History of Mathematics

Ancient Egyptian mathematics was developed and used in Ancient Egypt c. 3000 to c. 300 BCE, from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems, often involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyrus. From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, and algebra, such as the false position method and quadratic equations.

Written evidence of the use of mathematics dates back to at least 3200 BC with the ivory labels found in Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers.^[18] Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.^[19] Archaeological evidence has suggested that the Ancient Egyptian counting system had origins in Sub-Saharan Africa.^[20] Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in Egyptian architecture and cosmological signs.^[20]

The earliest true mathematical documents date to the 12th Dynasty (c. 1990–1800 BC). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (c. 1650 BC) is said to be based on an older mathematical text from the 12th dynasty.^[22]

The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.^[9]

Babylonian mathematics were written using a sexagesimal (base-60) numeral system.^[12] From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of seconds and minutes of arc to denote fractions of a degree. It is likely the sexagesimal system was chosen because 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30.^[12] Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a place-value system, where digits written in the left column represented larger values, much as in the decimal system.^[13] The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different from multiplying integers, similar to modern notation.^[13] The notational system of the Babylonians was the best of any civilization until the Renaissance,^[14] and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places.^[14] The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.^[13] By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions.^[13] This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.^[13]

Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs.^[15] The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.^[16] Tablets from the Old Babylonian period also contain the earliest known statement of the Pythagorean theorem.^[17] However, as with Egyptian mathematics, Babylonian mathematics shows no awareness of the difference between exact and approximate solutions, or the solvability of a problem, and most importantly, no explicit statement of the need for proofs or logical principles.^[13]

They also used a form of Fourier analysis to compute an ephemeris (table of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.^[11] To make calculations of the movements of celestial bodies, the Babylonians used basic arithmetic and a coordinate system based on the ecliptic, the part of the heavens that the sun and planets travel through.

Greek mathematics allegedly began with Thales of Miletus (c. 624–548 BC). Very little is known about his life, although it is generally agreed that he was one of the Seven Wise Men of Greece. According to Proclus, he traveled to Babylon from where he learned mathematics and other subjects, coming up with the proof of what is now called Thales' Theorem.^[23]

Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.^[30]

An equally enigmatic figure is Pythagoras of Samos (c. 580–500 BC), who supposedly visited Egypt and Babylon,^[24] and ultimately settled in Croton, Magna Graecia, where he started a kind of brotherhood. Pythagoreans supposedly believed that "all is number" and were keen in looking for mathematical relations between numbers and things.^[25] Pythagoras himself was given credit for many later discoveries, including the construction of the five regular solids.

Almost half of the material in Euclid's Elements is customarily attributed to the Pythagoreans, including the discovery of irrationals, attributed to Hippasus (c. 530–450 BC) and Theodorus (fl. 450 BC).^[26]  It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The greatest mathematician associated with the group, however, may have been Archytas (c. 435-360 BC), who solved the problem of doubling the cube, identified the harmonic mean, and possibly contributed to optics and mechanics.^[26] Other mathematicians active in this period, not fully affiliated with any school, include Hippocrates of Chios (c. 470–410 BC), Theaetetus (c. 417–369 BC), and Eudoxus (c. 408–355 BC).

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum),^[39] who probably discovered them while identifying sides of the pentagram.^[40] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. Hippasus, in the 5th century BC, however, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction.

Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans 'for having produced an element in the universe which denied the... doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.'^[41] Whatever the consequence to Hippasus himself, his discovery posed a very serious problem to Pythagorean mathematics, since it shattered the assumption that number and geometry were inseparable–a foundation of their theory.

Plato is important in the history of mathematics for inspiring and guiding others.^[31] His Platonic Academy, in Athens, became the mathematical center of the world in the 4th century BC, and it was from this school that the leading mathematicians of the day, such as Eudoxus of Cnidus, came.^[32] Plato also discussed the foundations of mathematics,^[33] clarified some of the definitions (e.g. that of a line as "breadthless length"), and reorganized the assumptions.^[34] The analytic method is ascribed to Plato, while a formula for obtaining Pythagorean triples bears his name.^[32]

The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well.^[77] It also defined the concepts of circumference, diameter, radius, and volume.^[78]

The Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.^[68] Of particular note is the use in Chinese mathematics of a decimal positional notation system, the so-called "rod numerals" in which distinct ciphers were used for numbers between 1 and 10, and additional ciphers for powers of ten.^[69] Thus, the number 123 would be written using the symbol for "1", followed by the symbol for "100", then the symbol for "2" followed by the symbol for "10", followed by the symbol for "3". This was the most advanced number system in the world at the time, apparently in use several centuries before the common era and well before the development of the Indian numeral system.^[76] Rod numerals allowed the representation of numbers as large as desired and allowed calculations to be carried out on the suan pan, or Chinese abacus. It is presumed that officials used the multiplication table to calculate land surface area, yields of crops and the amounts of taxes owed.^[68]

The Hellenistic era began in the late 4th century BC, following Alexander the Great's conquest of the Eastern Mediterranean, Egypt, Mesopotamia, the Iranian plateau, Central Asia, and parts of India, leading to the spread of the Greek language and culture across these regions. Greek became the lingua franca of scholarship throughout the Hellenistic world, and the mathematics of the Classical period merged with Egyptian and Babylonian mathematics to give rise to Hellenistic mathematics.^[27]

Greek mathematics and astronomy reached its acme during the Hellenistic and early Roman periods, and much of the work represented by authors such as Euclid (fl. 300 BC), Archimedes (c. 287–212 BC), Apollonius (c. 240–190 BC), Hipparchus (c. 190–120 BC), and Ptolemy (c. 100–170 AD) was of a very advanced level and rarely mastered outside a small circle.

Several centers of learning appeared during the Hellenistic period, of which the most important one was the Mouseion in Alexandria, Egypt, which attracted scholars from across the Hellenistic world (mostly Greek, but also Egyptian, Jewish, Persian, among others).^[28] Although few in number, Hellenistic mathematicians actively communicated with each other; publication consisted of passing and copying someone's work among colleagues.^[29]

In the 3rd century BC, the premier center of mathematical education and research was the Musaeum of Alexandria.^[36] It was there that Euclid (c. 300 BC) taught, and wrote the Elements, widely considered the most successful and influential textbook of all time.^[35]

Considered the "father of geometry", Euclid is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.

The Elements introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.^[37] In addition to the familiar theorems of Euclidean geometry, the Elements was meant as an introductory textbook to all mathematical subjects of the time, such as number theory, algebra and solid geometry,^[37] including proofs that the square root of two is irrational and that there are infinitely many prime numbers. Euclid also wrote extensively on other subjects, such as conic sections, optics, spherical geometry, and mechanics, but only half of his writings survive.^[38]

The Euclidean algorithm is one of the oldest algorithms in common use.^[93] It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. Centuries later, Euclid's algorithm was discovered independently both in India and in China,^[94] primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.

Archimedes of Syracuse is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,^[42] Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems.^[43] These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.^[44]

Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever,^[45] the widespread use of the concept of center of gravity,^[46] and the enunciation of the law of buoyancy or Archimedes' principle.

Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed.

Apollonius of Perga (c. 262–190 BC) made significant advances to the study of conic sections, showing that one can obtain all three varieties of conic section by varying the angle of the plane that cuts a double-napped cone.^[47] He also coined the terminology in use today for conic sections, namely parabola ("place beside" or "comparison"), "ellipse" ("deficiency"), and "hyperbola" ("a throw beyond").^[48] His work Conics is one of the best known and preserved mathematical works from antiquity, and in it he derives many theorems concerning conic sections that would prove invaluable to later mathematicians and astronomers studying planetary motion, such as Isaac Newton.^[49] While neither Apollonius nor any other Greek mathematicians made the leap to coordinate geometry, Apollonius' treatment of curves is in some ways similar to the modern treatment, and some of his work seems to anticipate the development of analytical geometry by Descartes some 1800 years later.^[50]

In 212 BC, the Emperor Qin Shi Huang commanded all books in the Qin Empire other than officially sanctioned ones be burned. This decree was not universally obeyed, but as a consequence of this order little is known about ancient Chinese mathematics before this date. After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. 

After the book burning of 212 BC, the Han dynasty (202 BC–220 AD) produced works of mathematics which presumably expanded on works that are now lost. The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles.^[79] It created mathematical proof for the Pythagorean theorem,^[81] and a mathematical formula for Gaussian elimination.^[80] The treatise also provides values of π,^[79] which Chinese mathematicians originally approximated as 3 until Liu Xin (d. 23 AD) provided a figure of 3.1457 and subsequently Zhang Heng (78–139) approximated pi as 3.1724,^[82] as well as 3.162 by taking the square root of 10.^[83]

Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art but may well contain much older material.^[84] The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers.

