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History of Numbers Timeline

(c. 3000 BCE–Present)

The history of numbers traces how humans developed counting, numerical notation, arithmetic, and advanced mathematical systems across civilizations. Less

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Ancient numeral systems

Egyptian base-10 system

3100 BC

The earliest known base-10 system dates to around 3100 BC in Egypt. Egyptian numerals provided a decimal framework for recording quantities, administration, and calculation.

Image source: Egyptian numerals

Egyptian base-10 system

Archimedes presents positional notation for huge numbers

300 BC

In The Sand Reckoner, Archimedes demonstrated a positional numeral system capable of expressing extremely large numbers. His method extended numerical notation beyond ordinary counting scales.

Khmer numerals document decimal notation

7 AD

By the seventh century, the decimal numeral concept had clearly reached Cambodia in the form of Khmer numerals. Later documentation shows related ideas spreading to China and the Islamic world.

Image source: Khmer numerals

Khmer numerals document decimal notation

Hindu–Arabic numerals reach Europe through Islamic sources

1000 AD

The Hindu–Arabic numeral concept began reaching Europe through Islamic sources around the year 1000. Its adoption eventually displaced Roman numerals as Europe's dominant system of numerical representation.

Image source: Hindu–Arabic numeral system

Hindu–Arabic numerals reach Europe through Islamic sources

Decimal fractions gain modern notation

1600 AD

By the seventeenth century, mathematicians generally used decimal fractions with notation resembling modern practice. Decimal notation made fractional quantities easier to calculate and communicate.

Image source: Decimal

Decimal fractions gain modern notation

Sánchez reports finger abacus systems

1961 AD

In 1961, George I. Sánchez reported a base-4 and base-5 finger abacus. The report documented alternative bodily and manual methods for representing and manipulating numbers.

Image source: Abacus

Sánchez reports finger abacus systems

Real and complex numbers

Babylonian approximation of square root of 2

1800 BC

Babylonian mathematicians recorded highly accurate approximations of irrational quantities such as the square root of 2 on clay tablets. Tablet YBC 7289 gives an approximation comparable to six decimal places.

Heron considers square roots of negative numbers

1 AD

The earliest fleeting reference to square roots of negative numbers occurred in the work of Heron of Alexandria. He encountered such a quantity while considering the volume of an impossible frustum of a pyramid.

Image source: Hero of Alexandria

Heron considers square roots of negative numbers

Tartaglia and Cardano develop polynomial root formulas

1500 AD

Sixteenth-century Italian mathematicians including Niccolò Fontana Tartaglia and Gerolamo Cardano discovered closed formulas for roots of third- and fourth-degree polynomials. These formulas made quantities involving square roots of negative numbers more prominent.

Image source: Complex number

Tartaglia and Cardano develop polynomial root formulas

Descartes introduces the concept of real numbers

1600 AD

René Descartes introduced the concept of real numbers in the seventeenth century. The idea grouped ordinary positive and negative quantities with rational and irrational values on a continuous number line.

Image source: Real number

Descartes introduces the concept of real numbers

Descartes coins the term imaginary

1637 AD

In 1637, René Descartes is sometimes credited with coining the term imaginary for quantities involving square roots of negative numbers. He intended the term as a derogatory description of their puzzling status.

Image source: Imaginary number

Descartes coins the term imaginary

Bernoulli identifies the exponential-growth base

1683 AD

While studying compound interest in 1683, Jacob Bernoulli found that as compounding intervals became shorter, exponential growth converged to a base of approximately 2.71828. This constant later became known as Euler's number, e.

Wallis depicts complex numbers geometrically

1685 AD

Wallis's De algebra tractatus contained an early appearance of the graphic representation of complex numbers, predating the later formal geometric interpretation.

Euler studies irrational numbers

1700 AD

Irrational numbers began to be studied systematically in the eighteenth century, particularly through the work of Leonhard Euler. Euler connected irrationality with nonfinite simple continued fractions and proved that e is irrational.

Image source: Irrational number

Euler studies irrational numbers

Lambert proves pi is irrational

1761 AD

Johann Lambert proved in 1761 that π is irrational. The result established that π cannot be expressed as a ratio of two integers.

Image source: Johann Heinrich Lambert

Lambert proves pi is irrational

Wessel gives complex numbers a geometric interpretation

1799 AD

Caspar Wessel described a geometrical interpretation of complex numbers in 1799. His account helped establish that complex numbers could be represented as points or directed quantities in a plane.

Liouville establishes transcendental numbers

1844 AD - 1851 AD

Joseph Liouville first established the existence of transcendental numbers in work published in 1844 and 1851. These numbers are not roots of any nonzero polynomial with integer coefficients.

Puiseux distinguishes poles and branch points

1850 AD

In 1850, Victor Alexandre Puiseux took a key step in distinguishing poles from branch points and introduced the concept of essential singular points in complex analysis.

Dedekind and others rigorously define real numbers

1869 AD - 1883 AD

The second half of the nineteenth century brought rigorous definitions of real and irrational numbers. Contributions came from Augustin-Louis Cauchy, Charles Méray in 1869, Karl Weierstrass and Eduard Heine in 1872, Richard Dedekind in 1872, and Georg Cantor in 1883.

Hermite proves e is transcendental

1873 AD

Charles Hermite proved in 1873 that e is transcendental. Thus Euler's number is not the root of any nonzero polynomial with integer coefficients.

Lindemann proves pi is transcendental

1882 AD

Ferdinand von Lindemann proved in 1882 that π is transcendental. This result resolved the ancient problem of squaring the circle with straightedge and compass.

