The area of study known as the history of mathematics is primarily an investigation调查研究 into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation符号、乐谱、注释 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美索不达米亚（两河流域） states of Sumer苏美尔（古地区名）, Akkad古巴比伦阿卡德地区 and Assyria亚旭（亚洲南部之古国）, together with Ancient Egypt and Ebla埃卜拉 began using arithmetic, algebra and geometry for purposes of taxation征税, commerce, trade and also in the field of astronomy and to formulate规划、用公式表示 calendars and record time.
The most ancient mathematical texts available are from Mesopotamia and Egypt – Plimpton普林顿 322 (Babylonian巴比伦人 c. 1900 BC), the Rhind莱茵德 Mathematical Papyrus纸莎草纸 (Egyptian c. 2000–1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian c. 1890 BC). All of these texts mention the so-called所谓的 Pythagorean毕达哥拉斯的 triples三倍数、三个一组 and so, by inference推理、推论, the Pythagorean theorem, seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a "demonstrative说明的、证明的 discipline" begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction". Greek mathematics greatly refined改进、精炼 the methods (especially through the introduction of deductive演绎的、推论的 reasoning and mathematical rigor严密、严格 in proofs) and expanded the subject matter of mathematics. Although they made virtually事实上、几乎 no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying调查、测量, structural结构的、建筑的 engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a place value system位置体系 and the first use of negative numbers负数. The Hindu–Arabic印度-阿拉伯 numeral system and the rules for the use of its operations, in use throughout the world today evolved over the course of the first millennium千年期、公元纪元 AD in India and were transmitted to the Western world via Islamic mathematics through the work of Muḥammad ibn Mūsā al-Khwārizmī. Islamic mathematics, in turn, developed and expanded the mathematics known to these civilizations. Contemporaneous同时期的、同时代的 with but independent of these traditions were the mathematics developed by the Maya civilization of Mexico and Central America, where the concept of zero was given a standard symbol in Maya numerals.
Many Greek and Arabic texts on mathematics were translated into Latin拉丁语 from the 12th century onward, leading to further development of mathematics in Medieval中世纪的 Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation停滞. Beginning in Renaissance文艺复兴 Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus微积分 during the course of the 17th century. At the end of the 19th century the International Congress国会、代表大会 of Mathematicians was founded and continues to spearhead带头、做先锋 advances in the field.