The second in a series. See the November 20 entry, “What is algebra?” for the first.
DECEMBER 13, 2011. Two key features of algebra as we understand the word today are:
1. Reasoning about numbers by recognizing patterns across numbers;
2. Solving a problem by introducing a term for an unknown and then, starting with what is known, reasoning to determine its value.
We first see the emergence of both features of algebra in the mathematics of ancient Babylonia, around 2,000 BCE.
Several hundred of the many thousands of Babylonian’s cuneiform-inscribed clay tablets that have been found are devoted to mathematics. They show that those ancient mathematicians had systematic procedures for solving geometric problems involving the determination of lengths and areas of figures. Today, we would solve those kinds of problems using linear and quadratic equations and indeterminate systems of linear equations. Their methods amounted to a form of geometric algebra that could be applied to solve problems beyond overtly geometric examples such as calculating the perimeters or areas of various plane figures or the volumes of solid objects: arithmetic problems arising in trade and commerce, for example, and other financial transactions such as inheritance. In addition, the Babylonians considered problems that seemed to have had no practical application, pursuing them purely for recreation. Although they described their procedures in terms of specific lengths and areas, they did so in a way that made it clear they applied in general, and in that sense they were starting to think algebraically, by recognizing patterns across quantities.
Moreover, some of their writings show the second characteristic feature of algebra, namely introducing an unknown and then reasoning to find its value. In their case, however, the unknown was not numeric but geometric – an unknown line on which they performed geometrical operations to get the answer.
In reasoning with unknown quantities, the Babylonians went further than other early civilizations with a mathematical tradition, such as the Egyptians, the Chinese, and the early Greeks, all of the first millennium BCE. Our knowledge of the mathematics of those peoples comes from works such as the Rhind papyrus, The Nine Chapters of the Mathematical Art, and Euclid’s Elements, respectively. The approach described in those documents was, like that of the Babylonians, fundamentally geometric and exhibited reasoning about patterns of quantities, but we do not find the introduction of an unknown followed by an argument to determine its value.
It is with the work of the Greek mathematician Diophantus (ca. 210–290 CE) that we first find clearly recognizable algebra, where the unknowns represent numbers whose values are to be determined. Around 250 CE, Diophantus, who lived in Alexandria in Egypt, wrote a multi-volume work, Arithmetica, which its title notwithstanding was an algebra book. Its author used letters (literals) to denote the unknowns and to express equations, but that is a purely notational distinction. He also was one of the first mathematicians to use negative numbers in calculations. He showed how to solve equations by using two techniques called restoration and confrontation. In modern terms, these correspond more or less (but not precisely) to (1) adding a quantity to both sides of an equation to eliminate a negative term on one side, and (2) eliminating like terms from both sides. He used these techniques to solve polynomial equations involving powers up to 6.
Almost four hundred years later, the Indian mathematician Brahmagupta (598–668 CE) likewise displayed recognizable algebra, in his book Brahmasphutasiddhanta, where he described the first complete arithmetic solution (including zero and negative solutions) to quadratic equations.
Following Diophantus and Brahmagupta, the next major step in the development of algebra – and it was huge – took place in the period generally referred to as “Arabic mathematics” or “Muslim mathematics”, a significant outpouring of mathematical activity stretching from the 8th century to the end of the 16th. Indeed, the word algebra itself comes from the Arabic word al-jabr, which occurs in the title of a highly influential book by the Persian mathematician al-Khwārizmī, completed around 830: al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” but more loosely means “solving an equation.”
That period will be the focus of my next article on algebra.
profkeithdevlin.org 
Milo Gardner, who works on Egyptian math, recently wrote something I just want to quote here. This comes from a thread in “Math, Math Education, Math Culture” on LinkedIn: http://www.linkedin.com/groupItem?view=&gid=33207&type=member&item=81780525
Modern mathematics including paper folding offers distractions from the central dual multiplication definition conflict. Multiplication defined as been repeated addition and scaling of rational numbers co-existed as main stream Western Tradition ideas 4,000 years ago, and maintained the tradition for 3,500 years.
