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 introduces students to rational numbers, their representation on a number line, finding rational numbers between two given rational numbers, and the properties of addition, subtraction, multiplication, and division of rational numbers. Understanding rational numbers extends the idea of fractions and negative numbers into one unified system.
Typed Notes
What Are Rational Numbers?
A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. For example, 3/4, -5/7, 0/1, and -2/3 are all rational numbers. Natural numbers, whole numbers, and integers are all special cases of rational numbers. For instance, the integer -3 can be written as -3/1. Zero is also a rational number since 0 = 0/1. Rational numbers can be positive, negative, or zero.
Standard Form and Equivalent Rational Numbers
A rational number is in standard form when the denominator is positive and the HCF of the numerator and denominator is 1. For example, -6/8 in standard form is -3/4 (divide both by HCF = 2). Equivalent rational numbers are obtained by multiplying or dividing both numerator and denominator by the same non-zero integer. For example, 2/3 = 4/6 = 6/9. To compare two rational numbers, convert them to the same denominator or use cross-multiplication.
Operations on Rational Numbers
Addition: To add p/q and r/s, use (ps + rq)/qs. If denominators are same, add numerators directly. Subtraction: p/q − r/s = (ps − rq)/qs. Multiplication: p/q × r/s = pr/qs. Division: p/q ÷ r/s = p/q × s/r = ps/qr (multiply by reciprocal). Rational numbers are closed under addition, subtraction, and multiplication. Division by zero is not defined. Addition and multiplication are commutative and associative for rational numbers. Zero is the additive identity and 1 is the multiplicative identity.
Rational Numbers on a Number Line and Between Two Rationals
Rational numbers can be represented on a number line. Positive rational numbers lie to the right of zero and negative rational numbers lie to the left. To find rational numbers between two rational numbers, convert them to equivalent forms with a common denominator and list numbers in between. Alternatively, the mean method works: the average of two rational numbers always lies between them. For example, between 1/3 and 1/2, the mean is (1/3 + 1/2)/2 = 5/12, which lies between them. There are infinitely many rational numbers between any two rational numbers.
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 include positive numbers, negative numbers, fractions, and zero.
- 03Two rational numbers are equivalent if cross-multiplication gives equal products: a/b = c/d if a×d = b×c.
- 04A rational number p/q is in standard form when q is positive and HCF of |p| and |q| is 1.
- 05Infinitely many rational numbers exist between any two rational numbers.
Important Formulas
Questions & Solutions
Pehle khud try karo — phir solution kholo. Yahi tariqa topper banata hai.
Write -18/24 in standard form.
Add the rational numbers 3/5 and -7/10.
Multiply -4/9 by 3/8.
Find two rational numbers between 1/4 and 1/2.
Divide -5/6 by 10/3.
Rational numbers are of the form p/q (q ≠ 0) and include all integers, fractions, and their negatives. They follow properties of commutativity, associativity, and distributivity. Standard form requires a positive denominator and HCF of numerator and denominator equal to 1. Operations follow specific rules using LCM or cross-multiplication. 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 NOT a rational number?
Q2.What is the standard form of -36/48?
Q3.What is -2/3 + 4/5?
Q4.A rational number between 0 and 1 using the mean method is:
Mark karo — dashboard par progress dikhega.