What is the arithmetic mean of the three digit numbers formed by using 1 5 and 9

Hint: As we have to form 3 – digit numbers, the first digit cannot be 0. So, the first digit can be arranged in 4 ways. Now, as repetition is allowed, the second and the third digit can be arranged in 5 ways. The total numbers formed will be equal to the product of these values.

Complete step-by-step solution:
In this question, we are asked how many 3 digit numbers can be formed using the digits 0, 1, 3, 5, 7 where repetition is allowed.
Given digits: 0, 1, 3, 5, 7
Therefore, there are a total of 5 digits.
Now, as we have to form 3 digit numbers, let us draw three boxes.

First box will be hundreds, the second box will be tens and the third box will be ones.
Now, as the number is 3 – digit, the first box cannot use digit 0, because if we use 0 as first digit, the number will not be a 3 digit number. So, only 4 digits that are 1, 3, 5 and 7 can be used in the first box.

100+x2−20x=81x{\displaystyle 100+x^{2}-20x=81x}x2+100=101x{\displaystyle x^{2}+100=101x}

Let the roots of the equation be x = p and x = q. Then p+q2=5012{\displaystyle {\tfrac {p+q}{2}}=50{\tfrac {1}{2}}}

, pq=100{\displaystyle pq=100} and

p−q2=(p+q2)2−pq=255014−100=4912{\displaystyle {\frac {p-q}{2}}={\sqrt {\left({\frac {p+q}{2}}\right)^{2}-pq}}={\sqrt {2550{\tfrac {1}{4}}-100}}=49{\tfrac {1}{2}}}

So a root is given by

x=5012−4912=1{\displaystyle x=50{\tfrac {1}{2}}-49{\tfrac {1}{2}}=1}

Several authors have also published texts under the name of Kitāb al-jabr wal-muqābala, including Abū Ḥanīfa Dīnawarī, Abū Kāmil Shujāʿ ibn Aslam, Abū Muḥammad al-'Adlī, Abū Yūsuf al-Miṣṣīṣī, 'Abd al-Hamīd ibn Turk, Sind ibn 'Alī, Sahl ibn Bišr, and Sharaf al-Dīn al-Ṭūsī.

S. Gandz has described Al-Khwarizmi as the father of Algebra :

Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers.[47]

Victor J. Katz adds :

The first true algebra text which is still extant is the work on al-jabr and al-muqabala by Mohammad ibn Musa al-Khwarizmi, written in Baghdad around 825.[48]

J.J. O'Conner and E.F. 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., to all be treated as "algebraic objects". 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. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before.[49]

R. Rashed and Angela Armstrong write:

Al-Khwarizmi's text can be seen to be distinct not only from the Babylonian tablets, but also from Diophantus' Arithmetica. It no longer concerns a series of problems to be solved, 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. On the other hand, the idea of an equation for its own sake appears from the beginning and, one could say, 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.[50]

According to Swiss-American historian of mathematics, Florian Cajori, Al-Khwarizmi's algebra was different from the work of Indian mathematicians, for Indians had no rules like the ''restoration'' and ''reduction''.[51] Regarding the dissimilarity and significance of Al-Khwarizmi's algebraic work from that of Indian Mathematician Brahmagupta, Carl Benjamin Boyer wrote:

It is true that in two respects the work of al-Khowarizmi represented a retrogression from that of Diophantus. First, it is on a far more elementary level than that found in the Diophantine problems and, second, the algebra of al-Khowarizmi is thoroughly rhetorical, with none of the syncopation found in the Greek Arithmetica or in Brahmagupta's work. Even numbers were written out in words rather than symbols! It is quite unlikely that al-Khwarizmi knew of the work of Diophantus, but he must have been familiar with at least the astronomical and computational portions of Brahmagupta; yet neither al-Khwarizmi nor other Arabic scholars made use of syncopation or of negative numbers. Nevertheless, the Al-jabr comes closer to the elementary algebra of today than the works of either Diophantus or Brahmagupta, because the book is not concerned with difficult problems in indeterminant analysis but with a straight forward and elementary exposition of the solution of equations, especially that of second degree. 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.[52]

Page from a Latin translation, beginning with "Dixit algorizmi"

Arithmetic[edit]

Algorists vs. abacists, depicted in a sketch from 1508 CE

Al-Khwārizmī's second most influential work was on the subject of arithmetic, which survived in Latin translations but is lost in the original Arabic. His writings include the text kitāb al-ḥisāb al-hindī ('Book of Indian computation'[note 2]), and perhaps a more elementary text, kitab al-jam' wa'l-tafriq al-ḥisāb al-hindī ('Addition and subtraction in Indian arithmetic').[55] These texts described algorithms on decimal numbers (Hindu–Arabic numerals) that could be carried out on a dust board. Called takht in Arabic (Latin: tabula), a board covered with a thin layer of dust or sand was employed for calculations, on which figures could be written with a stylus and easily erased and replaced when necessary. Al-Khwarizmi's algorithms were used for almost three centuries, until replaced by Al-Uqlidisi's algorithms that could be carried out with pen and paper.[56]

As part of 12th century wave of Arabic science flowing into Europe via translations, these texts proved to be revolutionary in Europe.[57] Al-Khwarizmi's Latinized name, Algorismus, turned into the name of method used for computations, and survives in the modern term "algorithm". It gradually replaced the previous abacus-based methods used in Europe.[58]

Four Latin texts providing adaptions of Al-Khwarizmi's methods have survived, even though none of them is believed to be a literal translation:

