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Rational Numbers

    Properties of Rational Numbers

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8th class > Rational Numbers > Properties of Rational Numbers

Properties of Rational Numbers

What is the Closure Property ?

The closure property is a fundamental concept in mathematics that applies to certain sets and operations.

It states that:

If we perform a particular operation on elements of a set, the result will also be an element of the same set.

The closure property is applicable to operations like addition, subtraction, multiplication, and division, and it varies depending on the type of numbers or mathematical structures involved.

Closure

(i) Whole numbers

Let us revisit the closure property for four operations like (addition, subtraction, multiplication and division) on whole numbers in brief.

Instructions

(ii)Integers

Check for closure property under all the four operations for natural numbers.

Instructions

You have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However, integers are closed under addition, subtraction and multiplication but not under division. Let's check now for rational numbers.

(iii) Rational numbers

Recall that a number which can be written in the form pq, where p and q are integers and q ≠ 0 is called a rational number. For example,23,67,9−5 are all rational numbers. Since the numbers 0, –2, 4 can be written in the form pq, they are also rational numbers. (Check it!)

(a) You know how to add two rational numbers. Let us add a few pairs.

Instructions

38+−57

  • For adding the rational numbers: calculate the LCM i.e.
  • Taking the LCM of the denominators then multiply the numerator with the respective factors.
  • Calculating; we get the sum as
  • Therefore, the sum result of 38+−57 a rational number

We find that sum of two rational numbers is again a .

We say that rational numbers are under addition.

That is, for any two rational numbers a and b, a + b is also a rational number.

(b) Will the difference of two rational numbers be again a rational number?

Instructions

−57−23

  • Subtract the rational numbers by calculating the LCM i.e.
  • Taking the LCM of the denominators and then multiply the numerator and denominator for both numbers
  • Calculating- we get the result as
  • Therefore, the result of −57−23 a rational number

We find that rational numbers are under subtraction.

That is, for any two rational numbers a and b, a – b is also a rational number.

(c) Let us now see the product of two rational numbers.

Instructions

  • To multiply any two rational numbers
  • Multiply the numerators and the denominatores separately
  • Calculating- we get the result as
  • Therefore,the result of −23×45 a rational number

We say that rational numbers are under multiplication.

That is, for any two rational numbers a and b, a × b is also a rational number.

Instructions

(d) We note that −53÷25 gives us which further a rational number.
27÷53 gives us which further a rational number.
−38÷−29 gives us which further rational number.

Can you say that rational numbers are closed under division?

We find that for any rational number a, a ÷ 0 is not defined.

The division is not under closure property because division by zero is not defined. However, all numbers except zero are closed under division.

Fill in the blanks in the following table.

Checking for Closure Property

OperationAdditionSubtractionMultiplicationDivision
Rational NumbersYes
IntegersYesNo
Whole NumbersYes
Natural NumbersNo

Commutativity

(i) Whole numbers

Recall the commutativity of different operations for whole numbers by solving the below question.

Instructions

0+7=7+0=7
7−3=3−7
2+3=3+2
7×2×5=7×2×5
4÷2=2÷4
Result is commutative
Result is not commutative

Check for yourself the associativity of different operations for natural numbers.

(ii) Integers

Check the commutativity of different operations for integers:

Instructions

4+−2=−2+4
5−−3=−3−5
3×−5=−5×3
6÷3=3÷6
Result is commutative
Result is not commutative

(iii) Rational numbers

Addition

You know how to add two rational numbers. Let us add a few pairs here:

−23+57 = 121 and 57+−23 = 121

So, −23 = 57

Now check: Is −65 + −83 = −83 + −65 ?

We find that they are both equal. Try it for any two random rational numbers.

We find that two rational numbers can be added in any order.

We can say that:

Addition is commutative for rational numbers. That is, for any two rational numbers a and b,

a + b = b + a.

Subtraction

Is 23−54 = 54−23 ?

