Algebra. It's powerful. But it's not what it was
By Keith Devlin @KeithDevlin@fediscience.org, @profkeithdevlin.bsky.social
What is algebra? That question arose in the Q&A session at a recent screening of the new documentary movie Counted Out that I wrote about in last month’s post. Here is the answer I alluded to in my response, but was way too much to launch into.
Cover page of a manuscript of al-Khwārizmī’s Ninth Century book Algebra. Al-Khwārizmī has been falsely claimed to have been the inventor of algebra, but the ancient Greek mathematician Diophantus had written an algebra text in the Third Century, and in any case a form of algebra had been in use long before then. Moreover, in his introduction, al-Khwārizmī says clearly that the work he presents is a compilation of what was known at the time. However, his book Algebra was the first such in Arabic, it did put algebra “on the map”, and it did initiate the chain of books that leads to present day algebra. Moreover, Islamicate scholars who followed him did much of the early development of algebra. That’s quite a legacy. Images of al-Khwārizmī in circulation are all works of fiction.
To most non-mathematicians today, the word “algebra” conjures up an image of “calculation involving letters”. There is some truth in that, for school algebra at least. But it misses the essential distinction between algebra and arithmetic. Both can be used to solve numerical problems, and both involve calculation. The difference lies in the nature of the methods.
In arithmetic, we calculate with numbers we are given to obtain a numerical answer. With algebra we begin by giving the unknown numerical answer a name (often the letter x, in the modern school classroom), and then operate (arithmetically) on that name to determine the value of the unknown. (See the simple example below.)
For an example of an arithmetic problem, I could ask you what you get when you square 7 and add 2? Teachers sometimes begin by writing this as an “equation”, using a question mark to indicate what is being sought: “7^2 + 2 = ? ”. You can solve it by performing the calculations the problem statement itself provides; compute 7^2 to get 49, then add 2 to get 51.
But with a school algebra problem, the process is reversed. You are given the “answer-number” and you have to determine one of the input numbers. For example, for what number do you get 22 when you square it and subtract 3? Typically, you approach a problem like this by expressing it as an equation x^2 — 3 = 22, (where the “x” is a symbol for “the unknown number”). A teacher may first write this with a question mark, like this: “?^2 — 3 = 22“.
The teachers’ question-mark presentations highlight the fact that in algebra, we work backwards. That’s surely why many people (most of us, we suspect) initially find algebra difficult. As mathematician Jordan Ellenberg observes in the movie, it’s like ice skating; things proceed smoothly until you try to skate backwards, which is much harder. Many people never master “going backwards”, be it in the algebra class or at the ice rink!
In the case of (school) algebra, the steps you take to solve the problem are still arithmetic, but because you are working backwards, you have to be something of a detective, and reason logically about what you need to do with the given numbers to discover what the unknown (“input”) number is. Typically, that “logical reasoning” takes the form of manipulating the terms of the equation to isolate the unknown. For the equation above, you add 3 to both sides to give “?^2 = 22 + 3”, and now you can proceed with straightforward calculations: “?^2 = 25”, so the unknown is 5.
That simple description does a pretty good job of capturing what algebra was for the first several centuries of its existence, and covers (in principle) much of what is taught in the school algebra class today. But mathematicians have a much broader description. To us, algebra is at the same time:
a toolkit for solving quantitative or logical problems in mathematics, science, engineering, technology, finance, commerce, the law, and other domains;
an important branch of mathematics; and
a precise language for describing, studying, analyzing, discussing, and communicating issues in those domains and solving problems that arise in such activities.
Moreover, we would rank item 3 as the most important aspect of the subject. It’s that meaning we have in mind when we say algebra is the language of science and engineering. And item 3 is the reason the movie says that citizens having some mastery of algebra is as important in today’s world as mastery of our native human language.
