2.
The integers which are in the form of
p/q where q is not equal to 0 are
known as Rational Numbers.
Examples : 5/8; -3/14; 7/-15; -6/-11
3.
Rational numbers are closed under addition. That is, for any two rational
numbers a and b, a+b s also a rational number.
For Example - 8 + 3 = 11 ( a rational number. )
Rational numbers are closed under subtraction. That is, for any two rational
numbers a and b, a – b is also a rational number.
For Example - 25 – 11 = 14 ( a rational number. )
Rational numbers are closed under multiplication. That is, for any two
rational numbers a and b, a * b is also a rational number.
For Example - 4 * 2 = 8 (a rational number. )
Rational numbers are not closed under division. That is, for any rational
number a, a/0 is not defined.
For Example - 6/0 is not defined.
4.
• Rational numbers can be added in any order.
Therefore, addition is commutative for rational
numbers. For Example :-
• Subtraction is not commutative for rational
numbers. For Example -
Since, -7 is unequal to 7
Hence, L.H.S. Is unequal to R.H.S.
Therefore, it is proved that Subtraction is not commutative for rational numbers.
L.H.S. R.H.S.
- 3/8 + 1/7
L.C.M. = 56
= -21+8
= -13
1 /7 +(-3/8)
L.C.M. = 56
= 8+(-21)
= -13
L.H.S. R.H.S.
2/3 – 5/4
L.C.M. = 12
= 8 – 15
= -7
5/4 – 2/3
L.C.M. = 12
= 15 – 8
= 7
5.
• Rational numbers can be multiplied in any order.
Therefore, it is said that multiplication
is commutative for rational numbers.
FOR EXAMPLE –
Since, L.H.S = R.H.S.
Therefore, it is proved that rational numbers can be multiplied in any order.
• Rational numbers can not be divided in any order.
Therefore, division is
Not Commutative for rational numbers.
FOR EXAMPLE –
Since, L.H.S. is not equal to R.H.S.
Therefore, it is proved that rational numbers can not be divided in any order.
L.H.S. R.H.S.
-7/3*6/5 = -42/15 6/5*(7/3) = -42/15
L.H.S. R.H.S.
(-5/4) / 3/7
= -5/4*7/3
= -35/12
3/7 / (-5/4)
= 3/7*4/-5
= -12/35
6.
Addition is associative for rational numbers.
That is for any three rational numbers a, b and c, :
a + (b + c) = (a + b) + c.
For Example
Since, -9/10 = -9/10
Hence, L.H.S. = R.H.S.
Therefore, the property
has been proved.
Subtraction is Not Associative for rational numbers
Multiplication is associative for rational numbers.
That is for any rational numbers a, b and c :
a* (b*c) = (a*b) * c
For Example –
Since, -5/21 = -5/21
Hence, L.H.S. = R.H.S
Division is Not Associative for Rational numbers.
ASSOCIATIVE PROPERTY
L.H.S. R.H.S.
-2/3+[3/5+(-5/6)]
= -2/3+(-7/30)
= -27/30
= -9/10
[-2/3+3/5]+(-5/6)
=-1/15+(-5/6)
=-27/30
=-9/10
L.H.S. R.H.S.
-2/3* (5/4*2/7)
= -2/3 * 10/28
= -2/3 * 5/14
= -10/42
= -5/21
(-2/3*5/4) * 2/7
= -10/12 * 2/7
= -5/6 * 2/7
= -10/42
= -5/21
7.
DISTRIBUTIVE LAW
For all rational numbers a, b and c,
a (b+c) = ab + ac
a (b-c) = ab – ac
For Example –
Since, L.H.S. = R.H.S.
Hence, Distributive Law Is Proved.
L.H.S. R.H.S.
4 (2+6)
= 4 (8)
= 32
4*2 + 4*6
= 8 = 24
= 32
8.
Additive Inverse
Additive inverse is also known as negative of a
number.
For any rational number a/b, a/b+(-a/b)= (-a/b)+a/b = 0
Therefore, -a/b is the additive inverse of a/b and a/b is
the
Additive Inverse of (-a/b).
Reciprocal
Rational number c/d is called the reciprocal or
Multiplicative Inverse of another rational number a/b
if a/b * c/d = 1