A History of Mathematics. Arabic numerals Unfortunately, our editorial approach may not be able to accommodate all contributions. If you prefer to suggest your own revision of the article, you can go to edit mode requires login. It is nkmero to note that in Arabic, jabr means the setting of a bone and is related to the word reduction. Studies in Honor of Constantine K.
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Algorithm example[ edit ] One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as: High-level description: If there are no numbers in the set then there is no highest number. Assume the first number in the set is the largest number in the set.
For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set. When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set. Quasi- formal description: Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code : Algorithm LargestNumber Input: A list of numbers L.
Output: The largest number in the list L. Heath , with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 for this is possible ; 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 for this is possible ; 7 is left, but 7 cannot be subtracted from 7.
He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length s successively q times along longer length l until the remaining portion r is less than the shorter length s. A location is symbolized by upper case letter s , e. S, A, etc. Derived from Knuth —4. Depending on the two numbers "Inelegant" may compute the g.
E2: [Is the remainder zero? Use remainder r to measure what was previously smaller number s; L serves as a temporary location. A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source  uses and Knuth suggested , Another interesting case is the two relatively prime numbers and But "exceptional cases"  must be identified and tested. Yes to all. What happens when one number is zero, both numbers are zero?
What happens if negative numbers are entered? Fractional numbers? If the input numbers, i. A notable failure due to exceptions is the Ariane 5 Flight rocket failure June 4, Algorithm analysis  indicates why this is the case: "Elegant" does two conditional tests in every subtraction loop, whereas "Inelegant" only does one.
Can the algorithms be improved? The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm;  rather, it can only be done heuristically ; i. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis. Main article: Analysis of algorithms It is frequently important to know how much of a particular resource such as time or storage is theoretically required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers estimates ; for example, the sorting algorithm above has a time requirement of O n , using the big O notation with n as the length of the list.
At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore, it is said to have a space requirement of O 1 , if the space required to store the input numbers is not counted, or O n if it is counted.
For example, a binary search algorithm with cost O log n outperforms a sequential search cost O n when used for table lookups on sorted lists or arrays. Formal versus empirical[ edit ] Main articles: Empirical algorithmics , Profiling computer programming , and Program optimization The analysis, and study of algorithms is a discipline of computer science , and is often practiced abstractly without the use of a specific programming language or implementation.
In this sense, algorithm analysis resembles other mathematical disciplines in that it focuses on the underlying properties of the algorithm and not on the specifics of any particular implementation.
Usually pseudocode is used for analysis as it is the simplest and most general representation. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences unless n is extremely large but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical.
Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign. Empirical testing is useful because it may uncover unexpected interactions that affect performance. Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
Muhammad ibn Musa al-Khwarizmi
Robertson wrote in the MacTutor History of Mathematics archive : Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry. Algebra was a unifying theory which allowed rational numbers , irrational numbers , geometrical magnitudes, etc. It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject.
It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Brill, Leiden, the Netherlands. Muhammad ibn Musa al-Khwarizmi — Wikipedia The Arabs in general loved a good clear argument from premise to conclusion, as well as systematic organization — respects in which neither Diophantus nor the Hindus excelled. Chronographie de Mar Elie bar Sinaya. Perhaps one of the most significant advances made alvoritmi Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra.
ALGORITMI DE NUMERO INDORUM PDF
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