A timeline of **numerals** and ** arithmetic **.

- c. 20,000 BC — Nile Valley, Ishango Bone: suggested, though disputed, as the earliest reference to prime numbers as also a common number.
^{ [1] } - c. 3400 BC — the Sumerians invent the first so-known numeral system,
^{[ dubious – discuss ]}and a system of weights and measures. - c. 3100 BC — Egypt, earliest known decimal system allows indefinite counting by way of introducing new symbols, .
^{[ }*citation needed*] - c. 2800 BC — Indus Valley civilization on the Indian subcontinent, earliest use of decimal ratios in a uniform system of ancient weights and measures, the smallest unit of measurement used is 1.704 millimetres and the smallest unit of mass used is 28 grams.
^{[ citation needed ]} - c. 2000 BC — Mesopotamia, the Babylonians use a base-60 decimal system, and compute the first known approximate value of π at 3.125.
^{[ citation needed ]}

- c. 1000 BC — Vulgar fractions used by the Egyptians.
- second half of 1st millennium BC — The Lo Shu Square, the unique normal magic square of order three, was discovered in China.
- c. 400 BC — Jaina mathematicians in India write the “Surya Prajinapti”, a mathematical text which classifies all numbers into three sets: enumerable, innumerable and infinite. It also recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually.
- c. 300 BC — Brahmi numerals are conceived in India.
- 300 BC — Mesopotamia, the Babylonians invent the earliest calculator, the abacus.
^{[ dubious – discuss ]}^{[ citation needed ]} - c. 300 BC — Indian mathematician Pingala writes the “Chhandah-shastra”, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a binary numeral system, along with the first use of Fibonacci numbers and Pascal's triangle.
- c. 250 BC — late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number).
- 150 BC — Jain mathematicians in India write the “Sthananga Sutra”, which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.
- 50 BC — Indian numerals, the first positional notation base-10 numeral system, begins developing in India.

- 300 — the earliest known use of zero as a decimal digit in the Old World is introduced by Indian mathematicians.
- c. 400 — the Bakhshali manuscript uses numerals with a place-value system, using a dot as a place holder for zero .
- 550 — Hindu mathematicians give zero a numeral representation in the positional notation Indian numeral system.
- 628 — Brahmagupta writes the
*Brahma-sphuta-siddhanta*, where zero is clearly explained, and where the modern place-value Indian numeral system is fully developed. It also gives rules for manipulating both negative and positive numbers, methods for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and the Brahmagupta theorem. - 940 — Abu'l-Wafa al-Buzjani extracts roots using the Indian numeral system.
- 953 — The arithmetic of the Hindu–Arabic numeral system at first required the use of a dust board (a sort of handheld blackboard) because “the methods required moving the numbers around in the calculation and rubbing some out as the calculation proceeded.” Al-Uqlidisi modified these methods for pen and paper use. Eventually the advances enabled by the decimal system led to its standard use throughout the region and the world.

- c. 1000 — Pope Sylvester II introduces the abacus using the Hindu–Arabic numeral system to Europe.
- 1030 — Ali Ahmad Nasawi writes a treatise on the decimal and sexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.
^{ [2] } - 12th century — Indian numerals have been modified by Persian mathematicians al-Khwārizmī to form the modern Arabic numerals (used universally in the modern world.)
- 12th century — the Arabic numerals reach Europe through the Arabs.
- 1202 — Leonardo Fibonacci demonstrates the utility of Hindu–Arabic numeral system in his
*Book of the Abacus*. - c. 1400 — Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as pi. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.” He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include
*The Key of arithmetics, Discoveries in mathematics, The Decimal point*, and*The benefits of the zero*. The contents of the*Benefits of the Zero*are an introduction followed by five essays: “On whole number arithmetic”, “On fractional arithmetic”, “On astrology”, “On areas”, and “On finding the unknowns [unknown variables]”. He also wrote the*Thesis on the sine and the chord*and*Thesis on finding the first degree sine*. - 15th century — Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general.
^{ [3] } - 1427 — Al-Kashi completes
*The Key to Arithmetic*containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones. - 1478 — An anonymous author writes the Treviso Arithmetic.

