The Father of Algebra
By Ibrahim B. Syed,
Ph.D., D.Sc., F.A.C.R.
• Clinical Professor of Medicine
• University of Louisville School of
• And
• President
• Louisville, KY 40242
•A Portrait
Portrait of Al-Khwarizmi
• This is
taken from
a stamp
from the
(770 - 840 C.E.)
• Abu Abdullah
Muhammad Ibn
Musa al-Khwarizmi
was born at
Khwarizm (Kheva),
a town south of
the river Oxus in
present day
• RESEARCH ON BRAIN shows Algebra
concepts can be taught -as early as
in Kindergarten
• Second graders at The School at
Columbia Univ. are learning algebra
• Other countries: To 6th or 7th graders
• US: 25% of middle graders get
• In California, Algebra is taughtbeginning of the 8th grade
• Algebra is a branch of Maths.
• Describes relationships
between things that vary over
• Variables denoted by letters
and symbols.
• Algebra is a study in logic.
• Careers today demand skills like
problem solving, reasoning, decisionmaking, and applying solid strategies
• Algebra provides one with a wonderful
grounding in those skills.
• Colleges require it and some
employers demand it.
• Algebra is a very unique
discipline. It is very abstract.
• The abstract-ness of algebra
causes the brain to think in
totally new patterns.
• Algebra builds a better brain.
• When the brain is
stimulated to think, the
hair-like dendrites of the
brain grow more extensive
and more complex
enabling more
connections with other
brain cells. We often hear
that we use only a small
percentage of our brain's
capacity. The study of
algebra is a way to
increase our use of this
marvelous muscle. By
studying algebra, more
"highways" are "built"
upon which future "cargo"
is transported -- cargo
other than algebra.
Abu Jafar Muhammad Ibn
Musa al-Khwarizmi
• 21st Century is the age of
Information Technology (IT)
• Modern Computers are
indispensable in everyday life.
• Al-Khwarizmi is the grandfather of
Computer Science.
• He is the Father of Algebra.
• A Page from alKhwarizmi's Kitab
al-Jabr walMuqabala, the
oldest Arabic work
on algebra
produced in the
9th century
BAYT AL-HIKMA-Center for
Jafar Muhammad
Study and Research Abu
ibn Musa al-Khwarizmi
lived in Baghdad(Gift of
God) in the early ninth
century. Baghdad at that
time was at cultural
crossroads, and, under
the patronage of the
Abbasid caliphs, the socalled House of Wisdom
at Baghdad, produced a
Golden Age of Arabic
science and
mathematics. In
Baghdad, scholars
encountered and built
upon the ideas of ancient
Greek and Indian
• There, al-Khwarizmi encountered the
Indian numeral system (0, 1, 2, 3, 4, 5, 6, 7,
8, 9), and he wrote a treatise on what we
call Arabic numerals. It was translated
into Latin in the twelfth century as
Algoritmi de numero Indorum (that is, AlKhwarizmi on the Hindu Art of Reckoning)
and was crucial in the introduction of
Arabic numerals to medieval Europe. It
may well represent the first use of zero as
a positional place holder. From that title,
we have the word "algorithm."
ALGEBRA- A Practical
System for solving
problems in
Cases of Inheritance
Tax collection
Lawsuits and
Algebra-A Practical System for
solving real life Problems in the
Islamic empire at that time
• In all their dealings with one
• Where the measuring of lands.
• The digging of canals.
• Geometrical computations.
• Other objects of various sorts
and kinds are concerned.
• Distributive: x(y+z) = xy+xz, (x+y)z
= xz+yz
• Associative: (x+y)+z = x+(y+z)
(xy)z = x(yz)
• Commutative: x+y = y+x
xy = yx
• Identity: x+0 = 0+x = x x 1 = 1x = x
• Inverse: x+(-x) = (-x)+x = 0
• x(x-1) = (x-1)x = 1
Al-Khwarizmi's most important work:
al-Kitab al-mukhtasar fi hisab
al-jabr w'al-muqabala
or The Compendious Book on
Calculation by Completion
[or Restoring] and Balancing.
