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Mention the Commutativity, Associative and Distributive properties of Rational Numbers.

  • Last Updated : 28 Sep, 2021

The number system includes different types of numbers for example prime numbers, odd numbers, even numbers, rational numbers, whole numbers, etc. These numbers can be expressed in the form of figures as well as words accordingly. For example, the numbers like 40 and 65 expressed in the form of figures can also be written as forty and sixty-five.

A Number system or numeral system is defined as elementary system to express numbers and figures. It is the unique way of representation of numbers in arithmetic and algebraic structure.

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Numbers are used in various arithmetic values applicable to carry out various arithmetic operations like addition, subtraction, multiplication, etc which are applicable in daily lives for the purpose of calculation. The value of a number is determined by the digit, its place value in the number, and the base of the number system.



Numbers generally are also known as numerals are the mathematical values used for counting, measurements, labeling, and measuring fundamental quantities.

Numbers are the mathematical values or figures used for the purpose of measuring or calculating quantities. It is represented by numerals as 2,4,7, etc. Some examples of numbers are integers, whole numbers, natural numbers, rational and irrational numbers, etc.

Types Of Numbers

There are different types of numbers categorized into sets by the real number system. The types are described below:

  • Natural numbers: Natural numbers are the positive numbers that count from 1 to infinity. The set of natural numbers is represented by ‘N’. It is the numbers we generally use for counting. The set of natural numbers can be represented as N = 1, 2, 3, 4, 5, 6, 7,…
  • Whole numbers: Whole numbers are positive numbers including zero, which counts from 0 to infinity. Whole numbers do not include fractions or decimals. The set of whole numbers is represented by ‘W’. The set can be represented as W = 0, 1, 2, 3, 4, 5,…
  • Integers: Integers are the set of numbers including all the positive counting numbers, zero as well as all negative counting numbers which count from negative infinity to positive infinity. The set doesn’t include fractions and decimals. The set of integers is denoted by ‘Z’. The set of integers can be represented as Z = …..,-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…
  • Decimal numbers: Any numeral value that consists of a decimal point is a decimal number. It can be expressed as 2.5, 0.567, etc.
  • Real number: Real numbers are the set numbers that do not include any imaginary value. It includes all the positive integers, negative integers, fractions, and decimal values. It is generally denoted by ‘R’.
  • Complex number: Complex numbers are a set of numbers that include imaginary numbers. It can be expressed as a+bi where “a” and “b” are real numbers. It is denoted by ‘C’.
  • Rational numbers: Rational numbers are the numbers that can be expressed as the ratio of two integers. It includes all the integers and can be expressed in terms of fractions or decimals. It is denoted by ‘Q’.
  • Irrational numbers: Irrational numbers are numbers that cannot be expressed in fractions or ratios of integers. It can be written in decimals and have endless non-repeating digits after the decimal point. It is denoted by ‘P’.

What are Rational Numbers?

Rational numbers are of the form p/q, where p and q are integers and q ≠ 0. Because of the underlying structure of numbers, p/q form, most individuals find it difficult to distinguish between fractions and rational numbers.

When a rational number is divided, the output is in decimal form, which can be either ending or repeating. 3, 4, 5, and so on are some examples of rational numbers as they can be expressed in fraction form as 3/1, 4/1, and 5/1.

Properties of Rational numbers

The main properties of numbers are:

  • Closure property
  • Commutative property
  • Associative property
  • Distributive property
  • Identity element property
  • Inverse element property

Closure Property

In this property of real numbers, we can add or multiply any two real numbers that will also result in a real number.



Example:

2 + 5 = 7 and 80 + 40 = 120 for addition

6 × 5 = 30 for multiplication

Commutative Property

It states that the operation of addition or multiplication on the number does not matter what is the order, it will give us the same result even after swapping or reversing their position.

Or we can say that the placement of adding or multiplying numbers can be changed but it will give the same results.

This property is valid for addition and multiplication not for subtraction and division.

                          x + y = y + x                                    or                        x.y = y.x

Example:

If we add 6 in 2 or add 2 in 6 results will be same                   If we multiply both the real number



                      7 + 2 = 9 = 2 + 7                                                           7 × 5 = 35 = 5 × 7                                                                                                                                        

Associative Property

This property states that when three or more numbers are added (or multiplied) or the sum(or product) is the same regardless of the grouping of the addends (or multiplicands).

