CLASS-7
INTRODUCTION OF RATIONAL NUMBERS

RATIONAL NUMBERS -

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, a rational number is any number that can be written in the form 𝑝/𝑞​, where 𝑝 and 𝑞 are integers and 𝑞 ≠ 0.

Here are some key properties of rational numbers:-

1. Closure under Operations:- The sum, difference, product, and quotient (except division by zero) of any two rational numbers is also a rational number.
2. Repeating or Terminating Decimals:- Every rational number can be represented as either a terminating decimal (e.g., 3/4 = 0.75) or a repeating decimal (e.g., 1/3 = 0.3‾ ).
3. Equivalent Fractions:- Rational numbers can have an infinite number of equivalent fractions. For example, 1/2 = 2/4 = 3/6 =….................
4. Ordering:- Rational numbers can be ordered on the number line. For any two rational numbers 𝑝/𝑞​ and 𝑟/𝑠​, if 𝑝/𝑞 < 𝑟/𝑠​, then 𝑝𝑠 < 𝑟𝑞.

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In other words, a rational number is any number that can be written in the form 𝑝𝑞qp​, where 𝑝p and 𝑞q are integers and 𝑞q is not equal to zero.

Rational numbers include integers, fractions, and terminating or repeating decimals. Here are some examples of rational numbers:

1. Integers: −3, 0, 1, 7

                    1         3          5

  2. Fractions: ------, ------, − ------

                    2         4          6

                                                   1              3                   7

  3. Terminating decimals: 0.25 = -----, 0.6 = -----, −1.75 = − -----

                                        4              5                    4

                                     1               2                      1

  4. Repeating decimals: 0.3‾= -----, 0.6‾= ------, 0.142857‾= -----

                                     3               3                      7

It's important to note that every integer is also a rational number, as it can be expressed as a fraction with a denominator of 1 (e.g., 5 = 5​/1). Additionally, irrational numbers, such as √2​ or π, cannot be expressed as fractions of integers and are not considered rational numbers.

Rational numbers are closed under addition, subtraction, multiplication, and division, meaning that the result of any arithmetic operation on rational numbers is also a rational number. They form an important part of the number system and are used extensively in mathematics, science, and everyday life.

ª  IMPORTANT TO KNOW:-

1. Every integer (positive, negative or zero) is a rational number.

2. Every fraction is a rational number.

The rational number(-p/q) can be written as – (p/q).

 p/q is a positive rational number and -p/q is a negative number.