The 3rd century BC is generally regarded as the "Golden Age" of Greek mathematics, with advances in pure mathematics henceforth in relative decline.^[51] Nevertheless, in the centuries that followed significant advances were made in applied mathematics, most notably trigonometry, largely to address the needs of astronomers.^[51] Hipparchus of Nicaea (c. 190–120 BC) is considered the founder of trigonometry for compiling the first known trigonometric table, and to him is also due the systematic use of the 360 degree circle.^[52]

In the 2nd century AD, the Greco-Egyptian astronomer Ptolemy (from Alexandria, Egypt) constructed detailed trigonometric tables (Ptolemy's table of chords) in Book 1, chapter 11 of his Almagest. Ptolemy used chord length to define his trigonometric functions, a minor difference from the sine convention we use today. Centuries passed before more detailed tables were produced, and Ptolemy's treatise remained in use for performing trigonometric calculations in astronomy throughout the next 1200 years in the medieval Byzantine, Islamic, and, later, Western European worlds. Ptolemy is also credited with Ptolemy's theorem for deriving trigonometric quantities, and the most accurate value of π outside of China until the medieval period, 3.1416.^[63]

In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1). The earliest known statement of the theorem is by the Chinese mathematician Sun-tzu in the Sun-tzu Suan-ching in the 3rd century CE.

Following a period of stagnation after Ptolemy, the period between 250 and 350 AD is sometimes referred to as the "Silver Age" of Greek mathematics.^[53] During this period, Diophantus made significant advances in algebra, particularly indeterminate analysis, which is also known as "Diophantine analysis".^[54] The study of Diophantine equations and Diophantine approximations is a significant area of research to this day. His main work was the Arithmetica, a collection of 150 algebraic problems dealing with exact solutions to determinate and indeterminate equations.^[55] The Arithmetica had a significant influence on later mathematicians, such as Pierre de Fermat, who arrived at his famous Last Theorem after trying to generalize a problem he had read in the Arithmetica (that of dividing a square into two squares).^[56] Diophantus also made significant advances in notation, the Arithmetica being the first instance of algebraic symbolism and syncopation.^[55]

Ancient Egyptian numerals were of base 10. They used hieroglyphs for the digits and were not positional. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system.

The concept of zero as a written digit in the decimal place value notation was developed in India.^[65] A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.^[66] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.^[67] Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:

A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

The first woman mathematician recorded by history was Hypatia of Alexandria (AD 350–415). She wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria had her stripped publicly and executed. Her death is sometimes taken as the end of the era of the Alexandrian Greek mathematics, although work did continue in Athens for another century with figures such as Proclus, Simplicius and Eutocius.^[57] Although Proclus and Simplicius were more philosophers than mathematicians, their commentaries on earlier works are valuable sources on Greek mathematics. The closure of the neo-Platonic Academy of Athens by the emperor Justinian in 529 AD is traditionally held as marking the end of the era of Greek mathematics, although the Greek tradition continued unbroken in the Byzantine empire with mathematicians such as Anthemius of Tralles and Isidore of Miletus, the architects of the Hagia Sophia.^[58] Nevertheless, Byzantine mathematics consisted mostly of commentaries, with little in the way of innovation, and the centers of mathematical innovation were to be found elsewhere by this time.^[59]

The modern sine convention is first attested in the Surya Siddhanta (showing strong Hellenistic influence)^[64], and its properties were further documented by the 5th century (AD) Indian mathematician and astronomer Aryabhata.^[60] The Surya Siddhanta describes rules to calculate the motions of various planets and the moon relative to various constellations, diameters of various planets, and calculates the orbits of various astronomical bodies. The text is known for some of earliest known discussion of sexagesimal fractions and trigonometric functions.^[61]

Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration.^[62] Though about half of the entries are wrong, it is in the Aryabhatiya that the decimal place-value system first appears.

