What Are Rational Numbers?
Rational numbers are those numbers that can be expressed as a quotient (the result in a regular division equation). Even if you express the resulting number not as a fraction and it repeats infinitely, it can still a rational number. Zero is a rational number.
As per the description, the rational numbers include all integers, fractions and repeating decimals. For every rational number, we can write them in the form of c/q, where c and q are integer values.
For a number to be considered a rational number it must satisfy the following criteria:
- It can be expressed in the form of a simple fraction with numerator (c) divided by a denominator (q).
- Both the numerator and the denominator must be regular integers themselves. An integer can simply be described as whole number like 3, 6, or 15.
- The denominator (q) cannot be zero. The numerator or the denominator can be positive or negative as long as the denominator is not zero.
Examples of Rational Numbers
- Number 5 can be written as 5/1 where both 5 and 1 are integers.
- 0.5 can be written as ½, 5/10, 25/50 or 10/20 and in the form of all terminating decimals.
- √81 is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
- 0.8888888 is recurring decimals and is a rational number
Facts About Rational Numbers
- The numbers that can be expressed as a ratio of two numbers i.e in the form of c/q are referred to as rational numbers.
- Rational numbers includes numbers which are finite or are recurring in nature.
- Rational numbers consists of numbers that are perfect squares such as 4, 9, 16 25 etc.
- Both numerator and denominator of rational numbers are whole numbers, in which the denominator of rational numbers is not equivalent to zero.
- Example of rational numbers: 5/3= 1.66, 1/7 =0.1428, 8/6=1.33
What Are Irrational Numbers?
An irrational number is a number that cannot be expressed as a ratio of two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. Instead, the numbers in the decimal would go on forever, without repeating.
The irrational numbers can be expressed in the form of non-terminating fractions and in different ways. For example, the square roots which are not perfect squares will always result in an irrational number.
Examples of Irrational Numbers
- 5/0 is an irrational number, with the denominator as zero.
- π is an irrational number which has value 3.142…and is a never-ending and non-repeating number.
- √2 is an irrational number, as it cannot be simplified.
- 0.212112111…is a rational number as it is non-recurring and non-terminating.
Facts About Irrational Numbers
- Numbers that cannot be expressed as ratio of two numbers i.e in the form of c/q are referred to as Irrational numbers.
- These consist of numbers which are non-terminating and non-repeating in nature.
- The irrational numbers includes surds such as √2, √3, √5, √7 and so on.
- Irrational numbers cannot be represented in fractional form.
- Examples of irrational numbers: √7, √17, √5, √9
Difference Between Rational And Irrational Numbers In Tabular Form
BASIS OF COMPARISON | RATIONAL NUMBERS | IRRATIONAL NUMBERS |
Description | The numbers that can be expressed as a ratio of two numbers i.e in the form of c/q are referred to as rational numbers. | Numbers that cannot be expressed as ratio of two numbers i.e in the form of c/q are referred to as Irrational numbers. |
Nature Of Numbers | Rational numbers includes numbers which are finite or are recurring in nature. | These consist of numbers which are non-terminating and non-repeating in nature. |
Consist Of | Rational numbers consists of numbers that are perfect squares such as 4, 9, 16 25 etc. | The irrational numbers includes surds such as √2, √3, √5, √7 and so on. |
Representation | Both numerator and denominator of rational numbers are whole numbers, in which the denominator of rational numbers is not equivalent to zero. | Irrational numbers cannot be represented in fractional form. |
Examples | 5/3= 1.66, 1/7 =0.1428, 8/6=1.33 | √7, √17, √5, √9 |
Arithmetic Rules For Rational And Irrational Numbers
- The sum of two irrational numbers may be an irrational number or a rational number, as an example (√2+ 4), (π + 2) are irrational numbers and √2 + (-√2) = 0
- The product of an irrational number to a rational number is an irrational number, as an example, 2√5,2π are irrational number.
- The product of two irrational number may be a rational or irrational number, as an example √2×–√2=-2,√2×√3 = √6
- The product of two identical irrational numbers may be rational or irrational, as an example √2×√2 = 2,
- The division of two irrational numbers can be rational or irrational, as an example 2√2/3√2= 2/3 , 2√2/√3 etc.