Mathematics is nothing but a number game. A number is an arithmetical value that can be a figure, word or symbol indicating a quantity, which has many implications like in counting, measurements, calculations, labelling, etc. Numbers can be natural numbers, whole numbers, integers, real numbers, complex numbers. Real numbers are further divided into rational numbers and irrational numbers. Rational numbers are the numbers which are integers and fractions
On the other end, Irrational numbers are the numbers whose expression as a fraction is not possible. In this article, we are going to discuss the differences between rational and irrational numbers. Have a look.
Content: Rational Numbers Vs Irrational Numbers
|Basis for Comparison||Rational Numbers||Irrational Numbers|
|Meaning||Rational numbers refers to a number that can be expressed in a ratio of two integers.||An irrational number is one which can't be written as a ratio of two integers.|
|Fraction||Expressed in fraction, where denominator ≠ 0.||Cannot be expressed in fraction.|
|Decimal expansion||Finite or recurring decimals||Non-finite or non-recurring decimals.|
Definition of Rational Numbers
The term ratio is derived from the word ratio, which means the comparison of two quantities and expressed in simple fraction. A number is said to be rational if it can be written in the form of a fraction such as p/q where both p (numerator) and q (denominator) are integers and denominator is a natural number (a non-zero number). Integers, fractions including mixed fraction, recurring decimals, finite decimals, etc., are all rational numbers.
Examples of Rational Number
- 1/9 – Both numerator and denominator are integers.
- 7 – Can be expressed as 7/1, wherein 7 is the quotient of integers 7 and 1.
- √16 – As the square root can be simplified to 4, which is the quotient of fraction 4/1
- 0.5 – Can be written as 5/10 or 1/2 and all terminating decimals are rational.
- 0.3333333333 – All recurring decimals are rational.
Definition of Irrational Numbers
A number is said to be irrational when it cannot be simplified to any fraction of an integer (x) and a natural number (y). It can also be understood as a number which is irrational. The decimal expansion of the irrational number is neither finite nor recurring. It includes surds and special numbers like π (‘pi’ is the most common irrational number) and e. A surd is a non-perfect square or cube which cannot be further reduced to remove square root or cube root.
Examples of Irrational Number
- √2 – √2 cannot be simplified and so, it is irrational.
- √7/5 – The given number is a fraction, but it is not the only criteria to be called as the rational number. Both numerator and denominator need to integers and √7 is not an integer. Hence, the given number is irrational.
- 3/0 – Fraction with denominator zero, is irrational.
- π – As the decimal value of π is never-ending, never-repeating and never shows any pattern. Therefore, the value of pi is not exactly equal to any fraction. The number 22/7 is just and approximation.
- 0.3131131113 – The decimals are neither terminating nor recurring. So it cannot be expressed as a quotient of a fraction.
Key Differences Between Rational and Irrational Numbers
The difference between rational and irrational numbers can be drawn clearly on the following grounds
- Rational Number is defined as the number which can be written in a ratio of two integers. An irrational number is a number which cannot be expressed in a ratio of two integers.
- In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero. While an irrational number cannot be written in a fraction.
- The rational number includes numbers that are perfect squares like 9, 16, 25 and so on. On the other hand, an irrational number includes surds like 2, 3, 5, etc.
- The rational number includes only those decimals, which are finite and repeating. Conversely, irrational numbers include those numbers whose decimal expansion is infinite, non-repetitive and shows no pattern.
After reviewing the above points, it is quite clear that the expression of rational numbers can be possible in both fraction and decimal form. On the contrary, an irrational number can only be presented in decimal form but not in a fraction. All integers are rational numbers, but all non-integers are not irrational numbers.