수의 세계가 넓어진 순간들
셈을 위한 자연수는 빚을 표현하는 음수, 비율을 나타내는 유리수, 분수로 잡히지 않는 무리수, 빈틈 없는 실수선, 회전을 계산하는 복소수로 이어졌습니다. 열두 수 세계를 지도·포함 관계·시간 흐름의 세 방식으로 비교하며, 앞선 체계가 풀지 못한 문제가 다음 체계를 어떻게 불러냈는지 살펴봅니다.
Travel from rationals through irrationals and transcendentals to the completed real line, then onward through pure imaginary numbers and the complex plane — following the questions and distinct kinds of historical milestone that expanded number.
The arrows show the order in which humanity crossed into the next question, not set containment. Follow the cities as the number line becomes complete and then opens into a plane.
Rationals seemed sufficient until √2 opened a gap and transcendental numbers revealed a deeper frontier. Dedekind completed them as one real line.
The equation x² = −1 stops on the real line, but i opens a perpendicular direction. When the real and imaginary axes meet as a + bi, multiplication becomes scaling and rotation.
Select a number system to explore its details.
Decimal notation, quaternions, and transfinite numbers branch into further expeditions.
Each date may mark a different kind of event: an artifact, a surviving text, a publication, or a formal construction. Brahmagupta’s rules in 628, Cardano’s 1545 publication, and Wessel’s 1799 plane should be read separately from later transmission and adoption.
What a single historical milestone cannot show — *why this one came after that*, *why a millennium to accept*, *how it became the language of the universe*
Realizing "there are lengths the world cannot measure" came long before "emptiness deserves to be a number."
Start by questioning the 4,000 years. Cultures used common fractions, sexagesimal fractions, and decimal-like notation in parallel; 1585 is better treated as a European popularization milestone than an invention date.
China used negative numbers BCE, yet as late as the 18th century Euler still wondered whether negatives might be greater than infinity.
Square roots of negative numbers appeared on the way to a real answer in 1545; by 1926, i stood in the standard equation for the time evolution of a quantum state.
For the Greeks, 1 was μονάς (monas) — "the unit itself", not "a number among numbers."
Follow zero as its role changed from placeholder to calculable number across Indian, Islamicate, translation, and Mediterranean commercial networks.
One quaternion product, or separate dot and cross products? A real dispute over notation changed teaching and physical calculation.
Narrative order is not containment. ℚ and the irrationals ℝ∖ℚ together form the reals ℝ, while the transcendentals 𝕋 lie inside the irrationals. The real and imaginary axes meet as a + bi in the complex plane ℂ.