A whole number or the quotient of any whole numbers, excluding
zero as a denominator A whole number or the quotient of any whole
numbers, excluding zero as a denominator Examples - 5/8; -3/14;
7/-15; -6/-11 Examples - 5/8; -3/14; 7/-15; -6/-11 Natural numbers
They are counting numbers. Natural numbers They are counting
numbers. Integers - Natural numbers, their negative and 0 form the
system of integers. Integers - Natural numbers, their negative and
0 form the system of integers. Fractional numbers the positive
integer which are in the form of p/q where q is not equal to 0 are
known as fractional numbers. Fractional numbers the positive
integer which are in the form of p/q where q is not equal to 0 are
known as fractional numbers.
Slide 3
Closure Property Rational numbers are closed under addition.
That is, for any two rational numbers a and b, a+b s also a
rational number. 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.
Slide 4
Commutative Property 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
Slide 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
Slide 6
Associative property Addition is associative for rational
numbers. Addition is associative for rational numbers. That is for
any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
That is for any three rational numbers a, b and c, a + (b + c) = (a
+ b) + c. For Example For Example Since, -9/10 = -9/10 Hence,
L.H.S. = R.H.S. Therefore, the property has been proved. has been
proved. Subtraction is not associative for rational numbers. For
Example - Since, 19/30 is not equal to 29/30 Hence, L.H.S. is not
equal to R.H.S. Therefore, the property has been proved. L.H.S.
L.H.S. R.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 -2/3-[-4/5-1/2]
-2/3-[-4/5-1/2] = -2/3 + 13/10 =-20 +39 /30 = 19/30
[2/3-(-4/5)]-1/2 = 22/15 = 44 15/30 = 29/30 = 29/30
Slide 7
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. for Rational numbers. For Example
Since, Hence, L.H.S. Is Not equal to R.H.S. equal to R.H.S. L.H.S.
L.H.S. R.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 L.H.S. L.H.S. R.H.S. R.H.S. / (-1/3 / 2/5) = / -5/6
= -6/10 = -3/5 [ / (-1/2)] / 2/5 = -1 / 2/5 = -5/2
Slide 8
Distributive Law Distributivity of multiplication over addition
and subtraction : Distributivity of multiplication over addition
and subtraction : For all rational numbers a, b and c, For all
rational numbers a, b and c, a (b+c) = ab + ac a (b+c) = ab + ac a
(b-c) = ab ac a (b-c) = ab ac For Example For Example Since, L.H.S.
= R.H.S. Hence, distributive law is proved L.H.S. L.H.S. R.H.S.
R.H.S. 4 (2+6) 4 (2+6) = 4 (8) = 32 4*2 + 4*6 4*2 + 4*6 = 8 = 24 =
32
Slide 9
The Role Of Zero (0) Zero is called the identity for the
addition of rational numbers. It is the additive identity for
integers and whole numbers as well. Zero is called the identity for
the addition of rational numbers. It is the additive identity for
integers and whole numbers as well. Therefore, for any rational
number a, a+0 = 0+a = a Therefore, for any rational number a, a+0 =
0+a = a For Example - 2+0 = 0+2 = 2 For Example - 2+0 = 0+2 = 2
-5+0 = 0+(-5) = -5 -5+0 = 0+(-5) = -5 The role of one (1) The role
of one (1) 1 is the multiplicative identity for rational numbers. 1
is the multiplicative identity for rational numbers. Therefore, a*1
= 1*a = a for any rational number a. Therefore, a*1 = 1*a = a for
any rational number a. For Example - 2*1 = 2 For Example - 2*1 = 2
1*-10 = -10 1*-10 = -10
Slide 10
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
Slide 11
Some Problems On Rational Numbers Q 1 ) Verify that (-x) is the
same as x for x = 5/6 A 1 ) The additive inverse of x = 5/6 = -x =
-5/6 of x = 5/6 = -x = -5/6 Since, 5/6 + (-5/6) = 0 Since, 5/6 +
(-5/6) = 0 Hence, -(-x) = x. Hence, -(-x) = x. Q 2 ) Find any four
rational numbers between -5/6 and 5/8 A 2 ) Convert the given
numbers to rational numbers with same denominators : -5*4-6*4 =
-20/24 5*3/8*3 = 15/24 -5*4-6*4 = -20/24 5*3/8*3 = 15/24 Thus, we
have -19/24; -18/24;........13/24; 14/24 Thus, we have -19/24;
-18/24;........13/24; 14/24 Any four rational numbers can be
chosen. Any four rational numbers can be chosen. L.H.S. L.H.S.
R.H.S. R.H.S. -(-5/6) = +5/6 = 5/6 5/6 = + 5/6 = 5/6