---
title: Real Numbers - Definition, Types & Basic Properties
description: Learn about real numbers (ℝ), including rational and irrational numbers. Understand their properties like density, infinity, and basic operations.
url: https://astra-ai.co/videos/real-numbers/
locale: en
type: video-topic
keywords:
  - real numbers
  - irrational numbers
  - number sets
  - real number line
published: 2026-10-01T09:45:34.783Z
updated: 2026-10-04T21:53:17.676Z
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---

# Real Numbers

> Real numbers include rationals and irrationals on the number line. This topic explains the sets of numbers and how they fit together.

Real numbers are a set of numbers that include rational and irrational numbers. They are used to represent all points on the number line. The set of real numbers is denoted by ℝ.

## SUBSETS OF REAL NUMBERS

Real numbers can be divided into two main subsets:

- **RATIONAL NUMBERS (ℚ):** Numbers that can be written as a fraction p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -3/4, 5 which is 5/1).

- **IRRATIONAL NUMBERS:** Numbers that cannot be written as a simple fraction and have an infinite, non-repeating decimal representation (e.g., √2, π, e).

## PROPERTIES OF REAL NUMBERS

- **DENSITY:** Between any two distinct real numbers, there is always another real number.

- **INFINITY (UNBOUNDEDNESS):** The set ℝ is unbounded in both the positive and negative directions.

- **CLOSURE:** Real numbers are closed under basic arithmetic operations (addition, subtraction, multiplication, and division, except for division by 0).

## BASIC OPERATIONS

- **ADDITION:** 2.5 + 1.3 = 3.8

- **SUBTRACTION:** 7 – 4.2 = 2.8

- **MULTIPLICATION:** 1.5 × 3 = 4.5

- **DIVISION:** 5 ÷ 2 = 2.5

## CONCLUSION

Real numbers represent the entire number line and allow for the notation of all possible quantities. Their structure is the foundation of many mathematical fields, including analysis and algebra.