result is an integer)
(-7) x (-8) = 56 (Result is an integer)
Since multiplication of integers gives integers, we say integers are closed under multiplication.
In general, a × b is an integer, for all integers a and b.
Commutative property of multiplication states that swapping of terms will not change the product.
Let’s consider the following examples.
(-8) x (-12) = (-12) x (-8)
(-11) x 100 = 100 x (-11)
(-19) x 0 = 0 x (-19)
So, we can say that multiplication is commutative for integers.
In general, for any two integers a and b, we can say a x b = b x a
Associative property of multiplication states that the way of grouping of numbers will not change the result.
Let’s consider the following example.
(–6) x [(–4) x (–3)] and [(–6) x (–4)] x (–3)
In the first case (–4) and (–3) are grouped together and in the second (–6) and (–4) are grouped together.
(–6) x [(–4) x (–3)] = (-6) x 12 = (-72)
[(–6) x (–4)] x (–3) = 24 x (-3) = (-72)
In both the cases, we get –72.
So, Multiplication is associative for integers.
In general, for any three integers a, b and c (a × b) × c = a × (b × c)
Distributive property of multiplication explains the distributing ability of an operation over another mathematical operation within a bracket. It can be either distributive property of multiplication over addition or distributive property of multiplication over subtraction.
Let’s consider the following example.
(–2) × (3 + 5) = –2 × 8 = –16 and;
[(–2) × 3] + [(–2) × 5] = (– 6) + (–10) = –16
So, (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 5]
In general, for any integers a, b and c, a × (b + c) = a × b + a × c
This property of multiplication states that the product of any integer (positive or negative) and zero is zero.
Let’s consider the following examples.
(-98) x 0 = 0
0 x 67 = 0
So, we can say that multiplication of any integer and zero gives zero.
In general, for any integer a, a × 0 = 0 × a = 0
Multiplicative identity property states that when we multiply one to any integer, we will get the integer itself as the product.
Let’s observe the following examples:
(– 16) x 1 = – 16
1 x (– 81) = – 81
The above examples show that 1 is the multiplicative identity for integers also.
In general, for any integer a we have, a × 1 = 1 × a = a
Example: Find 15 × 17
Solution: 15 × 17 can be written as 15 × (10 + 7).
= 15 × (10 + 7) = 15 × 10 + 15 × 7 = 150 + 105 = 255
Example: Find (-22) × 98
Solution: (-22) x 98 can be written as (-22) x (100 - 2)
= [(-22) × 100] – [(–22) × 2] = (–2200) – (– 44)
= –2200 + 44
= -2156
Example: Find (–16) × (–10) × 8
Solution: [(–16) × (–10)] × 8
= 160 x 8
= 1280
Example: Verify (–40) × [11 + (–3)] = [(–40) × 11] + [(–40) × (–3)]
Solution: L.H.S = (–40) × [11 + (–3)]
= (-40) x 8
= -320
R.H.S = [(–40) × 11] + [(–40) × (–3)]
= -440 + 120
= -320
L.H.S = R.H.S
Hence verified
Example: [(– 10) × 8] + (– 6) is equal to (a) 100 (b) –90 (c) – 86 (d) 86
Solution: [(– 10) × 8] + (– 6)
= (-80) + (-6)
= -86
Correct answer is (c) = -86
Example: (– 15) × 40 = – 40 × _______.
Solution: (– 15) × 40 = – 40 × 15.
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