This book is an explanation of the
solution to quadratic and linear equations
of six varieties. Al-jabr refers to the
process of moving a subtracted quantity
to the other side of an equation;
al-muqabala involves subtracting
equal quantities from both sides of
an equation.
Textbook of Algebra
• Hisab al-jabr w'al-muqabala
was translated into Latin(Robert Chestetr) in 1145
as Liber algebrae et almucabala,
from which we have the word "algebra"
for the whole process.
• But don't expect al-Khwarizmi's al-jabr
to look anything like our algebra.
• Al-Kwharizmi's book is written
entirely in prose, with none of the
symbols we use today.
Some Problems-Formal
• "If from a square, I subtract four of its
roots and then take one-third of the
remainder, finding this equal to four of the
roots, the square will be 256. "
• He explained it in the following manner:
• "Since one-third of the remainder is equal
to four roots, one knows that the
remainder itself will equal 12 roots.
Therefore, add this to the four, giving 16
roots. This (16) is the root of the square.
The above can also be stated in terms of
modern notation as
• 1/3 (x2 - 4x ) = 4x." Therefore x = 16.
"A man is hired to work in a vineyard 30 days for
10 Dollars. He works six days. How much of the
agreed price should he receive?"
• It is evident that since days are one-fifth of the
whole time; and it is also evident that the man
should receive pay having the same relation to the
agreed price that the time he works bears to the
whole time, 30 days. The month, i.e., 30 days,
represents the measure, and ten represents the price.
Six days represents the quantity, and in asking what
part of the agreed price is due to the worker you ask
the cost. Therefore multiply the price 10 by the
quantity 6, which is inversely proportional to it.
Divide the product 60 by the measure 30, giving 2
He first reduces an equation (linear or quadratic)
to one of six standard forms:
1. Squares equal to roots. Example: ax2 = bx
2. Squares equal to numbers. Example: ax2 = b
3. Roots equal to numbers. Example: ax = b
4. Squares and roots equal to numbers.
Example: ax2 + bx = c e.g. x2 + 10 x = 39.
5. Squares and numbers equal to roots.
Example: ax2 + c = bx e.g. x2 + 21 = 10 x.
6. Roots and numbers equal to squares.
Example: ax2 = bx + c, e.g. 3x + 4 = x2.
Solve the equation: x2 + 10 x = 39
He (Al-Khwarizmi) writes :... a square and 10 roots
are equal to 39 units.
what is the square which combined with ten of its roots
will give a sum total of 39?
The manner of solving this type of equation is to take
one-half of the roots just mentioned. Now the roots in the
problem before us are 10.
Therefore take 5, which multiplied by itself gives 25, an
amount which you add to 39 giving 64.
Having taken then the square root of this which is 8,
subtract from it half the roots, 5 leaving 3. The number
three therefore represents one root of this square, which
itself, of course is 9. Nine therefore gives the square.
The Geometric Proof
• Al-Khwarizmi starts
with a square of side x,
which therefore represents
x2 (Figure 1).
• To the square we must
add 10x
and this is done by adding
four rectangles
each of breadth 10/4 and
length x to the square
(Figure 2).
• Figure 2
has area x2 + 10 x
which is equal to 39.
Geometric Proof
• We now complete
• the square by adding the four
little squares
• each of area. 5/2x5/2 = 25/4.
• Hence the outside square in
Fig 3 has area 4 x 25/4 + 39 =
25 + 39 = 64.
• The side of the square is
therefore 8.
• But the side is of length 5/2 +
x + 5/2
• so x + 5 = 8, giving x = 3.
Al-Khwarizmi's concept of algebra can
now be grasped with greater precision:
it concerns the theory of linear and
quadratic equations with a single
unknown and the elementary arithmetic
of relative binomials and trinomials. ...
The solution had to be general and
calculable at the same time and in a mathematical
fashion, that is, geometrically founded. ...
The restriction of degree, as well as that of the number of
unsophisticated terms, is instantly explained.
From its true emergence, algebra can be seen as a theory
of equations solved by means of radicals, and of
algebraic calculations on related expressions...
GEORGE SARTON(18841956) Author of Introduction
to History of Science
(3 Volumes)
Former Prof. At Harvard Univ.
