History of Mathematical Notation — Timeline

Tally marks & counting

Earliest notation: scratches, notches, and marks on bones or wood used for counting and record keeping (simple, non-positional systems).

Egyptian numerals

Hieroglyphic symbols for powers of ten. Good for inscription and accounting but not positional arithmetic.

Babylonian base-60 system

Cuneiform positional numerals in base 60. This legacy explains modern measures (minutes, degrees).

Greek & Roman notations

Greeks used alphabetic numerals and focused on geometric explanation. Romans used Roman numerals (I, V, X…), which are poor for large arithmetic.

Hindu–Arabic numerals & zero

Indian mathematicians formalized the 0–9 positional system and the concept of zero — a turning point that enabled efficient calculation.

Transmission in the Islamic world

Scholars like Al-Khwarizmi wrote treatises that spread Hindu–Arabic numerals and algebraic methods to the Mediterranean world.

European adoption & early symbols

Fibonacci's Liber Abaci popularized positional numerals in Europe. By the 1500s symbols like +, −, and = appear in the record; Robert Recorde introduced “=” in 1557.

Algebraic notation & letters

Mathematicians such as François Viète and René Descartes standardized using letters for unknowns (x, y) and notation for powers (x², x³).

Calculus notation

Newton and Leibniz developed calculus independently. Leibniz's dy/dx and ∫ proved more convenient and became standard for derivatives and integrals.

Function notation & constants

Euler popularized f(x) for functions and introduced standard symbols such as e, i, and π — notation that underpins modern analysis.

Set theory & formal symbols

Cantor and others developed set theory and formalized symbols like ∈, ⊂, ∪, ∩ — enabling rigorous foundations for modern mathematics.

Logic, notation for computing & modern expansions

20th-century logic, category theory, and computer science introduced further notation (∧, ∨, ⇒, λ, ⊗) and tailored symbols for new fields.