Part III. Mathematics Advancements by Muslims:
Introduction: Just as with science,
the Muslims learned from the Greeks, Egyptians, Indians, and Babylonians. Many
translations took place in the House of Wisdom in Baghdad, the capital of the
Abbasid Empire. The Muslim scholars there translated the works of the Greeks who
loved mathematics and geometry, including Euclid's work on geometry. They
borrowed from India a number system that had a zero and rewrote it as their own.
They borrowed from the Babylonians whose number system was based on 60 (just
like the minutes in an hour), and from the ancient Egyptians who had the math
and geometry skills to build incredible pyramids. So from the beginning, "Arabic
math" was a mixing of international knowledge. But the Muslims made additional
contributions of their own, and through their study and written work, they
preserved the knowledge of mathematics that otherwise might have been lost to
the world.
Arithmetic:
12 + 10 =
14  8 =
5 X 4 =
6 ÷ 2 =

Algebra:

2x = 14

x = ?


3x + 6 = 18

x = ?


x^{2} + xy = 10

x = 5

y = ?

Geometry:
A = _{}r^{2 }
^{}
a^{2} + b^{2} = c^{2}

Trigonometry:
Top image from Hyperion Cultural
Academy. 
Arab contributions:
 the numbers we use are called Arabic numbers (numerals) which is a
system of tens, with place values, and a zero to show an empty place:
1,302,005
 fractions: ^{1}/_{2}
 decimal fractions: 1.5

Arab contributions:
Algebra was first fully developed by Al Khwarism, the "father of
algebra". 
Arab contributions:
The Arabs translated and improved upon the Egyptian, Hebrew, and Greek
geometry. 
Arab contributions:
AlTusi, a Muslim, is the "father of trigonometry". 
The decimal (tens place) system first came from India.
Al Khwarismi reworked these numbers and gave us Arabic numerals. Much
later Europeans changed the Arabic numerals into the numerals we use
today.
AlKhwarizmi wrote about squares and square roots, first studied by the
Greeks and Egyptians.
 squares 3^{2} = 9 (3 X 3)
 square roots = 3
AlKhashi (from Persia, 15th century) invented decimal fractions:
5.25 
In Khwarizmi's own words what he wanted to teach:
"...what is easiest and most useful in arithmetic, such as men
constantly require in cases of inheritance, legacies, partition, lawsuits,
and trade, and in all their dealings with one another, or where the
measuring of lands, the digging of canals, geometrical computations, and
other objects of various sorts and kinds are concerned..." 
The Egyptians were very advanced in geometry and could build great
pyramids.
The Greeks loved geometry. The most famous Greek mathematician was
Euclid who wrote about geometry. The Arabs translated and improved upon
his work.
The Hebrews also had made important contributions to mathematics that
were studied by the Arabs. 
The idea of trigonometry was originally from the Greeks, by Hipparchus
in 140 BCE.
The Muslims further developed trigonometry from their work in
astronomy.
Today astronomers use trigonometry for calculating distances to stars,
and for measuring distances and heights of buildings, trees, etc.

