Which of the Following is a Rational Number?

Understanding the concept of rational numbers is essential 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, provide examples, and discuss how to determine if a number is rational or not.

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.”

Rational numbers can be positive, negative, or zero. They can also be whole numbers, integers, or decimals that terminate or repeat. For example, 2, -5, 0, 1/2, 0.75, and -3.333… are all rational numbers.

Identifying Rational Numbers

There are several methods to determine if a number is rational or not. Let’s explore some of these methods:

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, making it a rational number.

Similarly, the number 0.5 can be written as 1/2, and -2 can be written as -2/1. In both cases, the numbers can be expressed as fractions, making them rational.

Method 2: Terminating Decimals

A decimal is said to be terminating if it has a finite number of digits after the decimal point. Terminating decimals can always be expressed as fractions. For example, the number 0.75 can be written as 3/4, making it a rational number.

Similarly, the number 0.25 can be written as 1/4, and 0.5 can be written as 1/2. In all these cases, the decimals terminate, and they can be expressed as fractions, making them rational numbers.

Method 3: Repeating Decimals

A decimal is said to be repeating if it has a pattern of digits that repeats indefinitely. Repeating decimals can also be expressed as fractions. For example, the number 0.333… can be written as 1/3, making it a rational number.

Similarly, the number 0.666… can be written as 2/3, and 0.121212… can be written as 12/99. In all these cases, the decimals repeat, and they can be expressed as fractions, making them rational numbers.

Examples of Rational Numbers

Let’s explore some examples of rational numbers:

  • 2: This is a whole number and can be written as 2/1.
  • -5: This is a negative whole number and can be written as -5/1.
  • 1/2: This is a fraction and is already in the form of a ratio.
  • 0.75: This is a terminating decimal and can be written as 3/4.
  • -3.333…: This is a repeating decimal and can be written as -10/3.

Non-Rational Numbers

Not all numbers are rational. There are two types of numbers that are not rational: irrational numbers and imaginary numbers.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as fractions. They are non-repeating and non-terminating decimals. Examples of irrational numbers include π (pi), √2 (square root of 2), and e (Euler’s number).

These numbers have decimal representations that go on forever without repeating. For example, π is approximately 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679…

Imaginary Numbers

Imaginary numbers are numbers that involve the imaginary unit, denoted by the letter “i.” The imaginary unit is defined as the square root of -1. Imaginary numbers are not rational because they cannot be expressed as fractions or decimals.

Examples of imaginary numbers include 2i, -3i, and 5i. These numbers are used in complex numbers, which are a combination of real and imaginary numbers.

Summary

Rational numbers are numbers that can be expressed as a fraction, where the numerator and denominator are both integers. They can be positive, negative, or zero, and can be whole numbers, integers, or decimals that terminate or repeat. Rational numbers can be identified by representing them as fractions, or by determining if their decimal representation terminates or repeats.

On the other hand, irrational numbers and imaginary numbers are not rational. Irrational numbers are non-repeating and non-terminating decimals, while imaginary numbers involve the imaginary unit “i” and cannot be expressed as fractions or decimals.

Q&A

1. Is 0 a rational number?

Yes, 0 is a rational number. It can be expressed as 0/1, making it a fraction.

2. Is √3 a rational number?

No, √3 is an irrational number. It cannot be expressed as a fraction or a terminating or repeating decimal.

3. Is 1.25 a rational number?

Yes, 1.25 is a rational number. It can be written as 5/4, making it a fraction.

4. Is 5i a rational number?

No, 5i is an imaginary number. It involves the imaginary unit “i” and cannot be expressed as a fraction or a decimal.

5. Is -7 a rational number?

Yes, -7 is a rational number. It can be expressed as -7/1, making it a fraction.