Rational numbers
Properties of rational numbers:
Closure properties:
Addition: A+B=Rational number (3/8+5/7 = 19/56)
Subtraction: A-B = Rational number(-5/7 - 2/3= -29/21)
[but in case of whole numbers subtraction closure property is applicable]
Multiplication: A*B =Rational number(4/5* -2/3 =-8/15)
Division: A/B= Rational number if exclude 0(-5/3 divided by 2/5 = -25/6 )
Commutativity property:
Rational numbers: rational numbers are integers ,express in fractions,
p/q form where q is not equal to 0 ,
For example 2,33,and -24 are the numbers which convert in to
2/1,33/1 and -24/1
Numbers
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fraction
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Rational numbers
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7
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7/1
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This is rational number
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0.12
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12/100
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Yes
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-0.1
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-1/10
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Yes
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0.875
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7/8
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Yes
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Rational number is a number which can be written as p/q form ,where p
and q are integers ,q is not equal to 0
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Properties of rational numbers:
Closure properties:
Addition: A+B=Rational number (3/8+5/7 = 19/56)
Subtraction: A-B = Rational number(-5/7 - 2/3= -29/21)
[but in case of whole numbers subtraction closure property is applicable]
Multiplication: A*B =Rational number(4/5* -2/3 =-8/15)
Division: A/B= Rational number if exclude 0(-5/3 divided by 2/5 = -25/6 )
Commutativity property:
Addition: A+B= B+A (1/2 + 3/4 = 3/4 + 1/2)
Subtraction: A-B not equal to B+A (5/6-2/3 not equal to 2/3 -5/6)
Multiplication: A*B =B*A (2/3 * 3/8 = 3/8 *2/3)
Division: A/B Not equal to B/A
Associativity property :
Addition: A+(B+C)=( A+B)+C (5/3+ (1/2 + 3/4 )= (5/3 + 1/2) +3/4
Subtraction: not associative property
Multiplication: A* (B *C) =( A*B)*C [4/5*(2/3 * 3/8) =(4/5* 3/8 )*2/3]
Division: not associative property
Distributive property:
Distributivity of multiplication over addition and substrate
A(B+C) =AB+AC
A(B-C) = AB- AC
In Rational numbers additive identity is 0 ( additive inverse means just change sign of the number example 3/4 additive inverse is -3/4, (3/4+(-3/4) = 0/4 = 0)
Multiplicative identity of Rational numbers is 1 ( multiplicative inverse just invert the number with out changing sign ,4/5 multiplicative inverse is 5/4 , 4/5*5/4 =1 )
Subtraction: A-B not equal to B+A (5/6-2/3 not equal to 2/3 -5/6)
Multiplication: A*B =B*A (2/3 * 3/8 = 3/8 *2/3)
Division: A/B Not equal to B/A
Associativity property :
Addition: A+(B+C)=( A+B)+C (5/3+ (1/2 + 3/4 )= (5/3 + 1/2) +3/4
Subtraction: not associative property
Multiplication: A* (B *C) =( A*B)*C [4/5*(2/3 * 3/8) =(4/5* 3/8 )*2/3]
Division: not associative property
Distributive property:
Distributivity of multiplication over addition and substrate
A(B+C) =AB+AC
A(B-C) = AB- AC
In Rational numbers additive identity is 0 ( additive inverse means just change sign of the number example 3/4 additive inverse is -3/4, (3/4+(-3/4) = 0/4 = 0)
Multiplicative identity of Rational numbers is 1 ( multiplicative inverse just invert the number with out changing sign ,4/5 multiplicative inverse is 5/4 , 4/5*5/4 =1 )
How to convert decimal number in to rational number:
0.234 , 1.25
0.234=234/1000
1.25 =125/100
How to find rational numbers between two numbers:
Method 1 by using formula:
Rational numbers between 1/14 and 3/10
Formula is ½(a+b)
Where a = 1/14
b = 3/10
½(1/14+3/10)
Take LCM for 1/14+3/10 = 5+21/70
=
26/70
½(1/14+3/10)
=1/2*26/70
= 13/7
13/70 is the rational number between 1/14 and 3/10
Method 2
find rational numbers between 3 and 4
convert 3 and 4 in to fractional numbers
3/1 and 4/1
Multiply both the numbers with 10
3*10/1*10 and 4*10/1*10
30/10 and 40/10
So rational numbers between these two numbers is
31/10, 32/10, 33/10, 34/10, 35/10, 36/10, 37/10, 38/10, 39/10
,40/10.
By this we get 10 rational numbers ,if we need more than 10
numbers ,we should multiply 3/1 and 4/1
with 100 ,they will become 300/100 and 400/100 ,then we can write 100 numbers .
Note not only with 10 or 100 we can multiply with any number
for example with 5 ,
3*5/1*5 and 4*5/1*5
15/5 and 20/5…..rational numbers between these two numbers
are 16/5 ,17/5 ,18/5, 19/5 and 20/5
Find 3 rational numbers between 3/5 and 4/5
Multiply with 6
3*6/5*6 and 4*6/5*6
18/30 and 24/30
19/30 ,20/30, 21/30
Method 3 convert
in to decimal numbers
3/5 = 0.6
4/5 = 0.8
0.6 to 0.8 in this 0.61 ,0.65,0.69 0.7 and 0.75 are the terminating
numbers ,
Fractional numbers with terminating ends are rational numbers, numbers with non terminating ends are Irrational
numbers
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How to locate rational numbers in number line:
5/3 and -5/3
Convert 5/3 in to mixed fraction 1 2/3 that means this comes
at that point
Like that -5/3 in to mixed fractions -1 2/3 this comes
negative sign side.
Divided number line based
first in to -4,-3,-2,-1,0,1,2,3,4,….
Again divide 0 to 1 in to small parts .this how we start 3/3
means 1 ,so 3/3 comes on point 1, before numbers are 1/3,2/3
So in between 0 and 1
we have 2 more parts.
That means ,in this case each one unit again divided in to 3 equal parts.
How to locate rational numbers in number line:
3/2 and -3/2
Convert 3/2 in to mixed fraction 1 1/3 that means this comes
at that point
Like that -3/2 in to mixed fractions -1 1/3 this comes
negative sign side.
Divided number line based
first in to -4,-3,-2,-1,0,1,2,3,4,….
Again divide 0 to 1 in to small parts .this how we start, 2/2
means 1 ,so 2/2 comes on point 1, before numbers are 1/2
So in between 0 and 1
we have 1 more parts.
That means ,in this case each one unit again divided in to 2 equal parts.
