Finding Rational Numbers Between 1 and 2: A Comprehensive Guide

Introduction

Rational numbers are a fundamental concept in mathematics. They can be expressed as the quotient or fraction (frac{p}{q}) of two integers, where (q) is non-zero. This article delves into various methods to determine rational numbers between 1 and 2, as per the National Council of Educational Research and Training (NCERT) class 9 curriculum.

NCERT Class 9 Method I: Using Fractions and Decimals

One straightforward approach to finding rational numbers between 1 and 2 is by using fractions and decimals. Some examples include:

1.5 or (frac{3}{2}) 1.25 or (frac{5}{4}) 1.4 or (frac{7}{5}) 1.125 or (frac{9}{8}) 1.1 or (frac{11}{10})

These numbers are valid because they are in the form of (frac{p}{q}) where (p) and (q) are integers, and (q) is non-zero.

NCERT Class 9 Method II: Arithmetic Progression Approach

Another method uses arithmetic progression (AP). Consider an arithmetic progression where the first term (a 1) and the seventh term (a_7 2).

Using the formula for the (n)-th term of an AP, (a_n a (n-1)d), we have:

1: (a 1)

2: (a_7 1 6d 2) or (6d 1) thus (d frac{1}{6}).

We can now express 1 and 2 using this common difference:

(1 frac{1}{6}) (2 frac{12}{6})

Hence, the five rational numbers between 1 and 2 are:

(frac{7}{6}) (frac{8}{6} frac{4}{3}) (frac{9}{6}) (frac{10}{6} frac{5}{3}) (frac{11}{6})

These numbers are evenly spaced and conform to the requirements of the problem.

General Method: Multiplying by (frac{N 1}{N 1})

The NCERT class 9 book suggests a general method to find (N) evenly spaced rational numbers between two consecutive integers. For example, to find 5 rational numbers between 1 and 2, we multiply by (frac{6}{6}).

Hence, we have:

1 (frac{6}{6}) 2 (frac{12}{6})

The intermediate numbers are found by incrementing the numerator while keeping the denominator the same:

(frac{7}{6}) (frac{8}{6} frac{4}{3}) (frac{9}{6}) (frac{10}{6} frac{5}{3}) (frac{11}{6})

As you can see, this method provides a systematic approach to finding rational numbers within any given range.

Conclusion

There are countless rational numbers between 1 and 2, and the methods outlined above are just a few ways to find them. Whether you use fractions and decimals, arithmetic progression, or systematic multiplication, the core idea remains the same: to express these numbers as the quotient of two integers.

Understanding and applying these methods not only enhances your mathematical skills but also helps in solving more complex problems in the future.