The Universal Language
and diverse history
of Al-Gebra
‫اﻟﺟب‬
Vocabulary through Morphology series
amplified with Etymology
Timeline
A knowledge of the history of mathematics can enhance student understanding of the
reasons these developments occurred. - Convergence
Mathematicians have always seen farther by standing on the shoulders of
those who came before them, to paraphrase Sir Isaac Newton, the great
seventeenth- and early eighteenth-century English mathematician and natural
philosopher.
Contents: Math developed over time
“How on earth did we get modern Algebra?”
A. 3 stages of algebraic expression ( ex- “out” + pressare “to press”)
B. Math ‘round the world, ancient 2 modern
C. Conceptual stages
The logical development so prevalent in our textbooks is often sterile because
it explains neither why people were interested in a particular algebraic topic in
the first place nor why our students should be interested in that topic today.
History, on the other hand, often demonstrates the reasons for both.
Rhetorical algebra (1650 BCE through 0 to 200 CE)
In 4,000 year old Babylonian rhetorical algebra,
equations are written in full sentences.
"The thing plus one equals two"
"The thing plus 1 equals 2"
x+1=2
Babylonian math used a sexagesimal system—of base 60
Before the seventeenth century since, mathematics was generally carried out
in words.
Syncopated algebra (200 CE - 1500 CE)
Algebra expressions, Stage 2:
Syncopated algebra uses some symbolism but does not yet contain all of the characteristics of symbolic algebra.
Diophantus' 6 surviving books of the 13 of the Arithmetica 270 CE (1621 Latin translation),
Then in 628 in Brahmagupta's Brahma Sphuta Siddhanta
He was a Hellenized Babylonian mathematician who lived in Alexandria,
Egypt.
He called a problem “absurd” whose only solutions were negative
numbers.
Diophantus was the first to introduce some kind of systematic
symbolism for polynomial equations.
Mathematical historian Kurt Vogel writes: “Diophantus was not, as he has often
been called, the father of algebra. Nevertheless, his remarkable, if
unsystematic, collection of indeterminate problems is a singular achievement
that was not fully appreciated and further developed until much later.”
Epitaph
‘Here lies Diophantus,' the wonder behold.
Through art algebraic, the stone tells how old:
'God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his father's age—chill fate took him.
After consoling his fate by the science of numbers for four more years, he reached his end.'
Symbolic algebra uses full symbolism (1500 CE to now)
Early steps toward fully Symbolic algebra can be seen
in the work of several Islamic mathematicians
such as 13th century’s Ibn al-Banna,
and 15th century’s al-Qalasadi,
though fully symbolic algebra culminates in 1619
with the work of Rene Descartes. In 1637, he
introduces the use of the letters z,
y, and x
for unknown quantities.
Yardstick, Measuring tape (s)
Cartesian coordinate system:
A Whole New World
The Magic Carpet and the
Invisible Curtain. Perceptive Colors
Thakkar, Harsh.
blog.
https://perceptivecolors.wordpress.com/2018/08/14/the-magic-carpet-and-the-invisible
-curtain/, tunethoughts25@gmail.com
Ancient World Mathematicians
https://www.britannica.com/science/algebra#/media/1/14885/57444
Egyptian Rhind Papyrus 1900 BCE
Solves linear equations of the form
x+ax=b
and x+ax+bx=c,
where a, b, and c are known
and
x is the unknown, is referred to as "
In some problems the author "
aha" or heap
.
checks" his solution, writing one of the earliest known simple proofs.
Transcribed by Ahmes in 1650 BCE
This algebra of proportions is the only algebra evident in the extant Egyptian
mathematical papyri.
Only two surviving papyri contain collections of mathematical problems and
their solutions: the Rhind Mathematical Papyrus, named for the Scotsman
Alexander H. Rhind (1833–1863) who purchased it at Luxor in 1858, and the
Moscow Mathematical Papyrus, purchased in 1893 by Vladimir S.
Golenishchev (1856–1947) who later sold it to the Moscow Museum of Fine
Arts. The former was copied about 1650 BCE by the scribe A’h-mose from an
original about 200 years older, while the latter also dates from the nineteenth
century BCE. Unfortunately, although many papyri have survived due to the
generally dry Egyptian climate, only a few short fragments of other original
Egyptian mathematical papyri are still extant.
