What are rational number in mathematics

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 What are rational numbers?

A rational number is a number that can be expressed as a ratio of two integers. In other words, it is a number that can be written in the form of a fraction, where the numerator and denominator are both integers. For example, 2/3, 7/4, and -5/6 are all rational numbers.


Rational numbers are an important concept in mathematics, and they have many useful properties. In this article, we will explain what rational numbers are, how they are used in mathematics, and some of their key properties.


What are Rational Numbers?


A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero. For example, the number 2/3 is a rational number because it can be expressed as the ratio of the integers 2 and 3. Similarly, the number -5/6 is a rational number because it can be expressed as the ratio of the integers -5 and 6.


Rational numbers are a subset of the real numbers, which includes all numbers that can be represented on a number line. The real numbers include both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed as a ratio of two integers, and they include numbers such as pi and the square root of 2.


Examples of Rational Numbers


Here are some examples of rational numbers:


- 1/2

- -3/4

- 7/8

- 2/3

- 0 (0 can be expressed as the ratio of 0 and 1, which are both integers)


Note that not all fractions are rational numbers. For example, the fraction 1/3 is not a rational number because it cannot be expressed as a ratio of two integers.


How are Rational Numbers Used in Mathematics?


Rational numbers are used in many areas of mathematics, including algebra, geometry, and number theory. Here are some examples of how rational numbers are used:


- In algebra, rational numbers are used to solve equations that involve fractions. For example, if you have an equation that involves the fraction 2/3, you can use algebraic methods to find the value of the variable that makes the equation true.

- In geometry, rational numbers are used to represent the coordinates of points on a plane. For example, if you have a point with coordinates (2/3, 4/5), you can use rational numbers to represent the x and y coordinates of the point.

- In number theory, rational numbers are used to study the properties of integers. For example, if you have an integer n, you can use rational numbers to study the factors of n and the properties of its divisors.


Properties of Rational Numbers


Rational numbers have many important properties that make them useful in mathematics. Here are some of the key properties of rational numbers:


1. Closure under addition and multiplication: If a and b are rational numbers, then a + b and ab are also rational numbers.


2. Commutativity and associativity: Addition and multiplication of rational numbers are both commutative and associative. That is, if a, b, and c are rational numbers, then:


a + b = b + a

ab = ba


(a + b) + c = a + (b + c)

(ab)c = a(bc)


3. Existence of identity and inverses: Rational numbers have an identity element (0 for addition and 1 for multiplication) and inverses (the additive inverse of a is -a, and the multiplicative inverse of a is 1/a, where a is not equal to zero).


4. Distributivity: Multiplication is distributive over addition. That is, if a, b, and c are rational numbers, then:


a(b

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