Table of Contents

## Introduction to Real Numbers

In mathematics, a real number is a number that represents a quantity along a line. The real numbers include the natural numbers, integers, rational numbers, and irrational numbers. Real numbers are often represented by a point on a number line. The number line can be thought of as a continuum of points, with each point representing a real number.

## Real Numbers

A real number is a number that represents a point on a number line. Real numbers include rational and irrational numbers. Rational numbers are numbers that can be expressed as a ratio of two integers, while irrational numbers are numbers that cannot be expressed as a rational number.

### Euclid’s Division Lemma

The Euclid’s Division Lemma is a mathematical theorem that helps in the division of one number by another. The theorem states that if a number is divided by another number, the remainder will be the same as the division of the second number by the first number.

### Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every number can be written as a product of prime numbers.

### Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a rational number. The decimal expansion of an irrational number never ends and never repeats. The most famous irrational number is pi, which is the ratio of a circle’s circumference to its diameter.

## Rational Numbers

- Rational numbers are numbers that can be expressed as the quotient of two integers, a numerator and a denominator. In other words, rational numbers are numbers that can be written in the form p/q, where p and q are both integers and q is not equal to 0.
- The set of all rational numbers is denoted by Q.
- Some examples of rational numbers include 1/2, 3/4, and -7/8.