1.1: Real Number System

The real number system includes all numbers used for counting, measuring, and comparing quantities. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers expand this set by including zero and negative whole numbers: ..., –3, –2, –1, 0, 1, 2, .... Rational numbers are those that can be expressed as the ratio of two integers m/n, where n≠0. This includes fractions like 1/2​, integers such as 46, and decimals that terminate, such as 0.17, or those that repeat, like 0.317171717. A repeating decimal can be rewritten as a fraction using algebraic techniques.

Irrational numbers cannot be written as a ratio of integers. Their decimal expansions go on forever without repeating. Examples include √2, √5, ��, e, and 3/��². These often arise from geometric problems, such as finding the diagonal of a square. Every real number corresponds to a point on the number line. This continuous representation allows rational and irrational numbers to coexist within a single mathematical framework, enabling precise calculations and measurements in both theoretical and applied mathematics.

The real number system includes all values that can be placed on the number line, including both rational and irrational numbers.

Rational numbers include values that can be written as fractions. For example, cutting a fruit into two equal parts gives one-half, a rational number. These also include negative fractions, like negative three-fourths.

Rational numbers include decimals that terminate, like 0.5, or repeat, like 0.33, with a bar over the digits.

Within rational numbers lies the subset of integers. Integers include negative numbers, zero, and positive numbers. Counting fruits in a tray gives natural numbers: 1, 2, 3, and so on. Adding zero to natural numbers forms the whole numbers. Extending these to include negatives gives the full set of integers.

Another part of the real number system, irrational numbers, represents values like the square root of two, which gives the diagonal of a unit square.

These numbers have non-repeating decimals that go on forever and cannot be written as fractions.

The real number system includes all values used to count, measure, or compare.