Knowra Real number Real number A real number is an element of the complete ordered field represented by points on the continuous number line. Real numbers include rational and irrational numbers but exclude non-real complex numbers.
Irrational number : A real number that cannot be expressed as a ratio of two integers. Irrational numbers fill the real line beyond the rational numbers.
Dedekind cut : A partition of the rational numbers into lower and upper sets that can define a real number. Dedekind cuts construct real numbers by locating boundaries among rationals.
Complex number : A number expressible as a+bi, where a and b are real and i² = −1. Complex numbers extend the reals with values that do not lie on the real line.
Real analysis : The branch of mathematics studying real numbers, sequences, functions, limits, and continuity. Real analysis develops the calculus and convergence theory grounded in the real continuum.
Completeness (real numbers) : The property that every nonempty real set bounded above has a least upper bound. Completeness distinguishes the real numbers from the rational numbers, which have gaps.
Cauchy sequence : A sequence whose terms become arbitrarily close to one another as the sequence progresses. Equivalence classes of rational Cauchy sequences provide another construction of the reals.
Rational number : A number expressible as a ratio of two integers with a nonzero denominator. Unlike the reals, the rationals alone are not complete and leave gaps.
Calculus : The mathematical study of change and accumulation using derivatives and integrals. Calculus relies on real-valued limits, functions, and quantities.
Ordered field : A field equipped with an order compatible with addition and multiplication. The real numbers form a complete ordered field, combining arithmetic with comparison.
Decimal representation : A representation of a number using digits and powers of ten. Infinite decimal expansions represent irrational reals as well as rationals.
Show all 20