In mathematics,the Euclidean algorithm,or Euclid's algorithm,is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers),the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid,who first described it in his Elements . It is an example of an algorithm,a step-by-step procedure for performing a calculation according to well-defined rules,and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form,and is a part of many other number-theoretic and cryptographic calculations.
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said,"Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers,or defined as generalizations of the integers.
Division is one of the four basic operations of arithmetic. The other operations are addition,subtraction,and multiplication. What is being divided is called the dividend,which is divided by the divisor,and the result is called the quotient.
In mathematics,division by zero,division where the divisor (denominator) is zero,is a unique and problematic special case. Using fraction notation,the general example can be written as ,where is the dividend (numerator).
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 Khandakhadyaka,a more practical text.
In arithmetic,long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps.
In mathematics,a multiply perfect number is a generalization of a perfect number.
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division,usually by examining its digits. Although there are divisibility tests for numbers in any radix,or base,and they are all different,this article presents rules and examples only for decimal,or base 10,numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.
Mathematics emerged independently in China by the 11th century BCE. The Chinese independently developed a real number system that includes significantly large and negative numbers,more than one numeral system,algebra,geometry,number theory and trigonometry.
Elementary arithmetic is a branch of mathematics involving addition,subtraction,multiplication,and division. Due to its low level of abstraction,broad range of application,and position as the foundation of all mathematics,elementary arithmetic is generally the first branch of mathematics taught in schools.
In arithmetic,the galley method,also known as the batello or the scratch method,was the most widely used method of division in use prior to 1600. The names galea and batello refer to a boat which the outline of the work was thought to resemble.
Alligation is an old and practical method of solving arithmetic problems related to mixtures of ingredients. There are two types of alligation:alligation medial,used to find the quantity of a mixture given the quantities of its ingredients,and alligation alternate,used to find the amount of each ingredient needed to make a mixture of a given quantity. Alligation medial is merely a matter of finding a weighted mean. Alligation alternate is more complicated and involves organizing the ingredients into high and low pairs which are then traded off. Alligation alternate provides answers when an algebraic solution is not possible. Note that in this class of problem,there may be multiple feasible answers.
The Akhmim wooden tablets,also known as the Cairo wooden tablets are two wooden writing tablets from ancient Egypt,solving arithmetical problems. They each measure around 18 by 10 inches and are covered with plaster. The tablets are inscribed on both sides. The hieroglyphic inscriptions on the first tablet include a list of servants,which is followed by a mathematical text. The text is dated to year 38 of an otherwise unnamed king's reign. The general dating to the early Egyptian Middle Kingdom combined with the high regnal year suggests that the tablets may date to the reign of the 12th Dynasty pharaoh Senusret I,c. 1950 BC. The second tablet also lists several servants and contains further mathematical texts.
Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in the Song dynasty and Yuan dynasty,culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.
In elementary algebra,the unitary method is a problem-solving technique taught to students as a method for solving word problems involving proportionality and units of measurement. It consists of first finding the value or proportional amount of a single unit,from the information given in the problem,and then multiplying the result by the number of units of the same kind,given in the problem,to obtain the result.
The digits of some specific integers permute or shift cyclically when they are multiplied by a number n. Examples are:
Jigu suanjing was the work of early Tang dynasty calendarist and mathematician Wang Xiaotong,written some time before the year 626,when he presented his work to the Emperor. Jigu Suanjing was included as one of the requisite texts for Imperial examination;the amount of time required for the study of Jigu Suanjing was three years,the same as for The Nine Chapters on the Mathematical Art and Haidao Suanjing.
Kriyakramakari (Kriyā-kramakarī) is an elaborate commentary in Sanskrit written by Sankara Variar and Narayana,two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics,on Bhaskara II's well-known textbook on mathematics Lilavati. Kriyakramakari,along with Yuktibhasa of Jyeshthadeva,is one of the main sources of information about the work and contributions of Sangamagrama Madhava,the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a 9th-century astronomer from Kerala.
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by = c where x and y are unknown quantities and a,b,and c are known quantities with integer values. The algorithm was originally invented by the Indian astronomer-mathematician Āryabhaṭa and is described very briefly in his Āryabhaṭīya. Āryabhaṭa did not give the algorithm the name Kuṭṭaka,and his description of the method was mostly obscure and incomprehensible. It was Bhāskara I who gave a detailed description of the algorithm with several examples from astronomy in his Āryabhatiyabhāṣya,who gave the algorithm the name Kuṭṭaka. In Sanskrit,the word Kuṭṭaka means pulverization,and it indicates the nature of the algorithm. The algorithm in essence is a process where the coefficients in a given linear Diophantine equation are broken up into smaller numbers to get a linear Diophantine equation with smaller coefficients. In general,it is easy to find integer solutions of linear Diophantine equations with small coefficients. From a solution to the reduced equation,a solution to the original equation can be determined. Many Indian mathematicians after Aryabhaṭa have discussed the Kuṭṭaka method with variations and refinements. The Kuṭṭaka method was considered to be so important that the entire subject of algebra used to be called Kuṭṭaka-ganita or simply Kuṭṭaka. Sometimes the subject of solving linear Diophantine equations is also called Kuṭṭaka.
Zhang Qiujian Suanjing is the only known work of the fifth century Chinese mathematician,Zhang Qiujian. It is one of ten mathematical books known collectively as Suanjing shishu. In 656 CE,when mathematics was included in the imperial examinations,these ten outstanding works were selected as textbooks. Jiuzhang suanshu and Sunzi Suanjing are two of these texts that precede Zhang Qiujian suanjing. All three works share a large number of common topics. In Zhang Qiujian suanjing one can find the continuation of the development of mathematics from the earlier two classics. Internal evidences suggest that book was compiled sometime between 466 and 485 CE.