In the 9th century, the mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote an important book on the Hindu–Arabic numerals and one on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi, were instrumental in spreading Indian mathematics and Indian numerals to the West. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,^[87] and he was the first to teach algebra in an elementary form and for its own sake.^[88] He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.^[89] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."^[90]

Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad Ibn Shujāʿ was a prominent Egyptian mathematician during the Islamic Golden Age. He is considered the first mathematician to systematically use and accept irrational numbers as solutions and coefficients to equations.^[91] His mathematical techniques were later adopted by Fibonacci, thus allowing Abu Kamil an important part in introducing algebra to Europe.^[92]

In the Pre-Columbian Americas, the Maya civilization that flourished in Mexico and Central America during the 1st millennium AD developed a unique tradition of mathematics that, due to its geographic isolation, was entirely independent of existing European, Egyptian, and Asian mathematics.^[92] Maya numerals used a base of twenty, the vigesimal system, instead of a base of ten that forms the basis of the decimal system used by most modern cultures.^[92] The Maya used mathematics to create the Maya calendar as well as to predict astronomical phenomena in their native Maya astronomy.^[92] While the concept of zero had to be inferred in the mathematics of many contemporary cultures, the Maya developed a standard symbol for it.^[92]

Abū Bakr Muḥammad ibn al Ḥasan al-Karajī was a 10th-century Persian mathematician and engineer who flourished at Baghdad. He was born in Karaj, a city near Tehran. His three principal surviving works are mathematical: Al-Badi' fi'l-hisab (Wonderful on calculation), Al-Fakhri fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi fi'l-hisab (Sufficient on calculation).

Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala, which survives from the medieval era in at least four copies.

His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials.

The high-water mark of Chinese mathematics occurred in the 13th century during the latter half of the Song dynasty (960–1279), with the development of Chinese algebra. The most important text from that period is the Precious Mirror of the Four Elements by Zhu Shijie (1249–1314), dealing with the solution of simultaneous higher order algebraic equations using a method similar to Horner's method.^[70] The Precious Mirror also contains a diagram of Pascal's triangle with coefficients of binomial expansions through the eighth power, though both appear in Chinese works as early as 1100.^[71] The Chinese also made use of the complex combinatorial diagram known as the magic square and magic circles, described in ancient times and perfected by Yang Hui (AD 1238–1298).^[71]

Japanese mathematics, Korean mathematics, and Vietnamese mathematics are traditionally viewed as stemming from Chinese mathematics and belonging to the Confucian-based East Asian cultural sphere.^[72] Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644).^[73] For instance, although Vietnamese mathematical treatises were written in either Chinese or the native Vietnamese Chữ Nôm script, all of them followed the Chinese format of presenting a collection of problems with algorithms for solving them, followed by numerical answers.^[74] Mathematics in Vietnam and Korea were mostly associated with the professional court bureaucracy of mathematicians and astronomers, whereas in Japan it was more prevalent in the realm of private schools.^[75]

Europeans learned of Arabic numerals about the 10th century, though their spread was a gradual process. Two centuries later, in the Algerian city of Béjaïa, the Italian scholar Fibonacci first encountered the numerals; his work was crucial in making them known throughout Europe. European trade, books, and colonialism helped popularize the adoption of Arabic numerals around the world. The numerals have found worldwide use significantly beyond the contemporary spread of the Latin alphabet, and have become common in the writing systems where other numeral systems existed previously, such as Chinese and Japanese numerals. The first mentions of the numerals from 1 to 9 in the West are found in the Codex Vigilanus of 976, an illuminated collection of various historical documents covering a period from antiquity to the 10th century in Hispania.^[68]

In the 12th century, European scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's The Compendious Book on Calculation by Completion and Balancing, translated into Latin by Robert of Chester, and the complete text of Euclid's Elements, translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona.^[95] These and other new sources sparked a renewal of mathematics.

Leonardo of Pisa, now known as Fibonacci, serendipitously learned about the Hindu–Arabic numerals on a trip to what is now Béjaïa, Algeria with his merchant father. (Europe was still using Roman numerals.) There, he observed a system of arithmetic (specifically algorism) which due to the positional notation of Hindu–Arabic numerals was much more efficient and greatly facilitated commerce. He soon realised the many advantages of the Hindu-Arabic system, which, unlike the Roman numerals used at the time, allowed easy calculation using a place-value system. Leonardo wrote Liber Abaci in 1202 (updated in 1254) introducing the technique to Europe and beginning a long period of popularizing it. The book also brought to Europe what is now known as the Fibonacci sequence (known to Indian mathematicians for hundreds of years before that)^[96] which Fibonacci used as an unremarkable example.

Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of π.^[86] The Kerala school has made a number of contributions to the fields of infinite series and calculus.