China, beyond the abacus used by abacist
The earliest known magic squares appeared in China,[17] as early as 650 BCE.[18]
Around 1250, Li Zhi (or Li Ye) wrote Ts'e-yuan hai-ching,
Sea-Mirror of the Circle
or
Measurements
Yang Hui’s suanfa 1275; “Yang Hui’s Mathematical Methods”
In 1303, Chu Shih-chieh wrote《四元玉鑒》Ssy-yüan yü-chien,
the
Precious Mirror of the Four Elements,
the peak in the development of Chinese algebra.
Ch'in
Chiu-shao
India
"Modus Indorum," the method of the Indians.
The main contribution of Indian mathematicians was the elaboration of the
10
decimal (deci- means
), positional numeral system.
Indian arithmetic, developed consistent and correct rules for operating with positive and negative
numbers and for treating zero like any other number (means “empty,” from Sanskrit word shoonya).
Aryabhata (476–550 A.D.) authored Aryabhatiya, giving rules.
Brahmagupta (fl. 628) brought us the Brahma Sphuta Siddhanta.
Bhaskara II (AD 1114–1185).
Islamic algebra
In the 8th century, Islam had a cultural awakening, and research in mathematics and the sciences increased.
Baghdad had its Bait al-Hikma or "House of Wisdom."
In 953, Al-Karaji is the “first person to completely free algebra from geometrical operations and to replace them with the arithmetical type
of
operations which are at the core of algebra today.
Omar Khayyám’s book included equations of the third degree. He also saw a strong relationship between Geometry
and Algebra, describing Geometric algebra and algebraic geometry.
3 theories about the origins of Arabic Algebra: influences emphasized from
1. Hindus, 2. Mesopotamian or Persian-Syriac and 3. Hellenistic-Egyptian.
Many scholars believe that it is the result of a combination of all three
sources.[49]
Al-Jabr is an Arabic word, published in Persia in 820
‫ﻛﺗﺎب اﻟﺟﺑر واﻟﻣﻘﺎﺑﻠﺔ‬
,or Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala
can be translated as The Compendious Book on Calculation by Completion and Balancing
restoration", "completion", described as
'reduction' and 'balancing' of subtracted terms
According to Al-Khwarizmi, the word algebra is "
that is a trans- position (trans- means across) to other sides of the equation (cancellation of like terms).
He was the first to teach algebra in an elementary form and for its own sake.
Each of six chapters deals with a different type of formula. He treated an equation for its own sake and "in a generic
manner, not just for solving a problem, but to define an infinite class of problems.
Al-Khwarizmi's work established algebra as a mathematical discipline that is independent of arithmetic and geometry.
For Halloween, also "reuniter of broken bones" or "bonesetter.
R. Rashed and Angela Armstrong
Symbolic notation from the Maghreb (N. Africa)
During the 12th century, Al-Hassār developed the modern symbolic
mathematical notation for fractions, where a horizontal bar separates a
numerat/or, means 'number,’ from the Latin "numerus" or “enumerate” to count,
or the parts of ?
_______ (out of)
& the de/nominat/or, prefix de- means DOWN (in some cases, a reduced form of d _ _ -),
means “to name'" / namer from the Latin "denomino" or the naming the whole "thing/s" we are counting
Italy
In 1202; Leonardo Fibonacci wrote Liber abaci, “Book of the Abacus,” the first European
work on Indian and Arabian mathematics, which introduced Hindu-Arabic numerals
to Europe.
The first 7 chapters dealt with the notation, explaining the principle of place value, by
which the position of a figure determines whether it is a unit (1), 10, 100.
animal horns, spirals of
sunflower heads, pine
cones, in the arrangement
of leaf buds on a stem,
related logarithmic
(equiangular) spiral in
snail shells, and the regular
descent (genealogy) of the
male bee.
The techniques were then applied to such practical problems as profit margin, barter,
money changing, conversion of weights and measures, partnerships, and interest
(BANKING).