Modern dictionaries give a definition that summarizes the above description in a superficial way; for example, the Apple Dictionary app on my Mac says:
al·ge·bra| ˈaljəbrə | noun the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations: courses in algebra, geometry, and Newtonian physics. • a system of algebra based on given axioms.
Note the generality of the statement, and the reference to axioms.
Wikipedia, which is generally reliable for mathematics entries, says (July 31, 2024):
Algebra is the branch of mathematics that studies algebraic structures and the manipulation of statements within those structures. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.
Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field investigating variables that appear in several linear equations, so-called systems of linear equations. It tries to discover the values that solve all equations at the same time.
Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several binary operations defined on that set. It is a generalization of elementary and linear algebra since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups, rings, and fields, based on the number of operations they use and the laws they follow. Universal algebra constitutes a further level of generalization that is not limited to binary operations and investigates more abstract patterns that characterize different classes of algebraic structures.
For all its present power, the origins of algebra were very modest. An early precursor emerged somewhere between Europe and India more than two thousand years ago as an arithmetic technique used by merchants, surveyors, lawyers, government secretaries, and civil engineers, to solve numerical problems that arose in their daily work. Major advances were made in ancient Greece and First-Millennium India. (Also, clear antecedents to algebra have been found in Mesopotamian clay tablets dating back some 4,000 years.)
By the ninth century, the method was known and used in the international trading hub of Baghdad, where it was referred to by a short Arabic phrase containing the word al-jabr. (Hence today’s name.) It wasn’t yet a subject and there were no textbooks; just a technique for solving numerical problems where you started out by naming an unknown quantity. Its status changed when the caliph (“king”) commissioned a scholar called al-Khwārizmī to write a practical algebra textbook from which future artisans could to learn how to use the method. (Commemorating the influence of his book, today’s word algorithm derives from his name.)
That book set in motion eight-hundred-years of growth of what eventually led to the new science summarized above—a process of development that stretched across many countries and different cultures, and involved hundreds of mathematicians.
François Viète (1540-1603), who took the first step leading algebra outside of arithmetic.
Up to the sixteenth century, the growth of algebra consisted essentially of the development of more and more techniques for solving equations. Then, the French mathematician François Viète developed an algebra not of numbers but of geometric objects. To do that, he had to use letters to denote both known quantities and unknowns, which set the stage for an altogether new direction, where algebra is separate from arithmetic.
In the following century, two other Frenchmen, Pierre de Fermat, and René Descartes took Viète’s framework and applied it to numbers, and Descartes showed how to apply the new algebra to solve problems in geometry. Not long afterwards, Newton and Leibniz made use of it in calculus. By that point, algebra looked very much like today’s algebra.
But, it was becoming clear that this new algebra had the potential to become something much broader than a problem-solving technique; namely a general language of mathematics and science. Realizing that goal took a further three-hundred years of development, but in the end, by the start of the twentieth century, algebra (what we sometimes refer to as “modern algebra”) had achieved that status.
The reason that final step took so long was that the newer uses of algebra developed in the seventeenth century involved techniques that pushed the underlying concept of number well beyond the one that had been in use since ancient times. Those final three hundred years of development involved a revolution in how we conceive of numbers. (It was, then, not a development of algebra per se, rather of the underlying substrate on which it operated.)
Most mathematicians have some familiarity with the more recent, item 3 story, where modern, axiomatic mathematics was developed. Few know about the first part. They may think they do, because it’s easy to look back at modern translations of works by medieval mathematicians and “understand” what they were doing. (This was definitely true for me before I looked into this history in detail.) But our very familiarity with modern algebra misleads us when we look at (say) a medieval equation—even if (in fact, especially if) that equation is written out in modern fashion using today’s (‘algebraic’) formalism. (For one thing, medieval mathematicians did not have equations in our sense; see below.)
Since anyone today who reads (or reads about) pre-eighteenth algebra works will almost certainly do so with today’s conception of algebra, some historians of mathematics refer to the pre-eighteenth century algebra as “pre-modern algebra” and the more recent algebra as “modern algebra”, or just “algebra”.
