Overview of the Number System

The number system is a fundamental concept in mathematics, encompassing various types of numbers used for different purposes. Here is an overview of the number system, covering natural numbers, whole numbers, integers, irrational numbers, rational numbers, fractions, decimals, and their relationships.

Natural Numbers

Natural numbers, also known as counting numbers, consist of positive integers starting from 1. They are used for counting and ordering objects. For example, 1, 2, 3, 4, and so on. However, they exclude 0, fractions, decimals, and negative numbers.

Whole Numbers

Whole numbers are a subset of natural numbers, including 0. They comprise all positive integers and 0. Unlike natural numbers, they allow 0 and have a smaller lower bound. Whole numbers are represented on the right side of the number line and can be subjected to arithmetic operations like addition, subtraction, multiplication, and division.

Integers

Integers include both positive and negative whole numbers, as well as 0. They are defined as the smallest set of numbers containing 0, 1, and every number that can be obtained by adding or subtracting integers. In other words, integers are the set of all natural numbers with their negative counterparts.

Irrational Numbers

Irrational numbers are those that cannot be expressed as a finite or infinite sequence of integers. Examples of irrational numbers include pi, e, and the square roots of non-square integers. These numbers have no finite recurring decimal expansion, making them difficult to deal with mathematically.

Rational Numbers

Rational numbers are a special class of numbers that can be expressed as quotients of two integers, a/b, where a and b are integers and b is not equal to 0. They include integers and fractions. They can be represented as either terminating decimals or non-terminating repeating decimals. Both natural numbers and whole numbers are a subset of rational numbers.

Fractions

Fractions are a representation of parts of a whole. They are a subset of rational numbers and can be expressed as a/b, where a is the numerator and b is the denominator, both being integers.

Decimals

Decimals are a way of representing fractions. They consist of the digit(s) to the right of the decimal point, followed by an infinite sequence that may or may not be repeating. For example, 0.333... is a repeating decimal equivalent to the fraction 1/3.