Download PPT on rational numbers

• A whole number or the quotient of any whole
numbers, excluding zero as a denominator
Examples - 5/8; -3/14; 7/-15; -6/-11
• Natural numbers – They are counting numbers.
• 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.
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.
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.
Commutative Property
Rational numbers can be added in any order. Therefore, addition
is commutative for rational numbers.
For Example
–
L.H.S.
R.H.S.
Subtraction is not
commutative for rational
numbers.
For Example -
- 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
5/4 – 2/3
L.C.M. = 12
L.C.M. = 12
= 8 – 15
= 15 – 8
= -7
=7
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.
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 = 6/5*(7/3) =
-42/15
-42/15
L.H.S.
R.H.S.
(-5/4) / 3/7 3/7 / (-5/4)
= -5/4*7/3 = 3/7*4/-5
= -35/12
= -12/35
Associative property
Addition is associative for rational numbers.
That is for any three rational numbers a, b and c, a + (b + c) =
(a + b) + c.
L.H.S.
R.H.S.
For Example
Since, -9/10 = -9/10
-2/3+[3/5+(-5/6)]
[-2/3+3/5]+(-5/6)
Hence, L.H.S. = R.H.S.
= -2/3+(-7/30)
=-1/15+(-5/6)
Therefore, the property
= -27/30
=-27/30
has been proved.
= -9/10
=-9/10
Subtraction is not associative for rational numbers.
For Example -2/3-[-4/5-1/2] [2/3-(-4/5)]-1/2
Since, 19/30 is not equal to
= -2/3 + 13/10 = 22/15 – ½
29/30
=-20 +39 /30
= 44 – 15/30
Hence, L.H.S. is not equal
= 19/30
= 29/30
to R.H.S. Therefore, the
property has been proved.
Multiplication is associative for rational numbers. That is
for any rational numbers a, b and c
a* (b*c) = (a*b) * c
L.H.S.
R.H.S.
For Example –
-2/3* (5/4*2/7) (-2/3*5/4) * 2/7
Since, -5/21 = -5/21
= -2/3 * 10/28 = -10/12 * 2/7
Hence, L.H.S. = R.H.S
= -2/3 * 5/14 = -5/6 * 2/7
= -10/42
= -10/42
= -5/21
= -5/21
Division is not associative
for Rational numbers.
L.H.S.
R.H.S.
For Example –
½ / (-1/3 / 2/5) [½ / (-1/2)] / 2/5
Since,
= ½ / -5/6
= -1 / 2/5
Hence, L.H.S. Is Not
= -6/10
= -5/2
equal to R.H.S.
= -3/5
= -5/2
Distributive Law
Distributivity of multiplication over addition and
subtraction :
For all rational numbers a, b and c,
a (b+c) = ab + ac
L.H.S.
R.H.S.
a (b-c) = ab – ac
4 (2+6)
4*2 + 4*6
For Example –
= 4 (8)
= 32
Since, L.H.S. = R.H.S.
Hence, distributive law is proved
= 8 = 24
= 32
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.
Therefore, for any rational number a, a+0 = 0+a = a
For Example - 2+0 = 0+2 = 2
-5+0 = 0+(-5) = -5
The role of one (1)
1 is the multiplicative identity for rational numbers.
Therefore, a*1 = 1*a = a for any rational number a.
For Example 2*1 = 2
1*-10 = -10
Some Problems On Rational Numbers
Q1) Verify that –(-x) is the same as x for x = 5/6
A1) The additive inverse
L.H.S.
R.H.S.
of x = 5/6 = -x = -5/6
-(-5/6)
5/6
Since, 5/6 + (-5/6) = 0
= +5/6
= + 5/6
Hence, -(-x) = x.
= 5/6
= 5/6
Q2) Find any four rational numbers between -5/6 and 5/8
A2) Convert the given numbers to rational numbers with same
denominators :
-5*4-6*4 = -20/24
5*3/8*3 = 15/24
Thus, we have -19/24; -18/24; ........13/24; 14/24
Any four rational numbers can be chosen.