The problem is, modern algebra is so powerful and versatile, that those of us trained in its use can apply our algebraic understanding to the writings of those medieval mathematicians and, apart from some seemingly minor anomalies, it all makes sense. But our modern understanding is not at all what those earlier mathematicians were thinking. The difference is significant. That “pre-modern algebra” versus “(modern) algebra” distinction is much more than a timeline marker.
In addition, since the numbers algebra built upon were also different, historians distinguish them too: “pre-modern numbers” versus “(modern) numbers”. (The mechanics of arithmetic remained essentially the same, as was the case for algebra; it was people’s conception of the numbers it worked on that changed.)
For most of history, what we today think of as a number (i.e., an object, or noun) was in fact the adjective in a multitude-species pair. People conceived numbers as real-world objects viewed from the perspective of quantity; so numbers came attached to objects (though for mathematicians they could be abstract objects). We are familiar with this today with currency and geometric angles. We have monetary amounts such as “5 dollars and 20 cents” or angles such as “15 degrees and 30 seconds”. The numerals in those expressions are adjectives that modify the nouns (those nouns being dollars, cents, degrees, and seconds, respectively).
Those multitude-species pairs were the entities that equations were made up of; they were their “numbers”. For example, the following pre-modern quadratic equation comes from an eleventh-century Arabic algebra textbook
“eighty-one shays equal eight māls and one hundred ninety-six dirhams”.
Here the English version consists of translations of the Arabic words from that textbook, apart from the terms “māls”, “dirhams”, and “shays”. The word shay referred to the unknown in the equation, māl is the term they used to denote the square of the unknown, and dirham denoted the unit term. The plural “s”s in our presentation of the equation are English additions (hence not italicized); Arabic does not designate plurals that way.
Contorted into modern notation, that pre-modern equation is this one:
81x = 8x^2 + 196
On the face of it, it looks like they used māl to refer to refer to what we call x^2, shay for our x, and dirham for our word “unit” (which we usually leave implicit, as they often did too). That’s true, but in a superficial, and misleading way. We can express our modern equation as a sentence like this
“Eighty-one multiplied by x equals eight multiplied by x^2 plus one hundred and ninety-six.”
But that is not an accurate modern rendering of the Arabic equation.
For one thing, there were no multiplications in their equations. Sticking to the modern notation, the Arabic term “eighty-one shays” corresponding to 81x is not a translation of “81 multiplied by x” and the term “eight māls” corresponding to 8x^2 is not a translation of “8 multiplied by x^2”. Multiplying unknowns by coefficients is something we can do in modern algebra but was not possible in medieval algebra. (They did not have coefficients; see momentarily.)
Rather, the terms “eighty-one shays”, “eight māls”, and “one hundred ninety-six dirhams denoted collections of objects. In fact, they borrowed those three Arabic nouns from everyday language, where māl meant an amount of money, a dirham was a silver coin, and shay literally meant “thing”. With those meanings, the equation can be rephrased like this:
“a collection of eighty-one shays has the same value as a collection
consisting of eight māls and one hundred ninety-six dirhams.”
Given the mercantile origin of some of the terms, we can think of their “equations” as being statements that two bags of coins of different denominations have the same value. Which is probably as close as we can get to their conception of an equation.
Moreover, the Arabic word they used to mean “equals” was associated with weight, as in two amounts “balance” when put in the scales. In fact, one Arabic author explained the simplification of equations by comparing an equation to coins in a balance.
But this comparison with trade, while helpful, also has a misleading element. We said above that they “borrowed” those words from everyday use. Their equations were not necessarily about money (though they could be). Rather, money, with its different coinages, was a particular instance of their conception of numbers. They used the words “shay” and “māl “ to denote the unknown and its square in an equation, respectively, and “dirham” to denote the (known) number component, regardless of the items under discussion.
