Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.[a]

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7]

Mathematicians seek and use patterns[8][9] to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.

Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[10]

Galileo Galilei (1564–1642) said, "The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth."[11] Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences".[12] Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions".[13] David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."[14] Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality."[15]

Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics, or mathematics for its own sake, without having any application in mind. Practical applications for what began as pure mathematics are often discovered.[16]


The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals,[17] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.

Greek mathematician Pythagoras (c. 570 BC – c. 495 BC), commonly credited with discovering the Pythagorean theorem

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.[18]

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[19] The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time.

In Babylonian mathematics, elementary arithmetic (addition, subtraction, multiplication and division) first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]

Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematics in its own right with Greek mathematics.[20]

Persian mathematician Al-Khwarizmi (c. 780 – c. 850), the inventor of algebra.

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics: most of them include the contributions from Persian mathematicians such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs."[21]


The word mathematics comes from Ancient Greek μάθημα (máthēma), meaning "that which is learnt",[22] "what one gets to know", hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times.[23] Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant "the mathematical art".

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "teachers" rather than "mathematicians" in the modern sense.

In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians.[24]

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek.[25] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math.[26]

Other Languages
Afrikaans: Wiskunde
Alemannisch: Mathematik
Ænglisc: Rīmcræft
العربية: رياضيات
aragonés: Matematicas
armãneashti: Mathematică
অসমীয়া: গণিত
asturianu: Matemátiques
Avañe'ẽ: Papapykuaa
Aymar aru: Jakhu
azərbaycanca: Riyaziyyat
تۆرکجه: ریاضیات
বাংলা: গণিত
Bahasa Banjar: Matamatika
Bân-lâm-gú: Sò͘-ha̍k
Basa Banyumasan: Matematika
башҡортса: Математика
беларуская: Матэматыка
беларуская (тарашкевіца)‎: Матэматыка
भोजपुरी: गणित
Bislama: Matematikis
български: Математика
Boarisch: Mathematik
བོད་ཡིག: རྩིས་རིག
bosanski: Matematika
brezhoneg: Matematik
català: Matemàtiques
Чӑвашла: Математика
Cebuano: Matematika
čeština: Matematika
Chamoru: Matematika
chiShona: Masvomhu
corsu: Matematica
Cymraeg: Mathemateg
dansk: Matematik
Deutsch: Mathematik
ދިވެހިބަސް: ރިޔާޟިއްޔާތު
dolnoserbski: Matematika
Ελληνικά: Μαθηματικά