Let's check. We have LCM as with:

23−54 = and 54−23 =

What about: 12−35 = 35−12 ?

where , 12−35 = and 35−12 =

You will find that subtraction is not commutative for rational numbers.

Note: Subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.

Multiplication

Check for some more such products.

−73×65 = −4215 = 65×−73

Check if −89×−47 = −47×−89.

You will find that multiplication is commutative for rational numbers.

In general, a × b = b × a for any two rational numbers a and b.

Division

Is −54÷37 = 37÷−54 ?

You will find that expressions on both sides are not equal.

So division is not commutative for rational numbers.

Instructions

Complete the following table:

Checking for Commutative [a × b = b × a]

OperationAdditionSubtractionMultiplicationDivision
Rational NumbersYes
IntegersNo
Whole NumbersYes
Natural NumbersNo

Associativity

(i) Whole numbers

Recall the associativity of the four operations for whole numbers:

Instructions

2+3+4=2+3+4
4−3−0=4−3−0
7×2×5=7×2×5
8÷4÷2=8÷4÷2
Result is associative
Result is not associative

Check for yourself the associativity of different operations for natural numbers.

(ii) Integers

Associativity of the four operations for integers

Instructions

−2+3+−4=−2+3+−4
5−7−3=5−7−3
−6+−4+−5=−6+−4+−5
5×−7×−8=5×−7×−8
−10÷2÷−5=−10÷2÷−5
−4×−8×−5=−4×−8×−5
Result is associative
Result is not associative

(iii) Rational numbers

(a) Addition

Find −12+37+−43 and −12+37+−43 . Are the two sums equal?

Instructions

−12+37+−43

  • Addition of three rational numbers: Start by taking two numbers
  • Take the LCM i.e. and multiply the respective factors to the numerators
  • Calculating the sum of numerators
  • Further solve where the LCM is
  • We get the result as
  • We have found the answer.

−12+37+−43

  • Do the same procedure again
  • Take the LCM i.e. and multiply the respective factors to the numerator
  • Calculating the sum of the numerators
  • Further solve with the LCM being
  • We get the result as
  • We get the same result.

So the two rational numbers sums are .

We find that addition is for rational numbers. That is, for any three rational numbers a, b and c,

a + (b + c) = (a + b) +c.

(b) Subtraction

Instructions

You already know that subtraction is not associative for integers, then what about rational numbers.
Check if −23−45−12=23−−45−12
We have −23−45−12 = −23 - =
While 23−−45−12 = - 12 =
Thus, −23−45−12≠23−−45−12
Subtraction associative for rational numbers.

(c) Multiplication

Let us check the associativity for multiplication.

Instructions

−73×54×29 = −73×54×29
23×−67×45 = 23×−67×45
Result is associative
Result is not associative

We observe that multiplication associative for rational numbers.

That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.

(d) Division

Recall that division is not associative for integers, then what about rational numbers?

Let us see if 12÷−13÷25=12÷−13÷25

Instructions

Division for rational numbers

  • Divide the two rational numbers
  • Reciprocal of 25 is
  • Solve the terms within the bracket
  • Further calculating
  • We get the result as −610 i.e. (Enter simplified form).
  • Taking RHS = 12÷−13÷25
  • Reciprocal of −13 is
  • Using the reciprocal to solve
  • Multiplying the numerators and denominators, we get
  • Is LHS = RHS ?
  • We have reached a conclusion.

You will find that division associative for rational numbers.

Complete the following table:

Checking for Associative [a × (b × c) = (a × b) × c]

OperationAdditionSubtractionMultiplicationDivision
Rational NumbersNo
IntegersYes
Whole NumbersYes
Natural NumbersNo

Example 1: Find 37+−611+−821+522

Solution:

The LCM of the denominators is .

37+−611+−821+522 = 198462 + −252462 + −176462 + 105462 =

Thus, the sum is −125462

Example 2: Find −45×37×1516×−149

Solution: −45×37 = −4×35×7 =

1516×−149 = 15×−1416×9 =

Further, −1235×−3524 = (Put simplified form)

Thus, the product is 12

The role of zero (0)

Look at the following

2 + 0 = 0 + 2 = 2Addition of 0 to a whole number
– 5 + 0 = + = – 5Addition of 0 to an integer
−27 + = 0 +−27=−27Addition of 0 to a rational number

You have done such additions earlier also. Do a few more such additions.