Math historians report Egyptian fraction cultures formally used the paired multiplication definitions by 2050 BCE. Specifically, the Egyptian Middle Kingdom. Ahmes, a 1650 BCE scribe, recorded a 2/n table that scaled 2/3, 2/5, 2/7, …, to 2/101 to concise unit fraction series that followed a dual multiplication method.
Modern scholars scratched their collective heads during the 20th century when only reporting the additive side of the paired dual set of multiplication definitions. Ahmes 2/n table introduced 87 arithmetic, algebraic, geometric and weights and measures problems that required a dual understanding of the multiplication definitions.
Both sides of the multiplication definitions were needed by Ahmes, and Egyptian scribes, as scribes as late as Fibonacci in 1202 AD used to record the Liber Abaci, Latin speaking/writing Europe’s arithmetic, algebra, geometry and weights and measures instruction book for 250 years.
Of course, with the death of Egyptian fractions, and the birth of modern base 10 decimal arithmetic in 1600 AD, the ancient dual definition of multiplication conflict seemed to disappear. But has it?
I think not. Modern mathematical physics reports the same dual conflict in ways that would have made ancient Egyptian fraction scribes shake their heads.
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Maria,
Thank you for the post. Egyptian algebra has been labeled rhetorical algebra by math historians. Unknown values were rhetorically discussed before hard-to-read scribal shorthand calculations were written down.
RMP 32 is a case in point. The algebra is trivial:
x + (1/3 + 1/4)x =2
is solved today by
(19/12)x = 2
x = 24/19 = 1 + 5/19
Ahmes solved the problem the same way, adding three level numeration aspects (proto-number theory) that otherwise well-informed scholars failed to parse as Ahmes recorded.
One aspect of Ahmes’ work scaled 5/18 by 12/12 such that
.
60/228 = (38 + 19 + 2 + 1)/228 concluded
x = 1 + (1/6 + 1/12 + 1/114 + 1/228
A second aspect scaled the entire equation by 114 scaled
http://planetmath.org/encyclopedia/SCALEDEQUATIONSRMP32.html
A third aspect included a proof that scaled the remainders to 912.
Greek algebra followed the same scaled arithmetic logic that.recorded rational numbers in exact unit fraction series. .
By the time of Diophantus, indeterminate equation algebra flowered in ways that are explained by the arrival of the Chinese Remainder Theorem (CRT) on the Silk Road. Fibonacci included a medieval version of the CRT in the “Liber Abaci” that Diophantus would have recognized..
Fibonacci’s arithmetic included exact unit fraction series the used of an algorithm. Medieval number theory scaled rational numbers n/p such that (n/p – 1/m) = (mn = p)/mp set (mn -p) = 1 as often as possible, a notation introduced by Arab algebra around 800 AD.
Conclusion: Egyptian, Greek, Arab and medieval multiplication included a dual definition that scaled rational numbers and repeated addition, as Maria is discussing.
Best Regards,
Milo Gardner
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Thank you for your amazing post! It has long been really helpful.
I hope which you will proceed sharing your wisdom with us.
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Thank you Maria for your timely comments. Egyptian algebra was advanced as high as Babylonian algebra.
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I recently stumbled upon the video of your continuing studies lecture on this topic, from fall 2012 (I’m enjoying the series, thank you):
There and here, you use the terms “restoration” and “confrontation” in describing the methods of both Diophantus and Al-Khwarizmi. This seems to suggest they were doing basically the same things, but you don’t say so explicitly. How different or similar would you say their approaches were? Can we really know at this far remove?
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Glad you are enjoying the series. Scholars have figured out pretty well what the medieval algebraists were doing. It is hard for us to fully understand their methods because we are so familiar with numbers systems in a way they were not back then. For example, we are comfortable with using negative quantities in manipulating equations, but that was not the case back then. I consulted from time to time with Prof Jeffrey Oaks when writing about early algebra. This paper by him provides a good explanation of what they were doing.
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Thank you!
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