verify all the properties of operations on rational numbers
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Answers
Answer:
Addition of Rational Numbers
The closure property states that for any two rational numbers a and b, a + b is also a rational number.
\frac{1}{2}
2
1
+ \frac{3}{4}
4
3
= \frac{4 + 6}{8}
8
4+6
= \frac{10}{8}
8
10
Or, = \frac{5}{4}
4
5
The result is a rational number. So we say that rational numbers are closed under addition.
2) Subtraction of Rational Numbers
The closure property states that for any two rational numbers a and b, a – b is also a rational number.
\frac{1}{2}
2
1
– \frac{3}{4}
4
3
= \frac{4 – 6}{8}
8
4–6
= \frac{-2}{8}
8
−2
Or, = \frac{-1}{4}
4
−1
The result is a rational number. So the rational numbers are closed under subtraction.
3) Multiplication of Rational Numbers
The closure property states that for any two rational numbers a and b, a × b is also a rational number.
\frac{1}{2}
2
1
× \frac{3}{4}
4
3
= \frac{6}{8}
8
6
The result is a rational number. So rational numbers are closed under multiplication.
4) Division of Rational Numbers
The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.
\frac{1}{2}
2
1
÷ \frac{3}{4}
4
3
= \frac{1 ×4}{2 ×3}
2×3
1×4
= \frac{2}{3}
3
2
The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.
Commutative Property
1. Addition
For any two rational numbers a and b, a + b = b+ a
\frac{-2}{3}
3
−2
+ \frac{5}{7}
7
5
and \frac{5}{7}
7
5
+ \frac{-2}{3}
3
−2
= \frac{1}{21}
21
1
so, \frac{-2}{3}
3
−2
+ \frac{5}{7}
7
5
= \frac{5}{7}
7
5
+ \frac{-2}{3}
3
−2
We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.
2. Subtraction
For any two rational numbers a and b, a – b ≠ b – a. Given are the two rational numbers \frac{5}{3}
3
5
and \frac{1}{4}
4
1
,
\frac{5}{3}
3
5
– \frac{1}{4}
4
1
= \frac{20-3}{12}
12
20−3
= \frac{17}{12}
12
17
But, \frac{1}{4}
4
1
– \frac{5}{3}
3
5
= \frac{3-20}{12}
12
3−20
= \frac{-17}{12}
12
−17
So subtraction is not commutative for ratioanl numbers.
3. Multiplication
For any two rational numbers a and b, a × b = b × a
\frac{-7}{3}
3
−7
+ \frac{6}{5}
5
6
= \frac{6}{5}
5
6
+ \frac{-7}{3}
3
−7
= \frac{-42}{15}
15
−42
= \frac{-42}{15}
15
−42
We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.
4. Division
For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers \frac{5}{3}
3
5
and \frac{1}{4}
4
1
\frac{5}{3}
3
5
÷ \frac{1}{4}
4
1
= \frac{5×4}{3×1}
3×1
5×4
= \frac{20}{3}
3
20
But, \frac{1}{4}
4
1
÷ \frac{5}{3}
3
5
= \frac{1×3}{4×5}
4×5
1×3
= \frac{3}{20}
20
3
We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.
Associative Property
Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.
For example, given numbers are 5, -6 and \frac{2}{3}
3
2
( 5 – 6 ) + \frac{2}{3}
3
2
= -1 + \frac{2}{3}
3
2
= – \frac{1}{3}
3
1
Now, 5 + ( -6 + \frac{2}{3}
3
2
)
= – \frac{1}{3}
3
1
In both the groups the sum is the same.
Addition and multiplication are associative for rational numbers.
Subtraction and division are not associative for rational numbers.
Distributive Property
Distributive property states that for any three numbers x, y and z we have
x × ( y + z ) = (x × y) +( x × z)
Solved Examples for You
Question 1: …………….. are not associative for rational numbers.
Addition and multiplication
Subtraction and multiplication
Subtraction and division
I HOPE THIS WOULD HELP YOU
Closure property
1) Addition of Rational Numbers
The closure property states that for any two rational numbers a and b, a + b is also a rational number.
\frac{1}{2}
2
1
+ \frac{3}{4}
4
3
= \frac{4 + 6}{8}
8
4+6
= \frac{10}{8}
8
10
Or, = \frac{5}{4}
4
5
The result is a rational number. So we say that rational numbers are closed under addition.
