Irrational numbers are numbers that are in contradiction to rational numbers. Rational numbers can be expressed as the ratio of two numbers as x/y where x and y are integers and the denominator y is not equal to zero. But irrational numbers can note denoted in a fraction form using two integers.
The examples of irrational numbers are square roots of numbers that are not perfect squares. Another example of an irrational number is pi which is used as a mathematical constant.
The irrational numbers can be expressed as decimals, but the decimal part of the number does not terminate to a definite value. For example, the decimal representation of 1/3 = 0.3333.., but it does not terminate or give an exact value. So it is an irrational number. It can be said there is no finite value or digits in an irrational number.
It can be said that real numbers include both irrational and rational numbers. If R represents real numbers and Q represents rational numbers, then irrational numbers can be denoted by the expression R-Q.
Properties of Irrational Numbers:
- Irrational numbers are not fractions or integers.
- The decimal part of an irrational number is non-terminating.
- Irrational numbers are real numbers with approximate values but not finite values.
- The sum of an irrational number and a rational number will be an irrational number.
- The product of an irrational number and any nonzero rational number will be an irrational number.
- Irrational numbers are not perfect squares of any integer.
- Two irrational numbers when added, subtracted, multiplied, or divided with any other irrational number, the result can be either rational or irrational number.
What Are Rational Numbers?
A rational number is a number that can be represented in the fraction form as m/n where m and n are two integers and n is not equal to zero. We can say that any number in fraction form can be a rational number, where the denominator and numerator are integers and the denominator is not equal to zero. In addition to all the fractions, rational numbers also include all the integers that can be written in the form of a fraction. In this case, the integer will be the numerator and 1 is written as the denominator. This is an interesting concept that can be learned in a fun way from cuemath.com.
Some of the examples of rational numbers include 1/5, 2/7, -4/13, -32/45.
The integers 3, -14, 27 can also be considered as rational numbers because they can be expressed as 3/1,-14/1, and 27/1 respectively.
As per definition, a rational number has an integer as a numerator and a non-zero denominator. The number 0 is also a rational number because it can be represented in fractional forms such as 0/1, 0/2, 0/5, etc. where the denominator is not zero.
A rational number can be positive or negative. If both the numerators and denominators of a rational number are positive or negative then the rational number is positive. If one of the numerators or denominators is positive and the other is negative, then the rational number is negative.
For example, 3/7, -15/-28, 11/6 are positive rational numbers, and -2/3, 9/-16, -8/21 are negative rational numbers.
The standard form of a rational number is the fractional form where the numerator and denominator have no common factor other than 1 and the denominator is positive.
Properties of Rational Numbers
- If we multiply, add, or subtract any two rational numbers, the result will be a rational number.
- If both the numerator and denominator are multiplied or divided with the same non-zero integer, the value remains unchanged.
- Every fraction and integer can be expressed as a rational number.
- Zero is also a national number.