Reals measure continua that rationals alone cannot fill
Real numbers measure continuous one-dimensional quantities—length, time, temperature—where values can differ by arbitrarily small amounts. Each has an essentially unique infinite decimal expansion. Descartes coined real in the 1600s to contrast imaginaries like square roots of negatives. Calculus builds limits and derivatives on them.
Reals split into rationals and irrationals. Those that solve a polynomial equation with rational coefficients are called algebraic, a group that takes in every rational along with irrationals like √2 ≈ 1.414; the rest, π among them, are transcendental. Pictured geometrically, they are the points of the number line, on which the integers sit at equal spacing.
Informal pictures are not enough for rigorous proof, and a satisfactory formal definition ranks among the major achievements of nineteenth-century mathematics, laying the foundation of real analysis. Axiomatically, the reals are the Dedekind-complete ordered field, unique up to isomorphism, so any two such fields behave identically. Constructions from classes of Cauchy sequences of rationals, from Dedekind cuts, or from infinite decimals all meet those axioms and are therefore equivalent. That uniqueness is why mathematicians and physicists could compute with reals for centuries before the first formal definitions arrived in the late 1800s.
Being an ordered field means elementary arithmetic works: addition and multiplication are commutative and associative, multiplication distributes over addition, 0 and 1 serve as identities, every number has a negative, and every nonzero number has a reciprocal. Subtraction is adding the negative, division is multiplying by the reciprocal, and absolute value gives the distance from zero. The order is total, so for any two reals exactly one is smaller or they are equal, and it stays compatible with both addition and multiplication.
Source: Real number