2) Subtraction of Rational Numbers
The closure property states that for any two rational numbers a and b, a – b is also a rational number.
\frac{1}{2}
2
1
– \frac{3}{4}
4
3
= \frac{4 – 6}{8}
8
4–6
= \frac{-2}{8}
8
−2
Or, = \frac{-1}{4}
4
−1
The result is a rational number. So the rational numbers are closed under subtraction.
3) Multiplication of Rational Numbers
The closure property states that for any two rational numbers a and b, a × b is also a rational number.
\frac{1}{2}
2
1
× \frac{3}{4}
4
3
= \frac{6}{8}
8
6
The result is a rational number. So rational numbers are closed under multiplication.
4) Division of Rational Numbers
The closure property states that for any two rational numbers a and b, a ÷ b is also a rational number.
\frac{1}{2}
2
1
÷ \frac{3}{4}
4
3
= \frac{1 ×4}{2 ×3}
2×3
1×4
= \frac{2}{3}
3
2
The result is a rational number. But we know that any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division. But if we exclude 0, then all the rational numbers are closed under division.
Commutative Property
1. Addition
For any two rational numbers a and b, a + b = b+ a
\frac{-2}{3}
3
−2
+ \frac{5}{7}
7
5
and \frac{5}{7}
7
5
+ \frac{-2}{3}
3
−2
= \frac{1}{21}
21
1
so, \frac{-2}{3}
3
−2
+ \frac{5}{7}
7
5
= \frac{5}{7}
7
5
+ \frac{-2}{3}
3
−2
We see that the two rational numbers can be added in any order. So addition is commutative for rational numbers.
2. Subtraction
For any two rational numbers a and b, a – b ≠ b – a. Given are the two rational numbers \frac{5}{3}
3
5
and \frac{1}{4}
4
1
,
\frac{5}{3}
3
5
– \frac{1}{4}
4
1
= \frac{20-3}{12}
12
20−3
= \frac{17}{12}
12
17
But, \frac{1}{4}
4
1
– \frac{5}{3}
3
5
= \frac{3-20}{12}
12
3−20
= \frac{-17}{12}
12
−17
So subtraction is not commutative for ratioanl numbers.
3. Multiplication
For any two rational numbers a and b, a × b = b × a
\frac{-7}{3}
3
−7
+ \frac{6}{5}
5
6
= \frac{6}{5}
5
6
+ \frac{-7}{3}
3
−7
= \frac{-42}{15}
15
−42
= \frac{-42}{15}
15
−42
We see that the two ratrional numbers can be multiplied in any order. So multiplication is commutative for ratioanl numbers.
4. Division
For any two rational numbers a and b, a ÷ b ≠ b ÷ a. Given are the two rational numbers \frac{5}{3}
3
5
and \frac{1}{4}
4
1
\frac{5}{3}
3
5
÷ \frac{1}{4}
4
1
= \frac{5×4}{3×1}
3×1
5×4
= \frac{20}{3}
3
20
But, \frac{1}{4}
4
1
÷ \frac{5}{3}
3
5
= \frac{1×3}{4×5}
4×5
1×3
= \frac{3}{20}
20
3
We see that the expressions on both the sides are not equal. So divsion is not commutative for ratioanal numbers.
Associative Property
Take any three rational numbers a, b and c. Firstly add a and b and then add c to the sum. (a + b) + c. Now again add b and c and then a to the sum, a + (b + c). Is (a + b) + c and a + (b + c) same? Yes and this is how associative property works. It states that you can add or multiply numbers regardless of how they are grouped.
For example, given numbers are 5, -6 and \frac{2}{3}
3
2
( 5 – 6 ) + \frac{2}{3}
3
2
= -1 + \frac{2}{3}
3
2
= – \frac{1}{3}
3
1
Now, 5 + ( -6 + \frac{2}{3}
3
2
)
= – \frac{1}{3}
3
1
In both the groups the sum is the same.
Addition and multiplication are associative for rational numbers.
Subtraction and division are not associative for rational numbers.
Distributive Property
Distributive property states that for any three numbers x, y and z we have
x × ( y + z ) = (x × y) +( x × z)