Wrote on Al-Khwarizmi as
• ... the greatest mathematician
of the time, and if one takes
all the circumstances into
• one of the greatest of all
Al-Khwarizmi wrote on
• Algoritmi de numero Indorum (Al-Khwarizmi on
the Hindu Art of Reckoning) gave ALGORITHM
deriving from his name in the title of the book.
• He explained the use of ZERO
• He developed the decimal system
• Developed several arithmetical procedures
including operations on fractions.
• He developed in detail Trigonometric tables
containing Sine functions and tangent functions
• Developed calculus of two errors, which led him
to the concept of differentiation
Al-Kwarizmi's Books
• Kitab al-Jama wal-Tafreeq bil
Hisab al-Hindi and
• Kitab al-Jabr wa al-Muqabala
Were translated into Latin and
were used for several hundred
years in Europe
Al-Khwarizmi's works
influenced Leonardo of
• Fibonacci was the
European Mathematician
of the middle ages",
his full name was Leonardo
of Pisa.
• Discovered the enormous
practical advantages of
Zero and the Decimal
System compared to the
Roman numerals, which
were still current in
Western Europe.
• Al-Khwarizmi wrote on
• Calendars
• True positions of the sun, moon, and
• Spherical astronomy
• Parallax and eclipse calculations
• Visibility of the moon (21ST CENTURY
Muslims are confused on the sighting of the
• Wrote a book on Astronomical Tables-A
• Kitab surat al-ard (book of the
form of the earth)
• Gave latitudes and longitudes
for 2,402 cities and landmarks,
forming the basis for a world
• Corrected in detail Ptolemy's
views on Geography
• Supervised 70 geographers to
create a map of the then known
world which shows the pacific
coast of South America –about
700 years before Columbus
discovered America.
• Measured volume and
circumference of the earth.
• Wrote Kitab al-Tarikh and Kitab
al-Rukhmat (on sundials)
Al-Khwarizmi's World
• Al-Khwarizmi's Map
actually includes the
whole coast of Peru
and part of the coast of
Chile. We find rivers
and capes, in particular
two especially
characteristic capes
lying on the PeruEcuador coast; the
Promontorium or the
Cape of Satyrs, which
is Punta Aguja, and the
Notium Promontorium
or Southern Cape
which is Punta Pariña,
Turkish Admiral Piri Reis's World Map-1513 CE
is on the left side(Antarctica discovered in 1820).
On the Right Side is Muhammad Al-Idris's map
of 1154 CE
Satellite Photo Vs Piri Map
• Adelard of Bath (England)
was born in 1075. He studied
and taught in France and
visited Syria, Sicily and Spain
He died in 1160. He translated
several works on Mathematics
and Astronomy. Among the
most important works he
translated was the
Astronomical tables AlMajriti (1126). He translated
Al-Khwarizmi's tables and
other works on the abacus
and astrolabe. His
'Quaestiones naturales'
consists of 76 scientific
discussions derived from
Muslim sciences.
• Gerard born in 1114 in Cremona
(Italy). In Toledo, Spain he learnt
Arabic so he could translate
available Arabic works into Latin.
He died in 1187 in Toledo, Spain
(Andalusia). Among his
translations were the surgical part
of Al-Tasrif of Al-Zahravi
(Albucasis), the Kitab al-Mansuri
of AL-Razi (Rhazes) and the
Qanun of Ibn Sina (Avicenna),
Banu Musa's works, Al-Biruni's
commentry on Al-Khawarizmi
(after whom concept "Algorithm"
is named), the tables of Jabir b.
Aflah and Zarqali.
Al-Khwarizmi's books
translated into Latin
• Kitab al-Jam'a wal-Tafreeq bil Hisab
al-Hindi (on Arithmetic)
• Al-Maqala fi Hisab al-Jabr wa-alMuqabilah ( on Algebra) by
Englishman, Robert of Chester (1145
• Arabic numerals and number system
assisted progress in science,
accounting and bookkeeping.
• Baghdad, Damascus, Cairo
and Cordoba were the
centers of civilization. Here
Muslim scientists made
tremendous progress in
applied & theoretical science
and technology.
