Page 9 - SM inner class 7.cdr

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(i) Com mu ta tive prop erty un der ad di tion: (5 - ) 6 - 4 = - 1 4 = - 5 (RHS)
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Addition is commutative for integers. For any \ 5 - ( 6 - 4 ¹ ( 5 - 6 - 4
two integers a and b a, + b = b + a (iii) As so cia tive prop erty un der
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For ex am ple : 5 + -( 6 = 5 - 6 = - ; mul ti pli ca tion:
(-6 ) + 5 = - + 5 = -1 Multiplication is associative for integers. For
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\ 5 + - 6 = - 6 + 5 any three integers a b, and c,
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(a ´ ) b ´ = a ´ (b ´ ) c
c
(ii) Com mu ta tive prop erty un der
sub trac tion: For ex am ple :
6
´
Subtraction is not commutative for integers. [(-3 ) (-2 )] ´ 4 = ( ´ 4 ) = 24 (LHS)
´
´
=
For any two integers a and b a, - b ¹ b - a (-3 ) [(-2 ) ´ 4 ] (-3 ) (-8 ) = 24 (RHS)
´
\ [(-3 ) (-2 )] ´ 4 = [(-3 ) (-2 ) ´ 4 ]
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For ex am ple : 8 - -( 6 = 8 + 6 14;
=
(-6 ) - 8 = - - 8 = -14 (iv) As so cia tive prop erty un der di vi sion:
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\ 8 - - 6 ¹ - 6 - 8 Division is not associative for integers.
(iii) Com mu ta tive prop erty un der 4. Distributive Property
mul ti pli ca tion:
(i) Dis trib u tive prop erty of
Multiplication is commutative for integers. mul ti pli ca tion over ad di tion:
For any two integers a and b.
For any three integers a b, and c,
ab = ba
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a ´ ( b + c = ( a ´ b + ( a ´ c)
For ex am ple : 9 ´ -( 6 = -( 9 ´ 6 = - 54;
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For ex am ple : -2 4( + 3) = -2 7) = -14
(-6 ) ´ 9 = - ( ´ 9 ) = -54
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2
2
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4
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\ 9 ´ - 6 = - 6 ´ 9 = - ´ ) + - ´ )
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(
6
8
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(iv) Com mu ta tive prop erty un der = - ) + - )
di vi sion = -14
Division is not commutative for integers. (ii) Dis trib u tive prop erty of
For any two integers a and b a, ¸ b ¹ b ¸ a mul ti pli ca tion over sub trac tion:
For ex am ple : (-14 ) ¸ 2 = -7 For any three integers, a b, and c,
- 1
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2 ¸ - 14 = a ´ ( b - c = ( a ´ b - ( a ´ c)
)
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For ex am ple : -2 4( - 3) = -2 1) = -2
(
(-14 ) ¸ 2 ¹ 2 ¸ (-14 )
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(- ´4 ) (- ´3 ) = - ) - - )
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2
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8
2
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3. Associative Property
= -2
(i) As so cia tive prop erty un der ad di tion
The distributive property of multiplication
Addition is associative for integers. For any over the operations of addition and subtraction
three integers a b, and c is true in the case of integers.
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a + ( b + c = ( a + b + c
Identity Under Addition
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For Ex am ple : 5 + - +( 6 4 = 5 + - 2 = 3(LHS) Integer 0 is the identity under addition. That
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(5 - ) 6 + 4 = ( 1 + 4 = 3 (RHS) is, for any integer a,
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\ 5 + - + 4 = ( 5 - 6 + 4 a + 0 = 0 + a = a
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(ii) As so cia tive prop erty un der For ex am ple : 4 + 0 = 0 + 4 = 4
sub trac tion
Identity Under Multiplication
Subtraction is not associative for integers. For
The integer 1 is the identity under
any three integers a b, and c,
multiplication. That is, for any integer a,
)
)
a - ( b - c ¹ ( a - b - c
1 ´ a = a ´ 1 = a
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For Ex am ple : 5 - ( 6 - 4 = 5 - 2 = 3 (LHS)
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Mathematics In Focus - 7