Image source: Lindemann–Weierstrass theorem

Lindemann proves pi is transcendental

Number concepts

Pāṇini uses a null operator

500 BC

In the fifth century BC, Pāṇini used a null, or zero, operator in the Ashtadhyayi. This was an early algebraic treatment of absence or null value in a formal grammatical system.

Image source: Pāṇini

Pāṇini uses a null operator

Greek reference to negative numbers

3 AD

The first reference to negative numbers in a Western work appeared in Greece during the third century AD. The concept remained controversial and was not broadly accepted in Europe for many centuries.

Image source: Negative number

Greek reference to negative numbers

Bhaskara gives negative roots

1150 AD

In the twelfth century, Bhaskara gave negative roots for quadratic equations. He nevertheless cautioned that a negative value was inadequate in the case under discussion because people did not approve of negative roots.

Image source: Bhāskara II

Bhaskara gives negative roots

Fibonacci allows negative financial solutions

1202 AD

Fibonacci allowed negative solutions in financial problems in Liber Abaci, interpreting them as debts. He later used negative values to represent losses in Flos.

Image source: Liber Abaci

Fibonacci allows negative financial solutions

Nicolas Chuquet experiments with negative numbers

1484 AD

Nicolas Chuquet was an early European experimenter with negative numbers during the fifteenth century. His work helped preserve and develop ideas that European mathematicians generally resisted.

Image source: Nicolas Chuquet

Nicolas Chuquet experiments with negative numbers

European acceptance of negative and fractional numbers

1500 AD

The sixteenth century brought final European acceptance of negative integers and fractional numbers. This marked a major expansion of the kinds of quantities treated as legitimate numbers.

Wallis introduces infinity symbol

1655 AD

John Wallis introduced the symbol ∞ in a mathematical context in 1655. The symbol became widely associated with an infinite quantity.

European mathematicians accept negative results

1700 AD

As late as the eighteenth century, it was common practice to ignore negative results returned by equations on the assumption that they were meaningless. This illustrates the delayed acceptance of negative numbers in European mathematics.

Set theorists begin treating zero as a natural number

1800 AD

Traditionally, the natural numbers began with 1, and ancient Greeks did not even regard 0 as a number. During the nineteenth century, set theorists and other mathematicians increasingly included 0, relating it to the cardinality of the empty set.

Algebraic structures extend the concept of number

1800 AD

During the nineteenth century, mathematicians developed systems now called algebraic structures. These systems share selected properties of numbers and can be viewed as generalizations or extensions of the concept.

Cantor develops transfinite numbers and set theory

1895 AD

In 1895, Georg Cantor published a book on his new set theory. He introduced transfinite numbers and formulated the continuum hypothesis, extending the study of number beyond finite quantities.

Image source: Georg Cantor

Cantor develops transfinite numbers and set theory

Computer approximations reach trillions of digits

2000 AD

The numbers π and e have both been approximated by computer to trillions of digits. These computations demonstrate the continuing use of numerical methods to investigate constants with nonterminating expansions.

Number theory

Ibn al-Haytham discovers Wilson's theorem

1000 AD

Around 1000 AD, Ibn al-Haytham discovered the result later known as Wilson's theorem, a characterization of prime numbers through the factorial of one less than the prime.

Fibonacci describes trial division

1202 AD

In Liber Abaci, published in 1202, Fibonacci communicated Islamic mathematical contributions to Europe and became the first known European writer to describe the method of trial division.

Image source: Fibonacci sequence

Fibonacci describes trial division

Legendre conjectures the prime number theorem

1796 AD

In 1796, Adrien-Marie Legendre conjectured the prime number theorem, describing the asymptotic distribution of prime numbers. The theorem predicts how the number of primes below a given size grows.

Image source: Prime number theorem

Legendre conjectures the prime number theorem

Kummer advances ideal numbers

1800 AD

Ernst Kummer made a major advance in the theory of algebraic numbers and introduced ideal numbers. These ideas were later expressed as geometrical entities by Felix Klein in 1893.

Riemann formulates the Riemann hypothesis

1859 AD

Bernhard Riemann formulated the Riemann hypothesis in 1859. The conjecture concerns the distribution of prime numbers through the locations of the nontrivial zeros of the zeta function.

Image source: Riemann hypothesis

Riemann formulates the Riemann hypothesis

Klein gives ideal numbers geometric form

1893 AD

In 1893, Felix Klein expressed Kummer's ideal numbers as geometrical entities. This connected abstract algebraic number ideas with geometric representation.

Hadamard and de la Vallée-Poussin prove the prime number theorem

1896 AD

Jacques Hadamard and Charles de la Vallée-Poussin finally proved the prime number theorem in 1896. Their independent proofs established the predicted asymptotic distribution of primes.

Key Facts

  1. Early civilizations, including the Sumerians and Egyptians, created numerical systems for trade, taxation, construction, and record-keeping.
  2. The ancient Babylonians used a sophisticated base-60 system, which influences modern time and angle measurements.
  3. The ancient Greeks developed important theories about numbers, while the Romans popularized a numeral system based largely on letters.
  4. The Hindu-Arabic numeral system, including the concept of zero, originated in India and spread widely through the Islamic world and Europe.
  5. Modern mathematics uses numbers in diverse forms, including integers, fractions, irrational numbers, complex numbers, and digital binary code.

Source

This History of Numbers timeline was generated with the help of AI, using information found on the internet.

We work hard to keep these timelines accurate, but mistakes do get through. If you spot one, email us at [email protected] and we'll fix it for future visitors.

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