• While Europe festered in the
Dark Ages.
Muslim Impact on Europe
• Scholars and students from
various parts of the world and
Europe came to Cordoba to
• In the 9th century the library of
the monastery of St. Gall was
the largest in Europe with 36
volumes. At that time, that of
Cordoba contained over 500,000
Arabic Maths Worldwide
• Muslim mathematicians
invented geometrical algebra,
solved third and fourth degree
• The world witnessed a new
stage in the development of
mathematical science, driven
by the numerous translated
works from Arabic into
European languages.
Advancement of
Sciences in Europe
• The sciences, with Arab
mathematics as their essence,
flourished and developed in the
disciplines we know today.
• Without the number zero and
Arabic numerals in Europe, the
world as we know today would
have been different.
• Muslim mathematical study
concentrated in three areas: ongoing
progress in algebra, the development of
arithmetic algorithms, and the
increasing complexity in geometry.
• The number zero and decimal system in
Europe was the basis for the Scientific
• Problems that took days (using Roman
Numerals) to solve could now be solved
in minutes (using Arabic numerals).
• Al-Jabr wa-al-Muqabilah from whose title came
the name "Algebra"
• Kitab al-Jam'a wal-Tafreeq bil Hisab al-Hindi
(on Arithmetic, which survived in a Latin
translation but was lost in the original Arabic)
• Kitab Surat-al-Ard (on geography)
• Istikhraj Tarikh al-Yahud (about the Jewish
• Kitab al-Tarikh
• Kitab al-Rukhmat (about sun-dials)
• Europe would have been a lot
culturally and scientificallyhad it resisted the
globalization of mathematics,
science and technology at
that time.
• Today Western science and
technology are flowing to
many parts of the world. (Our
Ulama are resisting them)
Breaking the Boundaries
• The Renaissance, the
Enlightenment and the Industrial
Revolution were great
achievements. These
developments drew on the
experience of the Muslim world,
India and China.
• Today a mathematician in Boston
invokes algorithm to solve a
difficult computational problem,
then he/she is commemorating
The square root of math
• Al-Khwarizmi is one of many
whose works influenced the
European Renaissance, the
Enlightenment and the
Industrial Revolution.
• Modern prosperity is due to
science and technology,
which have delivered better
lives for people, longer lives,
and for larger populations.
13th Century Muslim World
• Led the whole world with its
development of the culture of
philosophy, science, mathematics,
astronomy, physics, chemistry and
• If the Muslim world had been able
to continue on the Qur'anic
commands on scientific research,
the cause of human progress would
have advanced by about 500 years.
• Algebra and algorithms are
enabling the building of
computers, and the creation
of encryption.
• The modern technology
industry would not exist
without the contributions of
Muslim mathematicians like
Ms. Carly Fiorina, Hewlett-Packard's
former Chairman and CEO
• In Minneapolis, MN on Sept. 26, 2001
said, "There was once a civilization that
was the greatest in the world.
• This civilization was driven by invention.
• Its architects designed buildings that
defied gravity.
• Its doctors examined the human body,
and found new cures for disease.
• Its astronomers looked into the heavens
named the stars, and paved the way for
space travel and exploration.
Ms. Carly Fiorina
• When other nations were afraid of
ideas, this civilization thrived on them,
and kept them alive.
• When censors threatened to wipe out
knowledge from past civilizations, this
civilization kept the knowledge alive,
and passed it on to others.
• While modern Western civilization
shares many of these traits, the
civilization I'am talking about was the
Islamic World, 800-1600 CE, which
included the Ottoman Empire, the
Courts of Baghdad, Damascus and
Cairo. (She forgot Cordoba)
Ms. Carly Fiorina
• We are unaware of our indebtedness to this other
(Islamic) Civilization, its gifts are very much part of
our heritage.
. Sufi poet-philosophers like Rumi challenged our
notions of self and truth.
• Leaders like Suleiman the Magnificent contributed to
our notions of tolerance and civic leadership based on
• It was leadership that harnessed the full
capabilities of a very diverse populationthat included Christian, Islamic and
Jewish traditions."

Al - Khwarizmi