 Difference Between Rational and Irrational Numbers Posted by

Rational and irrational numbers are Real numbers. A real number is a type of number in which a number can be written by infinite numbers of decimals. Such as 1 and 2 are two integers in real numbers there are infinite numbers between these two integers.  A real number can be positive or negative, massive or small, integers or fractions.

In this article, we discuss the rational and irrational numbers in detail.

What are Rational numbers?

A number that can be written in the form of a fraction or p/q, where q is not equal to zero, is known as a rational number. Keep in mind that all the integers are rational as each integer can be written in the form of a fraction with 1 as the denominator. We can also conclude that a number that is written in a ratio is known as a rational number as a ratio is written in a fraction.

The set of rational numbers is denoted by Q.

For example, 1/2, -3/4, 0.5 0r 5/10, 0.9 or 9/10, -0.6 or -6/10, etc.

Types of Rational numbers

Rational numbers have different types. Let us discuss them.

Integers

Each integer is a rational number as each integer can be written in the form of a fraction with 1 as the denominator.

For example,

2 or 2/1

4 or 4/1

9 or 9/1 etc.

Fractions

All the fractions that have integers on numerator and denominator places are said to be rational numbers. Fractions can be positive or negative.

For example,

1/4, 3/7, -5/4, -9/11, etc.

Terminating

A number having terminating decimals or finite decimals is said to be a rational number.

For example

2/5 = 0.4

1/10 = 0.1

20.2/2 = 10.1 etc.

Recurring or Non- terminating

A number that has infinite numbers of decimals is said to be recurring or non-terminating. A recurring or non-terminating number having a repeating pattern after a decimal point is said to be a rational number. It is also known as non-terminating repeating decimals.

For example,

1/3 = 0.33333333…

4/9 = 0.4444444…

9/11 = 0.818181… etc.

How to find Rational numbers?

We can find the rational number by keeping the types in mind. Let’s take some examples.

Example 1

Is 2/3 a rational number?

Solution

Step 1:Divide the fraction.

2/3 = 0.66666…

Step 2:Identify the type.

Since the decimals are recurring and repeating so it is a rational number.

2/3 = 0.666… is a rational number.

Rational or Irrational Calculator is a handy resource which can be used to calculate the rational or irrational number.

Example 2

Is 1/3 a rational number?

Solution

Step 1:Divide the fraction.

1/3 = 0.3333…

Step 2:Identify the type.

Since the decimals are recurring and repeating so it is a rational number.

1/3 = 0.3333… is a rational number.

Tips to remember Rational Number

• By definition, every number that can be written in the form of p/q is rational. But keep in mind that every number that can be written in fraction means the numbers could be integers, decimals that give p/q form after conversion.
• Every rational number, whole number, and integers are rational.
• Recurring and repeating decimals are rational numbers.
• Every negative fraction or integer is also rational.

What are Irrational Numbers?

A number that is not rational is said to be an irrational number. Irrational numbers are those real numbers that cannot be expressed in form of a fraction. A number that is non-terminating and non-repeating is said to be irrational.

The set of irrational numbers is denoted by Q’.

For example, sqrt (2), pi, e, etc. are irrational numbers.

Non-terminating and non-repeating

A number that has infinite numbers of decimals is said to be recurring or non-terminating. A recurring or non-terminating number having a non-repeating pattern after the decimal point is said to be an irrational number. It is also known as non-terminating repeating decimals.

For example, 3.14159269…, 2.718281…. etc.

Properties of Irrational Numbers

• The number that is non-terminating and non-repeating is said to be irrational.
• Irrational numbers are real numbers only.
• The Sum of rational and irrational is always irrational.
• Product of rational and irrational always irrational.
• The least common multiple of two irrational numbers may or may not exist.

Difference between Rational and Irrational numbers.

 Rational numbers Irrational numbers Rational numbers can be expressed in fractions such as p/q form. Irrational numbers cannot be expressed in fractions of integers. Examples are 0.5, 9/4, 100/20, 3/1, etc. Examples are √2, √5, pi, e, etc. These numbers can be terminating decimals. These numbers never have terminating decimals. These numbers are non-terminating and repeating patterns. These numbers are non-terminating and non-repeating patterns. It contains all the whole numbers, natural numbers, non-negative integers, negative integers. It is a different set and does not contain the whole numbers, natural numbers, non-negative integers, negative integers. Zero is a rational number. Zero is not irrational.

How to find Rational and Irrational Numbers?

Let us take an example in order to how to find rational and irrational numbers.

Example

Find the rational and irrational numbers from the following:

√2, √9, 4/5, √4, √5, e, 0.14141414141…

Solution

Step 1:Take all the numbers.

√2, √9, 4/5, √4, √5, e, 0.14141414141…

Step 2:Identify rational numbers.

Rational numbers are those numbers that can be written in the form of p/q, or when simplified we get terminating or recurring and repeating patterns of decimals. From these numbers,

√9, 4/5, √4, 0.14141414141… are rational numbers as,

√9 = 3 =3/1

4/5 = 0.8

√4 = 2 = 2/1

0.14141414141… = 0.14 as 14 is repeating.

Step 3:Identify irrational numbers.

Irrational numbers are those numbers that cannot be written in the form of p/q, or when simplified we get a non-terminating and non-repeating pattern of decimals. From these numbers,

√2, √5, e are irrational numbers as,

√2 = 1.414213562…

√5 = 2.23606797…

e = 2.7182818…

All of these are non-terminating and non-repeating so we conclude that these are irrational.

Summary

Rational and irrational numbers are real numbers. Rational numbers are those numbers that can be written in the form of p/q, where p & q are integers, and q is not equal to zero. Rational numbers have terminating decimals and recurring and repeating decimals. While on the other hand, irrational numbers are those numbers that cannot be written in the form of p/q, and irrational numbers are non-terminating and non-repeating decimals. The union of rational and irrational numbers is a real number.