Page 58 - classs 6 a_Neat

EXAMPLE 28: Consider the integers (-3), (-4) and (-8). EXAMPLE 37: Divide
We have: (-3) × [(-4) + (-8)) = (-3) × (-12) = 36. (i) (-115) by 23 (ii) 168 by (-7)
And, (- 3) × (- 4) + (- 3) × (- 8) = 12 + 24 = 36. (iii) (-272) by (-16) (iv)(-324) by (-27)
(-3) × [(-4) + (-8)) = (-3) × (-4) + (-3) × (-8). -115
(v) For any integer a, we have a x 1 = a. The integer 1 is called the multiplicative identityjor integers. SOLUTION: (i) (-115) ÷ 23 = = - 5 (ii) 168 ÷ (-7) = 168 = 24
23 -7

EXAMPLE 29: (i) (-3) × l = - 3 (ii) 7 × 1 = 7 -272
(vi) For any integer a, we have a × 0 = 0. (iii) (-272) ÷ (-16) = = 17 (iv) (-324) ÷ (-27) = -324 = 12
-16 -27
EXAMPLE 30: (i) 4 × 0 = 0 (ii) (-4) × 0 = 0 EXAMPLE 38: Fill in the blanks:
(i) - 273 + ...... = 1 (ii) ...... + 137 = - 2
EXAMPLE 31: Simplify: (iii)...... + 238 = 0 (iv) ...... + (-13) = - 5
(i) 8 × (-13) + 8 × 9 (ii) (-12) × 7 + (-12) × (-4) SOLUTION:
(iii) 9 × (-16) + (-12) × (-16) (iv) 10 × (-31) + 10 × (-9) (i) Clearly, we have -273 + (-273) = 1.
So, the required number= -273.
SOLTUION: Using the distributive law, we have: (ii) Required number= 137 × (-2) = -274.
(i) 8 × (-13) + 8 × 9 = 8 × [(-13) + 9) = 8 × (-4) = - 32. (iii) Required number = 238 × O = 0.
(ii) (-12) × 7 + (-12) × (-4) = (-12) × 17 + (-4)) = (-12) × 3 = - 36. (iv) Required number= (-13) × (-5) = 65.
(iii) 9 × (-16) + (-12) × (-16) = [9 + (-12)) × (-16) = (-3) × (-16) = 48.
(iv) 10 × (-31) + 10 × (-9) = 10 × [(-31) + (-9)] = 10 × (-40) = - 400. REMARK (i) When 0 is divided by any integer, the quotient is 0.
(ii) We cannot divide any integer by 0.
DIVISION ON INTEGERS
PROPERTIES OF DIVISION ON INTEGERS
We know that division of whole numbers is an inverse process of multiplication. We extend the same idea to
integers. (i) If a and bare integers then (a+ b) is not necessarily an integer.
EXAMPLE 39: (i) 15 and 4 are both integers, but (15.+ 4) is not an integer.
EXAMPLE 32: To divide·36 by (-9) means: what integer should be multiplied with (-9) to get 36? (ii)(-8) and 3 are both integers, but 1(-8) + 3) is not an integer.
Obviously, the answer is -4. (iii)if a is an integer and a ≠ 0 then (a ÷ a)= l.
:. 36 ÷ (-9) = - 4.
EXAMPLE 33: To divide (-40) by 8 means: what integer should be multiplied with 8 to get (-40)? EXAMPLE 40: (i) 9 ÷ 9 = 1 (ii) c-7) ÷ (-7) = 1
Obviously, the answer is -5. (iii) if a is an integer then (a ÷ 1) = a.
:. (-40) ÷ 8 = - 5.
EXAMPLE 34: To divide (-35) by (-5) means: what integer should be multiplied with -5 to get (-35 )? EXAMPLE 41: (i) 6 ÷ 1 = 6 (ii)(-3) ÷ 1 = (-3)
Obviously, the answer is 7. (iii) if a is a nonzero integer then (0 ÷ a) = 0, but (a+ 0) is not meaningful.
: . (-35) ÷ ( -5) = 7.
Thus, we have the following rules for division of integers. EXAMPLE 42: (i) 0 ÷ 5 = 0 (ii) 0 ÷ (-3) = 0
(iii)(5 ÷ 0) is not meaningful.
RULE 1 For dividing one integer by another, the two having unlike signs, we divide their values regardless of (iv) lf a, b, care integers then (a÷ b) ÷ c - ≠ - a ÷ (b ÷ c), unless c = l.
their signs and give a minus sign to the quotient.
EXAMPLE 43: Let a = 8, b = 4 and c = 2. Then,
EXAMPLE 35 (i) (-36) ÷ 9 = -4 (ii) 12 ÷ (-8) = - 9 (a ÷ b) ÷ c = (8 ÷ 4) ÷ 2 = (2 ÷ 2) = 1.
(iii) ( -132) ÷ 12 = - ll (iv) 144 ÷ (-18) = -8 a ÷ (b ÷ c) = 8 ÷ ( 4 ÷ 2) = (8 ÷ 2) = 4.
(a ÷ b) ÷ c - ≠ - a ÷ (b ÷ c).
RULE 2 For dividing one integer by another, the two having like signs, we divide their values regardless of If c = I then (a ÷ b) ÷ c = (8 ÷ 4) ÷ 1 = 2.
their signs and give a plus sign to the quotient. And a ÷ ( b ÷ c) = 8 ÷ ( 4 ÷ I) = 8 ÷ 4 = 2.
:. in this case (a ÷ b) ÷ c = a ÷ (b ÷ c).
EXAMPLE 36: (i) 42 ÷ 7 = 6 (ii) (-42) ÷ (-6) = 7 (vi) lf a, b, c are integers and a > b then
(iii) (-98) ÷ (-7) = 14 (iv) (-84) ÷ (-21) = 4 (i) (a ÷ c) > (b ÷ c), if c ts positive
(ii) (a ÷ c) < (b ÷ c), if c ts negative