The point is, right up to the seventeenth century, numbers were multitude-species pairs. The part of such a pair that we recognize today as the “number” (with today’s numbers being nouns) was in fact the adjective describing the amount of some entity.
Today, we recognize that structure when dealing with a number as a number of something, such as when we are handling money. For instance, we speak of “three dollars” or “four dollars and ten cents”. We cannot simply leave off the species (dollars, cents, or whatever). (Other than in elliptical fashion in an appropriate context.) Our use of currency shows that there are circumstances where we do still use multitude-species pairs in calculations. (We do not view “three dollars” as “three multiplied by dollar”.)
In our case, we do though conceive of abstract entities like “eighty-one”, “eight”, and “one-hundred ninety-six”, that exist independently of those multitude-species pairs. In the modern equation
81x = 8x^2 + 196
the first term does indeed denote the number 81 multiplied by the unknown (number) x, and the term 8x^2 denotes the number 8 multiplied by the unknown (number) x^2 (which we view as x multiplied by x). Every constituent is a number, and they are combined by arithmetic operations.
But the “equivalent” Arabic equation
“eighty-one shays equal eight māls and one hundred ninety-six dirhams”
cannot be parsed that way. The “eighty-one”, the “eight”, and the “one-hundred ninety-six” are not nouns, they are adjectives, that have meaning only when coupled with appropriate nouns (in this case “māls, “dirhams”, and “shays”, respectively).
Their concept of an equation was, then, very different from ours. It was built on a concept of “polynomial expression” (as a “bag of objects”) that was different from ours (as an expression for a number that explicitly shows how to calculate that number from the numbers constituent in the expression). Because of the roles played by 8 and x^2 in today’s equation, we say “8 is a coefficient of x^2” (in the equation), but it is a combination of two numbers. Their “eight māls” was a single number (a pre-modern number) consisting of a multitude of māls.
Their “bags of objects” concept of an equation was, clearly, a much simpler notion than our equations. We can easily understand their concept, but we doubt very much that a medieval mathematician would have known what to make of ours—which came many centuries later.
The key to today’s algebra was the creation of an abstract number system, specified by axioms, a process that was not completed until the early twentieth century. A valuable first step towards that solution was made by several seventeenth century mathematicians who defined numbers-as-objects from the multitude-species pairs that had served for so long.
Newton, for example, wrote down the proposal below in one of his Notebooks (1670). The species he was working with for calculus were line segments, which he called Quanta. To develop calculus, he looked at line segments from the perspective of their length; those were his multitude-species “numbers”, which he wanted to break free of. He wrote:
Number is the mode or affection of one quantum compared to another which is considered as One, whereby it’s Ratio or Proportion to that One is expressed. [Thus b/a is the number expressing the ratio of b to a.]
The brackets are his. He wrote “b/a” is vertical-stack form, not inline as here. His key claim was: “b/a is the number.” He had defined a ratio (hence a dimensionless object) to be his number concept.
Defining numbers to be ratios of Quanta was equivalent to adopting a unit Quantum. He formulated other variants in other Notebooks.
Newton also provided an explanation of negative numbers. Most mathematicians at the time refused to acknowledge negative numbers, for the obvious reasons that they don’t fit into the multitude-species picture. (They did, however, perform calculations where some of the intermediate steps were “negative.”) This explains why negative numbers remained banned long after mathematicians were freely making use of irrational and complex numbers.
The fact that the likes of Newton struggled to come up with a definition of number adequate for modern algebra surely indicates that a basic concept we today take for granted, and teach in elementary school, came only after a number of leading mathematicians wrestled with the issue for many years. Elementary arithmetic is far from elementary.
For further information about Viète, Fermat, and Descartes, see my April 2023 Devlin’s Angle post.
The above essay is abridged from the book “Let x be …” — The Story of Algebra, by Keith Devlin & Jeffrey Oaks, in preparation.