The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property.

That is, rearranging the numbers in such a manner that will not change their value.

                (x + y) + z = x + (y + z)                        and                           (x.y).z = x.(y.z)

Example: (8 + 5) + 6 = 8 + (5 + 6)                                     (8 × 5) × 6 = 8 × (5 × 6)

                            19 = 19                                                                             240 = 240

As you can see even after changing their order, it gives the same result in both the operations adding as well as multiplication.

Distributive Property



This property helps us to simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.      

                         x × (y + z) = x × y + x × z          and                      x × (y – z) = x × y – x × z  

Example:

Simplify 20 × (5 + 6)  

= 20 × 5 + 20 × 6

= 100 + 120

= 220

It applies same for the subtraction also.

Identity Element Property

This is an element that leaves other elements unchanged when combined with them. The identity element for the addition operation is 0 and for multiplication is 1.

For addition, a + 0 = a and for multiplication a.0 = 0        

Example:

For addition, if a = 6

a + 0 = 6 + 0 = 6

and for multiplication if a = 6

a.0 = 6.0 = 0

Inverse Element

The reciprocal for a number “a”, denoted by 1/a, is a number which when multiplied by “a” yields the multiplicative identity 1.

The multiplicative inverse of a fraction: a/b is b/a  

The additive inverse of a number “a” is the number that when added to “a”, gives result zero. This number is also known as the additive inverse or opposite (number), sign change, and negation.



Or we can say for a real number, it reverses its sign from positive number to negative and negative number to positive. Zero is itself additive inverse.

Example: Reciprocal of 9 is 1/9 and the additive inverse of 9 is -9

Mention the commutativity, associative and distributive properties of rational numbers. Also, check a × b = b × a and a + b = b + a for a = ½ and b = ¾

Solution: 

As we have exaplained above all the properties of Rational number which also include commutativity, associative and distributive properties 

so as per the question we have the value of a = 1/2 and b = 3/4

Therefore 

As per the commutativity property: It states that the operation of addition or multiplication on the number does not matter what is the order, it will give us the same result even after swapping or reversing their position.

                             a + b = b + a                            or                        a.b = b.a

Now we have a = 1/2 and b = 3/4

                   so 1/2 + 3/4 = 3/4 + 1/2                     or               1/2 . 3/4 = 3/4 . 1/2

                                 5/4 = 5/4                                                         3/8 =  3/8

Hence proved 

Similar Questions

Question 1: With the same value of  a = 1/2 and b = 3/4 and c = 2/3, prove associative property? 

Solution: 

As per the associative property 

This property states that when three or more numbers are added (or multiplied) or the sum(or product) is the same regardless of the grouping of the addends (or multiplicands).

The addition or multiplication in which order the operations are performed does not matter as long as the sequence of the numbers is not changed. This is defined as the associative property.

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

Now we have a = 1/2 and b = 3/4 and c = 2/3 

          (1/2 + 3/4 ) + 2/3 = 1/2 + (3/4 + 2/3)                   or                (1/2 . 3/4 ). 2/3 = 1/2 . (3/4 . 2/3)



                       5/4 + 2/3 = 1/2 + 17/12                                                        3/8 . 2/3 = 1/2 . 2/4

                            23/12 =  23/12                                                                        2/8 = 2/8

                                                                                                                              1/4 =  1/4

Hence Proved

Question 2: With the same value of a = 1/2 and b = 3/4 and c = 2/3, prove distributive property?

Solution:         

As per the distributive property 

This property helps us to simplify the multiplication of a number by a sum or difference. It distributes the expression as it simplifies the calculation.      

                        a × (b + c) = a × b + a × c                      and                      a × (b – c) = a × b – a × c

Now we have  a = 1/2 and b = 3/4 and c = 2/3 

     1/2 × ( 3/4 + 2/3 ) = (1/2 × 3/4) + (1/2 × 2/3)            and             1/2 × (3/4 – 2/3) = 1/2 × 3/4 – 1/2 × 2/3

               1/2 × 17/12 =  3/8 + 2/6                                                               1/2 × 1/12 = 3/8 – 2/6

                        17/24 = 17/24                                                                              1/24 = 1/24

Hence proved




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