Rational Numbers
Overview
Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q is not zero. This chapter extends our understanding from natural numbers and integers to rational numbers. We explore their properties such as closure, commutativity, associativity, and the role of identity and inverse elements. We also learn how to represent rational numbers on a number line and find rational numbers between any two given rational numbers.
Typed Notes
Properties of Rational Numbers
Rational numbers follow four main properties: Closure means performing an operation on two rational numbers gives a rational number (true for +, -, ×; division by zero excluded). Commutativity means a+b = b+a and a×b = b×a. Associativity means (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c). Distributivity means a×(b+c) = a×b + a×c. These properties make rational numbers very useful in calculations and algebra.
Identity and Inverse Elements
The additive identity is 0 because adding 0 to any rational number leaves it unchanged: a + 0 = a. The multiplicative identity is 1 because multiplying any rational number by 1 leaves it unchanged: a × 1 = a. The additive inverse of p/q is −p/q because p/q + (−p/q) = 0. The multiplicative inverse of p/q is q/p because p/q × q/p = 1. Every rational number has both an additive and a multiplicative inverse (except 0 has no multiplicative inverse).
Representation on Number Line and Finding Rational Numbers Between Two Rationals
Rational numbers can be plotted on a number line. Positive rationals lie to the right of 0 and negative rationals to the left. To find rational numbers between two rational numbers, use two methods: (1) Find the mean (average) of the two numbers — the result always lies between them. (2) Convert both fractions to equivalent fractions with a common denominator, then list integers between the numerators. Since infinitely many rationals exist between any two rationals, we can always find more by repeating the process.
Key Concepts
Revision ke liye ek nazar — exam se pehle yahi padhna hai.
- 01A rational number is expressed as p/q where p and q are integers and q ≠ 0.
- 02Rational numbers are closed under addition, subtraction, and multiplication but division by zero is not defined.
- 03The additive identity is 0 and multiplicative identity is 1 for rational numbers.
- 04The additive inverse of p/q is -p/q and the multiplicative inverse (reciprocal) is q/p (p ≠ 0).
- 05Between any two rational numbers, there are infinitely many rational numbers.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
Find the additive inverse and multiplicative inverse of 5/7.
Find two rational numbers between 1/3 and 1/2 using the mean method.
Verify that −3/5 × (2/7 + 1/3) = (−3/5 × 2/7) + (−3/5 × 1/3) using distributive law.
Represent −3/4 on a number line.
Using the equivalent fraction method, find three rational numbers between 2/5 and 3/5.
Rational numbers of the form p/q (q ≠ 0) satisfy closure, commutativity, associativity, and distributivity. Zero is the additive identity and one is the multiplicative identity. Every rational number has an additive inverse; every non-zero rational number has a multiplicative inverse. Rational numbers can be shown on a number line, and infinitely many rational numbers exist between any two rational numbers.
Chapter-end quiz
4 MCQsOption tap karo — turant pata chalega sahi ya galat, explanation ke saath.
Q1.Which of the following is the multiplicative identity for rational numbers?
Q2.What is the additive inverse of −7/9?
Q3.Which property is shown by: 2/3 × (1/4 + 1/5) = 2/3 × 1/4 + 2/3 × 1/5?
Q4.How many rational numbers exist between 1/4 and 1/3?
Mark karo — dashboard par progress dikhega.