- 1614 — John Napier publishes a table of Napierian logarithms in
*Mirifici Logarithmorum Canonis Descriptio*, - 1617 — Henry Briggs discusses decimal logarithms in
*Logarithmorum Chilias Prima*, - 1618 — John Napier publishes the first references to
*e*in a work on logarithms.

- 1758 —
*Arithmetika Horvatzka*, Croatia's first arithmetic textbook is published in Zagreb by Mihalj Šilobod Bolšić (1724–1787). - 1794 — Jurij Vega publishes
*Thesaurus Logarithmorum Completus*.

- 1706 — John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places.
- 1789 — Jurij Vega improves Machin's formula and computes π to 140 decimal places.
- 1949 — John von Neumann computes π to 2,037 decimal places using ENIAC.
- 1961 — Daniel Shanks and John Wrench compute π to 100,000 decimal places using an inverse-tangent identity and an IBM-7090 computer.
- 1987 — Yasumasa Kanada, David Bailey, Jonathan Borwein, and Peter Borwein use iterative modular equation approximations to elliptic integrals and a NEC SX-2 supercomputer to compute π to 134 million decimal places.
- 2002 — Yasumasa Kanada, Y. Ushiro, Hisayasu Kuroda, Makoto Kudoh and a team of nine more compute π to 1241.1 billion digits using a Hitachi 64-node supercomputer.

the system we use today consists of the same 10 numbers 0-9.

- Abacus – Calculating tool
- Alphabetic numeral system – Type of numeral system
- Attic numerals – Symbolic number notation used by the ancient Greeks
- Australian Aboriginal enumeration – Counting system used by Australian Aboriginals
- Counting rods – East Asian numeral system
- History of ancient numeral systems – Symbols representing numbers
- History of arithmetic – Branch of elementary mathematics
- History of mathematics
- History of numbers – Used to count, measure, and label
- History of the Hindu–Arabic numeral system
- Jeton – Coin-like counting token
- List of numeral system topics
- List of numeral systems
- Number theory – Mathematics of integer properties
- Prehistoric counting
- Relationship between mathematics and physics – Study of how mathematics and physics relate to each other
- Roman numerals – Numbers in the Roman numeral system
- Timeline of algorithms
- Timeline of mathematics

The **decimal** numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as *decimal notation*.

The **history of mathematics** deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the field of astronomy to record time and formulate calendars.

A **numeral system** is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

A **number** is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called *numerals*; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any non-negative integer using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, a *numeral* is not clearly distinguished from the *number* that it represents.

**0** (**zero**) is a number representing an empty quantity. Adding 0 to any number leaves that number unchanged. In mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 has the result 0, and consequently, division by zero has no meaning in arithmetic.

The ** Liber Abaci** or

A **numerical digit** or **numeral** is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.

**Brahmagupta** was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the *Brāhmasphuṭasiddhānta*, a theoretical treatise, and the *Khaṇḍakhādyaka*, a more practical text.

**Algorism** is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an **algorist**. This positional notation system has largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude, such as Roman numerals, and in some cases required a device such as an abacus.

**Positional notation** usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify. The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.

**Bhāskara** was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu–Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This commentary, *Āryabhaṭīyabhāṣya*, written in 629, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school: the *Mahābhāskarīya* and the *Laghubhāskarīya*.

**Indian mathematics** emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II,Varāhamihira, and Madhava. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

The Hindu–Arabic numeral system is a decimal place-value numeral system that uses a zero glyph as in "205".

The **Hindu–Arabic numeral system** is a positional base ten numeral system for representing integers; its extension to non-integers is the decimal numeral system, which is presently the most common numeral system.

The **history of mathematical notation** includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators. The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

The following is a timeline of key developments of geometry:

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

In mathematics, the **irrational numbers** are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being *incommensurable*, meaning that they share no "measure" in common, that is, there is no length, no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

- ↑ Rudman, Peter Strom (2007).
*How Mathematics Happened: The First 50,000 Years*. Prometheus Books. p. 64. ISBN 978-1-59102-477-4. - ↑ O'Connor, John J.; Robertson, Edmund F., "Abu l'Hasan Ali ibn Ahmad Al-Nasawi",
*MacTutor History of Mathematics Archive*, University of St Andrews - ↑ O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?",
*MacTutor History of Mathematics Archive*, University of St Andrews

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