Mathematics notes on Rational Numbers, conceptual and equation-free

1. Introduction

• Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero.

• Examples: numbers like 1/2, -3/4, 5, 0.

2. Properties of Rational Numbers

1. Closure Property

   • Adding, subtracting, or multiplying two rational numbers always gives a rational number.

   • Division of two rational numbers is also a rational number (except division by zero).

2. Commutative Property

   • The order of addition or multiplication does not affect the result.

3. Associative Property

   • Changing the grouping in addition or multiplication does not affect the result.

4. Existence of Additive Inverse

   • For every rational number, there is another rational number that adds up to zero.

5. Existence of Multiplicative Inverse

   • For every non-zero rational number, there is another rational number that multiplied gives one.

6. Distributive Property

   • Multiplication is distributive over addition for rational numbers.

3. Representation on Number Line

• Rational numbers can be represented on a number line just like integers.

• Steps to represent a rational number p/q:

  1. Divide the segment between 0 and 1 into q equal parts.

  2. Count p parts from 0 (to the right if positive, left if negative).

  3. Mark the point; that represents p/q on the number line.

Key Concept:

• Positive rational numbers are to the right of zero.

• Negative rational numbers are to the left of zero.

• Zero itself is a rational number (0/1).

4. Key Points to Remember

• Rational numbers include fractions, integers, and zero.

• They follow all arithmetic properties like closure, commutative, associative, distributive, and existence of inverses.

• Every rational number has a unique position on the number line.

• Rational numbers can be positive, negative, or zero.