What do you observe? You will find that when you add 0 to a whole number, the sum is again that whole number. This happens for integers and rational numbers also.

In general,

a + 0 = 0 + a = a, where a is a whole number

b + 0 = 0 + b = b, where b is an integer

c + 0 = 0 + c = c, where c is a rational number

Therefore, is called the identity for the ddition of rational numbers. It is the additive identity for integers and whole numbers as well.

Drag the boxes and put in the correct category

Instructions

4+0=0+4=4
−3+0=0+−3=−3
−47+0=0+−47=−47
6+0=0+6=6
−53+0=0+−53=−53
−9+0=0+−9=−9
Whole numbers
Integers
Rational numbers

1.What is the result of adding zero to the integer 7 ?

Solution: 7 + 0 =

2. If x = −15, what is x + 0 ?

Solution: x + 0 = + =

3. What is the sum of 35 and 0 ?

Solution: 35 + 0 =

4. If y = −712 , what number should be added to it to get a sum result of 0 ?

Solution: Let the number to be added is x.

Thus, −712 + x = 0 which gives us x =

The role of 1

We have,

Instructions

5×1=1×5
−27×1=1×−27
38×1=1×38
7×1=1×7
−45×1=1×−45
Whole numbers
Rational numbers

What do you find?

You will find that when you multiply any rational number with 1, you get back the same rational number as the product. Check this for a few more rational numbers. You will find: that,

a × 1 = 1 × a = a for any rational number a.

We say that 1 is the multiplicative identity for rational numbers.

Is 1 the multiplicative identity for integers?

And for whole numbers?

Distributivity of multiplication over addition for rational numbers

To understand this, consider the rational numbers −34 , 23 and −56.

Check the distributivity of multiplication over addition for −34 , 23 and −56.

Instructions

Check the distributive property

  • Evaluating: −34×23+−56, first add the numbers in the bracket. The value of the addition is by taking the LCM i.e. .
  • Now multiply both the numbers. We get the result: which can be simplified to
  • This same value can be achieved if we multiply −34 to all the terms within the bracket. Let's verify it.
  • Evaluating −34×23 we get: (Enter simplified number)
  • Now evaluating −34×−56 we get: (Enter simplified number)
  • Now adding both the number.
  • We get the value:
  • We have found that both the values are equal.
  • Thus, we have proved the following result.

Distributivity of Multiplication over Addition and Subtraction

For all rational numbers a, b and c,

a (b + c) = a b + a c

a (b – c) = a b – a c

Find using distributivity:

(i) 75×−312+75×512

Instructions

To solve this using distributive property, we can factor out the common term 7/5.
As 75×−312+75×512=75×−312+512.
First, let's simplify the expression inside the parentheses.
−312+512 = −3+512 = / = /.
Now, multiply the common factor 75 by the result : 75 × 16 = 730.
So, the result of distributive property = 730.

(ii) 916×412+916×−39

Instructions

To solve this using distributive property. Lets proceed step by step.
916 × 412 ;(simplify) 412 = /.
Now, calculate 916 × 13 = 948 = /.
2.916 × −39 ;(simplify) −39 = -(/).
Now, calculate 916 × −13 = −948 = -(/).
Now, add the results from step1 and step2.
= 316 + −316 = .
Therefore the value is 0.

Example 3: Evaluate the expression: 25×−37−114−37×35

Instructions

25×−37−114−37×35

  • We can adjust the terms using property.
  • We can also re-write the expression by replacing the subtraction sign with addition.
  • We can now take the common number from the first two brackets and keeping the remaining terms in a bracket.
  • The value within the bracket is .
  • Solving we get: which on further simplification becomes .
  • Hence, we found the answer.