His name is mainly known because of a problem in the Liber abaci, the Fibonacci
sequence. From it, scientists began to discover such sequences in nature …
1225, his Liber quadratorum, “Book of Square Numbers” is considered Fibonacci’s
masterpiece. Probably his most creative work was in congruent numbers—numbers
that give the same remainder when divided by a given number.
in the 13th century, the beginning of a revival in European algebra
Polynesian binary system
As far back as 1450 CE , about 5000 kilometers south of Hawaii, the people of a tiny island in French Polynesia,
Mangareva, combined two number systems (binary and decimal) into a novel binary system, in which 10 was multiplied by increasing
powers of two. The
Mangarevan language has special words for large groups. But their special
counting words are all decimal numbers multiplied by powers of two, which are 1, 2, 4, 8 … .
Specifically, takau equals 10; paua equals 20; tataua, 40; and varu, 80. This allowed them to cut down on
the number of digits involved in traditional binary systems.
While conventional mathematics uses a base 10 numbering system, binary uses base 2.
In base 10, numbers increase like this: 100 (1), 101 (10), 102 (100), etc.
Base 2 proceeds like this: 20 (1), 21 (2), 22 (4), etc.
With a base 2 system, all numbers can be represented by either a 1 or a 0 (a lightswitch turns
OFF / ON).
When we think of binary math, we think of computers.
Sadly, even the ____________ language itself is now threatened with extinction.
Andrea Bender and Sieghard Beller of the University of Auckland, published their
2013 results in the Proceedings of the National Academy of Sciences.
https://www.science.org/content/article/polynesians-may-have-invented-binary-math
https://www.iflscience.com/binary-math-was-used-polynesians-300-years-earlier-euro
peans-23761
Japan
In the 17th century, Kowa Seki 関 孝和 developed the idea of a determinant.
He created a new mathematical notation system,
including recreating major results in calculus.
He introduced Kanji to represent unknowns and variables in equations.
His reputation grew from the social reform he introduced
to develop the study of math in Japan
and make it widely accessible.
Another of Seki's contributions was the rectification of the circle, i.e. the
yenri, calculation of pi, correct to the 10th decimal place.
soroban
the determinant, a mathematical construction used to solve systems of linear equations by
eliminating dependent variables.
Kowa came to be known as “the arithmetical sage.”
http://outsiderjapan.pbworks.com/w/page/24496593/Wasan:%20Traditional%20Japanes
e%20Mathematics
German
In 1692 & 1694, Gottfried Leibniz was the first to use the concept of ‘function’ explicitly to denote several
geometric concepts derived from a curve, such as the perpendicular, abscissa, ordinate, tangent, and chord. Leibniz
realized that the coefficients of a system of linear equations could be arranged into an array, now called a
matrix, which can be manipulated to find the solution of the system, if any. He was a co-inventor of calculus.
Leibniz also discovered Boolean algebra and symbolic logic.
https://www.scribd.com/book/272143937/Taming-the-Unknown-A-History-of-Algebra-f
rom-Antiquity-to-the-Early-Twentieth-Century
Classical algebra
At the close of the 16th century came the start of the classical discipline of algebra, a flexible and operative symbolism,
marked by François Viète’s 1591 work. A main
innovation of Viète’s In artem analyticam isagoge
(“Introduction to the Analytic Art”) from ana- "up, throughout" and lysis "a loosening"was its
use of well-chosen symbols of one kind (vowels) for unknowns and of another kind (consonants)
for known quantities.
how to
determine unknown quantities by
means of those that are known.
1770, Leonhard Euler’s textbook, Elements of Algebra, defines algebra as:
Structural algebra
In the 1890s David Hilbert (oversaw almost 70 doctoral dissertations)
In 1910 Ernst Steinitz first established the simplest kinds of subfields that any field contains and established a
classification system. He then investigated how properties were passed from a field to any extension of it or to any
of its subfields. In this way, he was able to characterize all possible fields abstractly.
In the 1920s, Emmy Noether helped reorient algebra from the particular and applied solution of equations to the more
general and abstract. She was the greatest influence behind the consolidation of the structural image of algebra.
In 1930 a 2-volume textbook called Moderne Algebra, by the Dutch mathematician Bartel van der Waerden
was published, which presented a totally new image of the discipline. Its innovation lay in the unified picture it
presented of the discipline of algebra. The main task of algebra became to elucidate the properties of general
structures and the relationships among them.