emiliàn e rumagnòl: Matemâtica
эрзянь: Математика
español: Matemáticas
Esperanto: Matematiko
estremeñu: Matemáticas
euskara: Matematika
فارسی: ریاضیات
Fiji Hindi: Mathematics
føroyskt: Støddfrøði
français: Mathématiques
Frysk: Wiskunde
furlan: Matematiche
Gaeilge: Matamaitic
Gaelg: Maddaght
Gàidhlig: Matamataig
galego: Matemáticas
贛語: 數學
ગુજરાતી: ગણિત
客家語/Hak-kâ-ngî: Su-ho̍k
хальмг: Математика
한국어: 수학
Hausa: Lissafi
Hawaiʻi: Makemakika
Հայերեն: Մաթեմատիկա
हिन्दी: गणित
hrvatski: Matematika
Ilokano: Matematika
বিষ্ণুপ্রিয়া মণিপুরী: গণিত
Bahasa Indonesia: Matematika
interlingua: Mathematica
Interlingue: Matematica
isiXhosa: I-Mathematics
isiZulu: Imathemathiki
íslenska: Stærðfræði
italiano: Matematica
עברית: מתמטיקה
Basa Jawa: Matématika
kalaallisut: Matematikki
ಕನ್ನಡ: ಗಣಿತ
къарачай-малкъар: Математика
ქართული: მათემატიკა
kaszëbsczi: Matematika
қазақша: Математика
Kiswahili: Hisabati
Kreyòl ayisyen: Matematik
kurdî: Matematîk
Кыргызча: Математика
Ladino: Matemátika
Latina: Mathematica
latviešu: Matemātika
Lëtzebuergesch: Mathematik
lietuvių: Matematika
Ligure: Matematica
Limburgs: Mathematiek
Livvinkarjala: Matematiekku
la .lojban.: cmaci
Luganda: Ekibalangulo
lumbaart: Matemàtega
magyar: Matematika
मैथिली: गणित
македонски: Математика
Malagasy: Fanisana
മലയാളം: ഗണിതം
Malti: Matematika
मराठी: गणित
მარგალური: მათემატიკა
مصرى: رياضيات
مازِرونی: ریاضی
Bahasa Melayu: Matematik
Mìng-dĕ̤ng-ngṳ̄: Só-hŏk
Mirandés: Matemática
монгол: Математик
မြန်မာဘာသာ: သင်္ချာ
Nederlands: Wiskunde
Nedersaksies: Wiskunde
नेपाली: गणित
नेपाल भाषा: ल्याःज्या
日本語: 数学
Nordfriisk: Matematiik
Norfuk / Pitkern: Maethamatiks
norsk: Matematikk
norsk nynorsk: Matematikk
Nouormand: Caltchul
Novial: Matematike
occitan: Matematicas
олык марий: Математике
ଓଡ଼ିଆ: ଗଣିତ
Oromoo: Herrega
oʻzbekcha/ўзбекча: Matematika
ਪੰਜਾਬੀ: ਗਣਿਤ
पालि: गणितं
Pangasinan: Matematiks
پنجابی: میتھمیٹکس
Patois: Matimatix
ភាសាខ្មែរ: គណិតវិទ្យា
Picard: Matématikes
Piemontèis: Matemàtica
Tok Pisin: Ol matematik
Plattdüütsch: Mathematik
polski: Matematyka
português: Matemática
Qaraqalpaqsha: Matematika
qırımtatarca: Riyaziyat
română: Matematică
Runa Simi: Yupay yachay
русиньскый: Математіка
русский: Математика
саха тыла: Математика
Gagana Samoa: Matematika
संस्कृतम्: गणितम्
Seeltersk: Mathematik
Setswana: Mathematics
shqip: Matematika
sicilianu: Matimàtica
සිංහල: ගණිතය
Simple English: Mathematics
سنڌي: رياضي
SiSwati: Tekubala
slovenčina: Matematika
slovenščina: Matematika
ślůnski: Matymatyka
Soomaaliga: Xisaab
کوردی: ماتماتیک
Sranantongo: Sabi fu Teri
српски / srpski: Математика
srpskohrvatski / српскохрватски: Matematika
Basa Sunda: Matematika
svenska: Matematik
Tagalog: Matematika
தமிழ்: கணிதம்
Taqbaylit: Tusnakt
татарча/tatarça: Математика
తెలుగు: గణితము
тоҷикӣ: Риёзиёт
ತುಳು: ಗಣಿತ
Türkçe: Matematik
Türkmençe: Matematika
Thuɔŋjäŋ: Akuënkäŋ
ᨅᨔ ᨕᨘᨁᨗ: ᨆᨈᨛᨆᨈᨗᨀ
українська: Математика
اردو: ریاضی
Vahcuengh: Soqyoz
vèneto: Matemàtega
vepsän kel’: Matematik
Tiếng Việt: Toán học
Volapük: Matemat
文言: 數學
West-Vlams: Wiskunde
Winaray: Matematika
Wolof: Xayma
吴语: 数学
ייִדיש: מאטעמאטיק
Yorùbá: Mathimátíkì
粵語: 數學
Zazaki: Matematik
Zeêuws: Wiskunde
žemaitėška: Matematėka
中文: 数学
ГӀалгӀай: Математика
Lingua Franca Nova: Matematica