• Dixit Algorizmi (published in 1857 under the title Algoritmi de Numero Indorum[59])[60]
• Liber Alchoarismi de Practica Arismetice
• Liber Ysagogarum Alchorismi
• Liber Pulveris

Dixit Algorizmi ('Thus spake Al-Khwarizmi') is the starting phrase of a manuscript in the University of Cambridge library, which is generally referred to by its 1857 title Algoritmi de Numero Indorum. It is attributed to the Adelard of Bath, who had also translated the astronomical tables in 1126. It is perhaps the closest to Al-Khwarizmi's own writings.[60]

Al-Khwarizmi's work on arithmetic was responsible for introducing the Arabic numerals, based on the Hindu–Arabic numeral system developed in Indian mathematics, to the Western world. The term "algorithm" is derived from the algorism, the technique of performing arithmetic with Hindu-Arabic numerals developed by al-Khwārizmī. Both "algorithm" and "algorism" are derived from the Latinized forms of al-Khwārizmī's name, Algoritmi and Algorismi, respectively.

Astronomy[edit]

Page from Corpus Christi College MS 283. A Latin translation of al-Khwārizmī's Zīj.

Al-Khwārizmī's Zīj al-Sindhind[37] (Arabic: زيج السند هند, "astronomical tables of Siddhanta"[61]) is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind.[62] The word Sindhind is a corruption of the Sanskrit Siddhānta, which is the usual designation of an astronomical textbook. In fact, the mean motions in the tables of al-Khwarizmi are derived from those in the "corrected Brahmasiddhanta" (Brahmasphutasiddhanta) of Brahmagupta.[63]

The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.

The original Arabic version (written c. 820) is lost, but a version by the Spanish astronomer Maslamah Ibn Ahmad al-Majriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).[64] The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).

Trigonometry[edit]

Al-Khwārizmī's Zīj al-Sindhind also contained tables for the trigonometric functions of sines and cosine.[62] A related treatise on spherical trigonometry is also attributed to him.[49]

Al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents.[65][66]

Geography[edit]

Earliest extant map of the Nile, in al-Khwārazmī's Kitāb ṣūrat al- arḍ

A stamp issued 6 September 1983 in the Soviet Union, commemorating al-Khwārizmī's (approximate) 1200th birthday.

Al-Khwārizmī's third major work is his Kitāb Ṣūrat al-Arḍ (Arabic: كتاب صورة الأرض, "Book of the Description of the Earth"),[67] also known as his Geography, which was finished in 833. It is a major reworking of Ptolemy's second-century Geography, consisting of a list of 2402 coordinates of cities and other geographical features following a general introduction.[68]

There is only one surviving copy of Kitāb Ṣūrat al-Arḍ, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.[citation needed] The book opens with the list of latitudes and longitudes, in order of "weather zones", that is to say in blocks of latitudes and, in each weather zone, by order of longitude. As Paul Gallez[dubious – discuss] points out, this excellent system allows the deduction of many latitudes and longitudes where the only extant document is in such a bad condition as to make it practically illegible. Neither the Arabic copy nor the Latin translation include the map of the world itself; however, Hubert Daunicht was able to reconstruct the missing map from the list of coordinates. Daunicht read the latitudes and longitudes of the coastal points in the manuscript, or deduces them from the context where they were not legible. He transferred the points onto graph paper and connected them with straight lines, obtaining an approximation of the coastline as it was on the original map. He then does the same for the rivers and towns.[69]

Al-Khwārizmī corrected Ptolemy's gross overestimate for the length of the Mediterranean Sea[70] from the Canary Islands to the eastern shores of the Mediterranean; Ptolemy overestimated it at 63 degrees of longitude, while al-Khwārizmī almost correctly estimated it at nearly 50 degrees of longitude. He "also depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done."[71] Al-Khwārizmī's Prime Meridian at the Fortunate Isles was thus around 10° east of the line used by Marinus and Ptolemy. Most medieval Muslim gazetteers continued to use al-Khwārizmī's prime meridian.[70]

Jewish calendar[edit]

Al-Khwārizmī wrote several other works including a treatise on the Hebrew calendar, titled Risāla fi istikhrāj ta'rīkh al-yahūd (Arabic: رسالة في إستخراج تأريخ اليهود, "Extraction of the Jewish Era"). It describes the Metonic cycle, a 19-year intercalation cycle; the rules for determining on what day of the week the first day of the month Tishrei shall fall; calculates the interval between the Anno Mundi or Jewish year and the Seleucid era; and gives rules for determining the mean longitude of the sun and the moon using the Hebrew calendar. Similar material is found in the works of Abū Rayḥān al-Bīrūnī and Maimonides.[37]

Other works[edit]

Ibn al-Nadim's Kitāb al-Fihrist, an index of Arabic books, mentions al-Khwārizmī's Kitāb al-Taʾrīkh (Arabic: كتاب التأريخ), a book of annals. No direct manuscript survives; however, a copy had reached Nusaybin by the 11th century, where its metropolitan bishop, Mar Elias bar Shinaya, found it. Elias's chronicle quotes it from "the death of the Prophet" through to 169 AH, at which point Elias's text itself hits a lacuna.[72]

Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials; the Fihrist credits al-Khwārizmī with Kitāb ar-Rukhāma(t) (Arabic: كتاب الرخامة). Other papers, such as one on the determination of the direction of Mecca, are on the spherical astronomy.

Two texts deserve special interest on the morning width (Ma'rifat sa'at al-mashriq fī kull balad) and the determination of the azimuth from a height (Ma'rifat al-samt min qibal al-irtifā').

What is the mean of the numbers from 1 to 9?

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How many three

What is the arithmetic mean of the three