Bartel van der Waerden (1903-1996) earned his Ph.D. in 1926 from the University of
Amsterdam with the dissertation, “The algebraic foundations of the geometry of
numbers,” after studying also at the University of Göttingen, Germany, with Emmy
Noether (algebra) and Hellmuth Kneser (topology). After studying for a semester with
Emil Artin at the University of Hamburg, van der Waerden began writing his most
famous book, Moderne Algebra, basing Volume I (1930) on work of Noether and Artin
and Volume II (1931) on his own work in algebra. He was professor of mathematics at
the University of Leipzig from 1931 through the end of World War II in 1945 and at the
University of Zürich, Switzerland, from 1951 onward. Although he was interested in
mathematics history throughout his career, he published most of his work in this field
later in his career.
Modern or “Abstract algebra”
In the 19th century, Abstract algebra was developed.
Évariste Galois focused initially on what is now called Galois theory.
Now it focuses on constructibility issues…
NOTE: General purpose mathematical dictionaries—even large ones—usually
cover only a small fraction of math terms. For example, the ISI Multilingual
Glossary of Statistical Terms has around 5000 entries and it does not include all
the statistical terms in use. (In earliest uses, there’re around 500 statistical terms.)
4-D (time)
William Rowan Hamilton was a
nineteenth-century Irish mathematician, who
sought an analogous generalization for
three-dimensional space. In 1843 Hamilton’s
pursuits resulted in the quaternions, which
are four-dimensional.
The algebra of quaternions is often denoted by H
(for Hamilton).
In modern mathematical language, quaternions
form a four-dimensional associative normed division
algebra over the real numbers,
and therefore a ring …
https://numb3rth30ry.tumblr.com/post/183133403260/the-structure-underlying-quater
nion
Algebra differs from elementary Arithmetics
Those x’s and y’s that you learn to deal with in algebra denote numbers, but usually numbers in general, not particular numbers.
Be an
algebraist!
Doing algebra is a way of Thinking logically
in algebra you reason (logically) about numbers.
To set up a solution to do what you want, you have to think algebraically,
thinking about or across numbers in general, reasoning qualitatively about numbers.
In algebra, you introduce a term for an
unknown
number and reason logically to determine its value.
Thinking abstractly requires quite a lot of effort and training.
Fostering Algebraic Thinking: A Guide for
Teachers, Grades 6-10 1st Edition
by Mark Driscoll
https://profkeithdevlin.org/2011/11/20/what-is-algebra/
Which math ancestor,
place will your choice of
behavior honor this
year?
A Whole New World
What R your next math
goals?
. . . How will U achieve
them?
What is your WHY?
The Magic Carpet and the
Invisible Curtain. Perceptive Colors
Thakkar, Harsh.
blog.
https://perceptivecolors.wordpress.com/2018/08/14/the-magic-carpet-and-the-invisible
-curtain/, tunethoughts25@gmail.com
∞ [⫙4τℏξmατ!ℂ5 1∫ ∞ⅈℵ∑×ℏ∆∪∮τiβ
⌊ε]
Mathematics is beautiful.
Scared Of Math? Here's
One Way To Fight The Fear. NPR.
Kamenetz, Anya, 2018.
https://www.npr.org/sections/ed/2018/07/16/619328200/got-math-anxiety-here-s-oneway-to-calm-it-down
References
Bashmakova, I., and Smirnova, G. (2000), The Beginnings and Evolution of Algebra, Dolciani Mathematical Expositions 23. Translated
by Abe Shenitzer. The Mathematical Association of America.
Boyer, Carl B. (1991), A History of Mathematics (Second Edition ed.), John Wiley & Sons, Inc., ISBN 0471543977
Burton, David M. (1997), The History of Mathematics: An Introduction (Third Edition ed.), The McGraw-Hill Companies, Inc., ISBN
0070094659
Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0471180823
Corry, Leo (2023). algebra. Encyclopedia Britannica. https://www.britannica.com/science/algebra
Derbyshire, John (2006), Unknown Quantity: A Real And Imaginary History of Algebra, Joseph Henry Press, ISBN 030909657X
Flegg, Graham (1983), Numbers: Their History and Meaning, Dover publications, ISBN 0486421651
Heath, Thomas Little (1981a), A History of Greek Mathematics, Volume I, Dover publications, ISBN 0486240738
Heath, Thomas Little (1981b), A History of Greek Mathematics, Volume II, Dover publications, ISBN 0486240746
Stillwell, John (2004), Mathematics and its History (Second Edition ed.), Springer Science + Business Media Inc., ISBN 0387953361