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Q. Find (– 49) × 18; (–25) × (–31); 70 × (–19) + (–1) × 70 using distributive property.

We know, Distributive law : for any integers a, b and c,

(ii) Is (–15) × [(–7) – (–1)] = (–15) × (–7) – (–15) × (–1)?

L.H.S =  R.H.S =  Therefore, L.H.S = R.H.S

(ii) Is (–15) × [(–7) + (–1)] = (–15) × (–7) + (–15) × (–1)?

L.H.S =  R.H.S =  Therefore, L.H.S = R.H.S Hence, 

(i)  Is 10 × [(6 + (–2)] = 10 × 6 + 10 × (–2)?

L.H.S =  R.H.S =   Therefore,L.H.S = R.H.S Hence, 

(ii)  What will be the sign of the product if we multiply together:

                (a) 8 negative integers and 3 positive integers?

                (b) 5 negative integers and 4 positive integers?

                (c) (–1), twelve times?

                (d) (–1), 2m times, m is a natural number?

We know, If the number of negative integers in a product is even, then the product is a positive integer and if the number of negative integers in a product is odd, then the product is a negative integer And, any number of positive integers will always give a positive integer.  So, the sign of the product will be decided by the number of negative integers. (a) 8 negative integers and 3 positive...

(i) The product (–9) × (–5) × (– 6)×(–3) is positive whereas the product (–9) × ( –5) × 6 × (–3) is negative. Why?

We know, If the number of negative integers in a product is even, then the product is a positive integer and if the number of negative integers in a product is odd, then the product is a negative integer. Hence, the product is positive whereas the product is negative  

Q. Find: (–31) × (–100), (–25) × (–72), (–83) × (–28)

We know, Multiplication of two negative integers : 

(ii) Starting from (– 6) × 3, find (– 6) × (–7)

Given, Therefore,

(i) Starting from (–5) × 4, find (–5) × (– 6)

Given, Therefore,

2. Check if     (a)    25 × (–21) = (–25) × 21         (b)    (–23) × 20 = 23 × (–20)

Write five more such examples.

We know, when multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer. (a)              L.H.S =  R.H.S =  Therefore, L.H.S = R.H.S (b)    L.H.S =  R.H.S =  Therefore, L.H.S = R.H.S Five more examples:

1.   Find:
                (a) 15 × (–16)         (b) 21 × (–32)
                (c) (– 42) × 12        (d) –55 × 15

We know, multiplication of a positive and negative integer is given by: (a)          (b) (c)          (d)

1. Find:
                (i) 6 × (–19)
                (ii) 12 × (–32)
                (iii) 7 × (–22)

Multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer (i)                (ii)                  (iii)

2.  Write a pair of integers whose difference gives
   (a) a negative integer.                                           (b) zero.
   (c) an integer smaller than both the integers.      (d) an integer greater than only one of the integers.
   (e) an integer greater than both the integers.

(a) a negative integer :                                                   (b) zero :  (c) an integer smaller than both the integers.       (d) an integer smaller than only one of the integers.  (e) an integer greater than both the integers. 

1. Write a pair of integers whose sum gives
 (a) a negative integer                                                 (b) zero
 (c) an integer smaller than both the integers.           (d) an integer smaller than only one of the integers.
 (e) an integer greater than both the integers.

(a) a negative integer :                                                   (b) zero :  (c) an integer smaller than both the integers.       (d) an integer smaller than only one of the integers.  (e) an integer greater than both the integers. 

Q. We have done various patterns with numbers in our previous class.
Can you find a pattern for each of the following? If yes, complete them:
    (a)     7, 3, – 1, – 5, _____, _____, _____.
    (b)     – 2, – 4, – 6, – 8, _____, _____, _____.
    (c)     15, 10, 5, 0, _____, _____, _____.
    (d)     – 11, – 8, – 5, – 2, _____, _____, _____.
Make some more such patterns and ask your friends to complete them.

(a)     7, 3, – 1, – 5,   The pattern is :  (b)     – 2, – 4, – 6, – 8,   The pattern is :  (c)     15, 10, 5, 0,  The pattern is :   (d)     – 11, – 8, – 5, – 2,   The pattern is :  

2. Arrange 7, –5, 4, 0 and – 4 in ascending order and then mark them on a number line to check your answer.

The given number are:  7, –5, 4, 0 and – 4 Arranging them in ascending order (increasing order) On a number line, as we move towards right, the number increases.

1.  A number line representing integers is given below

–3 and –2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O?

First, we complete the number line. Now, the integers marked by: B = -6 D = -4 H = 0 J = 2 M = 5 O = 7
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