**Table of Contents**

**Rational Numbers**

Learn more about the characteristics that set rational numbers apart from other sorts of numbers on this page. In addition to being stated as fractions, terminating decimal numbers, and non-terminating but repeating decimal numbers, rational numbers also comprise integers, whole numbers, and natural numbers. The associative, commutative, distributive, and closure qualities are all attributes of rational numbers. Continue reading to find out more about these attributes.

**What are the properties of rational numbers?**

Any number that can be represented as p/q, where p and q are integers, and q 0, is considered rational. Rational numbers have six essential characteristics, one of which is

1.Closure.

2. Compatibilism.

3.Associativity.

4.Distributivity.

5. Identification Property.

6. The inverse property.

The four fundamental arithmetic operations in mathematics—addition, subtraction, multiplication, and division—can all be used to apply these properties. Now let’s explore these properties and how they affect various arithmetic operations.

**Closure Property**

The operations of addition, subtraction, and multiplication on the rational integers x and y always produce another rational number because they are closed under these operations. For illustration:

(7/6) + (2/5) = 47/30 (5/6) – (1/3) = 1/2

(2/5) * (3/7) = 6/35

However, since dividing by zero is undefined, division is not covered by the** **closure attribute. Notably, division is closed for all values other than zero.

**Commutative Property**

The addition and multiplication of the two rational integers 3 and 7 are always commutative. There is no commutative property for subtraction. By looking at the instances that have been solved, you can clearly understand this characteristic.

Law of Commutative Addition: x + y = x + y Ex: 3+7 = 7+3

Subtraction x-y≠y-x For example, 3-7=4 while 7-3=4.

It is not commutative to divide. x/y ≠y/x Ex: 3/7 1/2 = 6/7, but 1/2 3/7 = 14/3.

Ex: 3.4/7 =4/7.3 =12/49 Commutative Law of Multiplication: x.y = y.x

**Associative Property**

The Associative Property of Addition and Multiplication is followed by rational numbers. If x, y, and z are three rational numbers, then (x+y)+z = x+(y+z), and (xy)z = x(yz) for addition and multiplication, respectively.

As an illustration, (1/3 + 3/3) + 1/4 = 1/3 + (3/3 + 1/4) 13/12 = 13/12

**Distributive Property**

Assume we have x, y, and z, three rational numbers. The distributive principle of multiplication over addition therefore allows us to state that x. (y+z) = (x. y) + (x. z). We can use an example to demonstrate this characteristic.

Let’s use the example of x = 2/3, y = 1/5, and z = 4/7: 2/3. (1/5 + 4/7) = (2/3 . 1/5) + (2/3 . 4/7)

2/3 . (17/35) = 2/15 + 8/21

34/105 = 4/21 + 20/105

34/105 = 34/105

Thus, Right hand side is equal to the left hand side.

**Identity and Inverse Properties of Rational Numbers**

The Identity Property states that 0 is the additive identity and 1 is the multiplicative identity for rational numbers.

Examples:

2/3 + 0 = 2/3 [Additive Identity]

2/3 x 1 = 2/3 [Multiplicative Identity]

The Inverse Property states that for a rational number x/y, the additive inverse is -x/y and the multiplicative inverse is y/x.

Examples:

The additive inverse of 2/5 is -2/5. Hence, 2/5 + (-2/5) = 0.

The multiplicative inverse of 2/5 is 5/2. Hence, 2/5 x 5/2 = 1.

**Additive Property of Rational Numbers**

The additive identity property and the additive inverse property are the two basic additive properties of rational numbers. The examples below explain how to prove these two facts for every rational integer a/b, where b 0.

**Additive Identity Property**

According to the additive identity property for rational numbers, adding zero to any rational number (a/b) results in the creation of the actual rational number. This means that if any rational number is represented by a/b, then a/b + 0 = 0 + a/b = a/b, where 0 is regarded as the additive identity for rational numbers. This idea is shown by the example below:

The result of adding 0 to 3/7 is: 3/7 + 0 = 0 + 3/7 = 3/7.

**Additive Inverse Property**

Given any rational number a/b, the additive inverse property of rational numbers states that there exists a rational number (-a/b) such that their sum is equal to zero. The equation for this is a/b + (-a/b) = (-a/b) + a/b = 0.

The additive inverse of the rational number 3/7, for instance, is (-3/7).

We can deduce from the additive inverse property that (3/7) + (-3/7) = (-3/7) + 3/7 = 0.

**Frequently Asked Questions – FAQs**

**1. What is the result of multiplying two reasonable numbers?**

A rational number is created by multiplying or combining two rational numbers.

**2.** **What is a rational number’s distributive property?**

If a, b, and c are three rational integers, according to the distributive property, then an x (b+c) = (a x b) + (a x c)

**3. Only addition and multiplication are applicable to the commutative property of rational numbers. False or true?**

True. Only addition and multiplication, not subtraction or division, are subject to the commutative property of rational numbers.