A. Arabic Numerals
One of the greatest advances was the introduction of "Arabic"
numerals. The "Arabic" numerals
were influenced by India's mathematics. It is a system based on place values and
a decimal system of tens. This system had a zero to hold a place. These numbers
were much easier to use for calculation than the Roman system which used numbers, like I, V, X, L, C, M, etc.
Addition, subtraction, multiplication and division now became easy.
 Top: Western Arabic or
HinduArabic Numerals
 Below: Modern Arabic numerals
which developed from them
With Arabic numerals, simple fractions and decimal fractions
were also possible. Fractions and decimal fractions were also described by
Muslim mathematicians during the Middle Ages.
B. The Development of Algebra.
Al Khwarizmi wrote the first book on algebra. (The name
"algebra" was first used by him.)
Al Khwarizmi was born about 790 in Baghdad (now in Iraq) and
died about 850.
The word for "Algebra" comes from the Arabic word for "aljabr"
which means "restoration of balance" in both sides of an equation.. Algebra was
based on previous work from Greeks, Alexandrians in Egypt, and Hindus who had
preserved the work from ancient Egyptians and Babylonians.
In the ninth century, alKhwarizmi
wrote one of the first Arabic algebras with both proofs
and examples. Because of his work, he is called "the Father of Algebra."
AlKhwarizmi was a Persian born in the eighth century. He converted (changed)
Babylonian and Hindu numerals into a workable system that almost anyone could
use. He gave the name to his math as "aljabr" which we know as
"algebra".
A Latin translation of alKhwarizmi's book on algebra appeared
in Europe in the 12th century. In the early 13th century the new algebra
appeared in the writings of the famous Italian mathematician, Leonardo
Fibonacci. So, algebra was brought into Europe from ancient Babylon, Egypt and
India by the Arabs and then into Italy.
C. Geometry
The scholars at the House of Wisdom in Baghdad and at
universities in Cairo, Egypt also contributed to
geometry. Geometry was highly developed by the Greeks,
and the Muslims translated such great Greek thinkers as Euclid. Muslims used
their understanding of geometry into designing wheels of all kinds, especially
waterwheels and other systems for drawing up water, in improving farming
equipment, and in designing devices of war such as catapults and crossbows.
Geometry was also put to work in art, with beautiful geometric designs. Muslims
further defined Euclidian geometry, and pointed the way toward the discovery of
independent, nonEuclidean geometry developed in the most recent
centuries.
D. Trigonometry is also mostly a
Muslim creation. It is a branch of mathematics which studies plane and spherical
triangles. It developed from the need of astronomers to map points in the sky on
a heavenly sphere. Trigonometry's functions, involving
ratios such as sine and
cosine, tangent and
cotangent, were greatly developed and refined in the
Islamic lands.
E. Famous Muslim Mathematicians of the Middle
Ages
1. AlKhwarizmi (770  840
C.E.) was one of the greatest mathematicians who ever lived and
is called the "Father of Algebra". He also helped to bring "Arabic numerals"
into use into the Islamic Empire, as well as later into Europe. He also
demonstrated operations with fractions for the first time. Khwarizmi
influenced the growth of science and mathematics. Several of his books were
translated into many other languages, and were used as university textbooks
until the 16th century. His approach was systematic
and logical. He brought together the knowledge of his
time on various branches of science, especially
mathematics, and also added his original
contributions.
2. Omar Khayyam (1044  1123 C.E.): Another great Muslim mathematician was
Omar Khayyam. He is best
known today for his poetry, but his contribution to mathematics was great. He
showed how to express roots of cubic equations by line segments obtained by
intersecting conic sections. Khayyam was an outstanding poet, mathematician,
and astronomer. His work on algebra was known throughout Europe in the Middle
Ages, and he also contributed to a calendar reform. Khayyam refers in his
algebra book to Pascal's triangle. The algebra of Khayyam is geometrical,
solving linear and quadratic equations by methods appearing in Euclid's
Elements. Khayyam also gave important results on ratios giving a new
definition and extending Euclid's work to include the multiplication of
ratios. He poses the question of whether a ratio can be regarded as a number
but leaves the question unanswered.
3. AlKhashi was born in 1390 in
Kashan, Iran and died in 1450 in Samarkand (now Uzbek). He calculated ¼ (pi)
to 16 decimal places which was the best until about 1700. He considered himself the inventor of decimal fractions. He wrote The Reckoners' Key which summarizes arithmetic and contains work on algebra and
geometry.
4. AlBiruni (973  1048 C.E.)
was a philosopher, astronomer, pharmacologist (one who studies drugs and
herbs used for health), botanist
(one who studies plants),
geologist and mathematician. He translated Euclid's work into Sanskrit (an
Indian language), and calculated the earth's
circumference (distance around the earth)
and radius (distance to the center) with an accuracy that is
close to today's measurements.
5. Nasir AlDin AlTusi (1201  1274 C.E.)
pioneered spherical trigonometry which includes six fundamental
formulas for the solution of spherical rightangled triangles. One of his most
important mathematical contributions was the treatment of trigonometry as a
new mathematical discipline. He wrote on binomial coefficients which Pascal
later introduced. (He can be called the "Father of Trigonometry".) He was also
an astronomer philosopher, and medical scholar as well as a
mathematician.
Learn more about Arab mathematicians:
 See Hyperion Culture Academy's math
and science section which includes:
 HinduArabic
Numerals and how they came into Europe as "Arabic
numerals" are explained. See other links at the
bottom of this site. Also see how "Hindu"
numerals came to Persia and Arabia, and then how "Arabic" numbers came
into Europe.
 Read an overview of "Arab
Contributions to Medieval mathematics". It
includes the contributions of AlKhwarizmi, Abu Kamil Shuja (al'Hasib),
Abu'lWafa, alKarkhi, Omar Khayyam, and AlKashi.
 Several biographies of scientists and mathematicians
are found on on the Muslim
Scholars Homepage: AlKhwarizmi, AlKindi, Omar
Khayyam, AlBiruni, Nasir alDin, and others.
 Learn about Mathematicians
born in Iraq, including Al Khwarizmi.
 Biographies of
Mathematicians by Dr. Zahoor are listed.
Choose three or four of the most famous: alKhwarizmi (algebra) [alKhwarizmi is also
here], Omar
Khyyam, alBattani (trigonometry), alHaitham (known as Alhazen in the West, developed analytical geometry by
establishing linkage between algebra and geometry), alTusi (nonEuclidian geometry), and alBiruni (who determined the circumference of the earth).
 See a chart comparing modern
Arabic numerals with the earlier Arabic numerals developed in the Middle Ages (and influenced by the Hindu
numerals with the concept of place value and the "zero").
 AlKhwarizmi (father of algebra); read another biography of
AlKhwarizmi. Read about AlKhwarizmi and see
one of his famous works on completing
the square (shown below).
See AlKhwarizmi
for explanation.

Go to Part
One, Islamic Science and Math
Go to Part
Two: Chemistry, Astronomy, and Geography
Go to Part
Three, Islamic Sciences: Medicine, Botany,
Optics.
Go to Part
Four, Islamic Sciences: Why did the "Golden Age"
end?
You are here at Part Five, Islamic
Mathematics

