Numbers can be classified into various categories like real numbers, imaginary numbers, rational, irrational, whole numbers, natural numbers, even numbers, odd numbers, etc. Based on whether a real number can be represented in form of p/q, we can classify numbers as rational and irrational numbers. In this article, we discuss 7 differences between rational and irrational numbers.
What are Rational Numbers?
A rational number is a type of real number, which can be represented as a ratio of two numbers in the form of p/q where q is not equal to zero. Any fraction with a non-zero denominator can be considered as a rational number. Examples of rational numbers: 4, 3, 3/4, 5/8, 0. The number “0” is also a rational number.
What are irrational numbers?
Irrational numbers are real numbers that cannot be represented as a simple fraction in the form of p/q. In other words, we cannot form a ratio for an irrational number. Examples include surds, pie, Euler’s number, golden ratio, etc…
Difference between rational and irrational numbers
| COMPARISON | Rational Numbers | Irrational numbers |
| MEANINGS | Rational numbers are the numbers that can be represented as a ratio of two numbers. They exist in the form of p/q where q≠0. 2 can be represented 2/1. | Irrational numbers are numbers that cannot be represented as a ratio of two numbers. The number 3.14159265358979……….. cannot be represented in p/q form. |
| PROPERTIES | Closure Property: For two rational numbers say a and b the results of addition, subtraction and multiplication operations give a rational number. 2 + 3 = 5; 2*3 = 6; 2-3 = -1 Distributive Property: If a, b and c are three rational numbers, [a * (b + c)] = (a * b) + (a * c) Identity and inverse property: Identity property :0 is an additive identity and 1 is a multiplicative identity for rational numbers. additive identity: 2/4 + 0 = 2/4 inverse property: 2/4*1 = 2/4 Inverse property: For a rational number x/y, the additive inverse is -x/y, and y/x is the multiplicative inverse. a/b + (-a/b) = 0 [2/4 + (-2/4)] a/b*b/a = 1 Commutative Property: For rational numbers, addition and multiplication are commutative.a + b = b + a; a ×b = b × a Where a and b are rational. Associative Property:If x, y and z are rational, then for addition: x+(y + z)=(x + y)+z | •The addition of an irrational number and a rational number gives an irrational number. 3 + 3.141592…….. = 6.141592……(irrational) •Multiplication of any irrational number with any nonzero rational number results in an irrational number. √3 * 3 = 5.196………..(irrational) •The addition or the multiplication of two irrational numbers may be a rational number. √3 * √3 = 3 (rational) |
| NATURE OF | They mostly include those numbers which are finite and are recurring in nature. eg: 3.4, 4, 5.66, 7.32, 5 | They are non-terminating or non-recurring in nature. e.g: 3.141592……, 1.618033988……. |
| CONSISTS OF | They consist of numbers that are perfect squares. | They consist of surds such as √3, √2, √5, etc |
| FORM | Both the numerator and the denominators of the rational numbers are whole numbers, where the denominator of a rational number is not equal to zero. | They cannot be represented in fractional form. |
| EXAMPLES | 2/4, 3/4, 5/8 | √5, √3 |
| Application in real life | Any whole number like 1,2,3…. is used very frequently in economics, consumer data, etc. | pie is an irrational number used in the calculation of the area and perimeter of a circle, Golden ratio φ = 1.61803398874989…. Roots are frequently used in trigonometry, Euler’s number |
Simmilarities between rational and irrational numbers
Both rational and irrational numbers are real numbers. So they share the same properties of a real number.
- The Closure Property
- The Commutative Property
- The Associative Property
- The Distributive Property
