Which of the Following is a Rational Number?

Understanding the concept of rational numbers is fundamental in mathematics. Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. In this article, we will explore what rational numbers are, how to identify them, and provide examples to illustrate their properties. Let’s dive in!

What are Rational Numbers?

Rational numbers are a subset of real numbers that can be expressed as a fraction, where the numerator and denominator are both integers. The word “rational” comes from the Latin word “ratio,” which means “ratio” or “proportion.” This is fitting because rational numbers represent the ratio of two integers.

Rational numbers can be positive, negative, or zero. They can be written in the form a/b, where a and b are integers and b is not equal to zero. The numerator a represents the number of parts we have, and the denominator b represents the total number of equal parts the whole is divided into.

Identifying Rational Numbers

Identifying whether a number is rational or not can be done through various methods. Let’s explore some of the common techniques:

Method 1: Fraction Representation

The most straightforward way to identify a rational number is by representing it as a fraction. If a number can be expressed as a fraction, it is rational. For example, the number 3 can be written as 3/1, where the numerator is 3 and the denominator is 1. Similarly, the number -2 can be written as -2/1.

Let’s take another example: 0.75. To determine if it is rational, we can convert it to a fraction. Since 0.75 is equivalent to 75/100, we can simplify it by dividing both the numerator and denominator by their greatest common divisor, which is 25. Thus, 0.75 is rational and can be expressed as 3/4.

Method 2: Terminating or Repeating Decimals

Rational numbers can also be identified by their decimal representation. A rational number will always have a decimal that either terminates or repeats. Let’s consider the number 0.3333… (where the 3s repeat infinitely). This number can be expressed as 1/3, which is a rational number. Similarly, the number 0.5 terminates and can be written as 1/2.

On the other hand, irrational numbers, such as the square root of 2 (√2), have decimal representations that neither terminate nor repeat. For example, √2 is approximately 1.41421356… and the decimal goes on indefinitely without any repeating pattern.

Examples of Rational Numbers

Now that we understand how to identify rational numbers, let’s explore some examples:

Example 1: 2/3

The fraction 2/3 is a rational number. The numerator is 2, and the denominator is 3, both of which are integers. Therefore, 2/3 is a rational number.

Example 2: -5

The whole number -5 can be expressed as -5/1, where the numerator is -5 and the denominator is 1. Since both the numerator and denominator are integers, -5 is a rational number.

Example 3: 0.25

The decimal 0.25 can be written as a fraction by placing the digits after the decimal point over the appropriate power of 10. In this case, 0.25 is equivalent to 25/100. Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor, we get 1/4. Therefore, 0.25 is a rational number.

Properties of Rational Numbers

Rational numbers possess several interesting properties that make them unique. Let’s explore some of these properties:

Property 1: Closure Property

The closure property states that the sum, difference, product, or quotient of any two rational numbers is always a rational number. For example, if we add 2/3 and 1/4, we get 11/12, which is a rational number.

Property 2: Commutative and Associative Properties

Rational numbers follow the commutative and associative properties for addition and multiplication. The commutative property states that changing the order of the numbers being added or multiplied does not affect the result. For example, 2/3 + 1/4 is the same as 1/4 + 2/3.

The associative property states that changing the grouping of the numbers being added or multiplied does not affect the result. For example, (2/3 + 1/4) + 1/5 is the same as 2/3 + (1/4 + 1/5).

Property 3: Identity Elements

Rational numbers have identity elements for addition and multiplication. The identity element for addition is 0, as adding 0 to any rational number does not change its value. The identity element for multiplication is 1, as multiplying any rational number by 1 does not change its value.

Property 4: Inverse Elements

Every rational number has an additive inverse and a multiplicative inverse. The additive inverse of a rational number a/b is -a/b, which, when added to a/b, gives the identity element 0. The multiplicative inverse of a rational number a/b is b/a, which, when multiplied by a/b, gives the identity element 1.

Q&A

Q1: Is 0 a rational number?

A1: Yes, 0 is a rational number. It can be expressed as 0/1, where the numerator is 0 and the denominator is 1.

Q2: Is every integer a rational number?

A2: Yes, every integer is a rational number. Integers can be expressed as fractions with a denominator of 1. For example, the integer 5 can be written as 5/1, which is a rational number.

Q3: Is the square root of 9 a rational number?

A3: Yes, the square root of 9 (√9) is a rational number. The square root of 9 is 3, which can be expressed

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