Real numbers include all rational and irrational numbers, representing any value along the continuous number line.
The set of real numbers (ℝ) includes natural numbers, integers, rational numbers (fractions), and irrational numbers (non-repeating, non-terminating decimals like π and √2). Real numbers can be positive, negative, or zero. Every real number has a decimal representation. Real numbers are closed under addition, subtraction, multiplication, and division (except by zero). The real number system is fundamental to calculus, analysis, and all applied mathematics.
The concept of real numbers evolved over centuries. The Pythagoreans discovered irrational numbers (like √2) around 500 BCE, which was controversial. Richard Dedekind formalized real numbers using Dedekind cuts in 1872. Georg Cantor also developed a theory of real numbers using Cauchy sequences. The continuum hypothesis concerns the size of the real numbers.