Born : 953, Karaj, Persia
Died : 1029 (aged 75–76)
Main
interests : Mathematics, Engineering.
Abū Bakr Muḥammad ibn al Ḥasan al-Karajī (Persian: ابو بکر محمد بن الحسن الکرجی; c. 953 – c. 1029) was a
10th-century Persian mathematician and engineer who
flourished at Baghdad. He was born in Karaj, a city near Tehran.
His three principal surviving works are mathematical: Al-Badi'
fi'l-hisab (Wonderful on calculation), Al-Fakhri
fi'l-jabr wa'l-muqabala (Glorious on algebra), and Al-Kafi
fi'l-hisab (Sufficient on calculation).
Work
Al-Karaji wrote on mathematics and engineering. Some
consider him to be merely reworking the ideas of others (he was influenced
by Diophantus) but most regard him as more original, in particular
for the beginnings of freeing algebra from geometry. Among historians, his most
widely studied work is his algebra book al-fakhri fi al-jabr wa
al-muqabala, which survives from the medieval era in at least four copies.
In his book "Extraction of hidden waters" he
has mentioned that earth is spherical in shape but considers it the centre of
the universe long before Galileo Galilei, Johannes
Kepler or Isaac Newton, but long after Aristotle and Ptolemy.
He expounded the basic principles of hydrology and this book reveals a
profound knowledge of this science and has been described as the oldest extant
text in this field.
He systematically studied the algebra of exponents,
and was the first to realise that the sequence x, x^2, x^3,... could be
extended indefinitely; and the reciprocals 1/x, 1/(x^2), 1/(x^3),... . However,
since for example the product of a square and a cube would be expressed, in
words rather than in numbers, as a square-cube, the numerical property of
adding exponents was not clear.
His work
on algebra and polynomials gave the rules for arithmetic
operations for adding, subtracting and multiplying polynomials; though he was
restricted to dividing polynomials by monomials.
F. Woepcke was the first historian to realise the
importance of al-Karaji's work and later historians mostly agree with his
interpretation. He praised Al-Karaji for being the first who introduced the
theory of algebraic calculus.
Al-Karaji gave the first formulation of
the binomial coefficients and the first description of Pascal's
triangle. He is also credited with the discovery of the binomial theorem.
In a now lost work known only from subsequent
quotation by al-Samaw'al Al-Karaji introduced the idea of argument
by mathematical induction. As Katz says:
Another important idea introduced by al-Karaji and
continued by al-Samaw'al and others was that of an inductive argument for
dealing with certain arithmetic sequences. Thus al-Karaji used such an argument
to prove the result on the sums of integral cubes already known
to Aryabhata [...] Al-Karaji did not, however, state a general result
for arbitrary n. He stated his theorem for the particular integer 10 [...]
His proof, nevertheless, was clearly designed to be extendable to any other integer.
[...] Al-Karaji's argument includes in essence the two basic components of a
modern argument by induction, namely the truth of the statement
for n = 1 (1 = 13) and the deriving of the truth for n = k from that
of n = k - 1. Of course, this second component is not explicit
since, in some sense, al-Karaji's argument is in reverse; this is, he starts
from n = 10 and goes down to 1 rather than proceeding upward.
Nevertheless, his argument in al-Fakhri is the earliest extant proof
of the sum formula for integral cubes.
Sumber
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