John McWhorter, one of my favorite public intellectuals, writes (in his recent essay “Lets chill out about apostrophes”), “Writing does not entail immutable rules in the way that mathematics does.” I think he’d be happy to know that some of the rules that govern mathematical formulas are just as mutable as the rules of punctuation.
Many of the rules people associate with classroom math, such as the friendly FOIL and the infamous PEMDAS (both of which I’ll define and discuss below), are actually fairly recent innovations, designed to prevent students from falling into certain common errors, and those errors are themselves “illnesses of modernity” – errors made possible (and even inevitable) by relatively recent changes in the way humanity does algebra. So before we talk about rules and the errors those rules were designed to thwart, let’s talk about the once-controversial symbolic algebra revolution.
Even the way people write plain-old numbers hasn’t stood still over the past millennium. The “Hindu-Arabic” symbols 1234567890 have been fairly stable in Western mathematics since Leonardo of Pisa (better known as Fibonacci) imported them to Europe around the year 1200, but in modern Hindi those digits are
and in modern Arabic they’re
And intercultural differences persist within Western mathematics. In some parts of the West commas are used to separate digits into groups of three and a period is used as a decimal divider (decimal point), but elsewhere in the West it’s the other way round, so that the number one thousand two hundred and thirty-four and a half could be written as either 1,234.5 or as 1.234,5 depending on where you live.
Of course, the facts of math don’t change – two plus three equals five wherever and whenever you are. It’s the way we record such facts on the page that changes. Take the formula 2 + 3 = 5. I’ve already mentioned that the symbols 2, 3, and 5 are scarcely a thousand years old, but “+” and “=” are of even more recent vintage; they were introduced in the 1500s, and the original versions of those symbols strike the modern eye as grotesquely wide: −−−+−−− and ====== (without the spaces in between). The condensation of those symbols into their slim modern forms was part of a thorough-going cultural shift toward condensation and brevity within and beyond mathematics.
THE DAWN OF MODERN ALGEBRA
Algebra existed in the Arab world more than a dozen centuries ago (the word “algebra” is of Arabic origin, meaning “the restoration of what has been broken”) and precursors to algebra can be found elsewhere many centuries before that. But early algebra was made up of words, not symbols. Consider an example of what historians of mathematics call “rhetorical algebra”1: “To find two numbers whose sum will be the first of two given numbers and whose difference will be the second, form half the sum of the two given numbers and also half their difference.” (Back then all math problems were word problems!) Nowadays we would simply write “To solve x+y = a and x−y = b, take x = (a+b)/2 and y = (a−b)/2.” If (like me) you prefer the modern form, you can thank François Viète (1540-1603). Trained as a lawyer, Viète had broad interests, including astronomy and cryptography; he figured out that planets orbited in ellipses forty years before Johannes Kepler did, and his success as a cryptanalyst led some of his enemies to accuse him of witchcraft.
In 1591, Viète published In Artem Analyticam Isagoge (“Introduction to the Analytic Art”). He gave his Arabic forerunners their due but he wasn’t very nice about it: “Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary to introduce an entirely new form into it.” In this work Viète advocated a condensed style of representing relationships between quantities, using symbols with little or no relationship to the quantities they represented. One might compare an algebraic formula to a cryptogram: the information of the original message is present but you have to do some work to bring it forth from its disguise.
The goal of Viète’s analytic art was bold: “to leave no problem unsolved” (nullum non problema solvere). In this brashness Viète had a kindred spirit in the English thinker Francis Bacon who in his 1626 book New Atlantis foresaw a technological civilization whose goal would be nothing less than “the effecting of all things possible”. Bacon was, like Viète, a champion of concision; in his writing (such as his 1620 master work Novum Organum) Bacon avoided the long ornate sentences he saw as a blemish in much philosophic writing, favoring a style that sometimes took brevity too far.2
Once you know the “algebra code” and are comfortable with it, it takes little time to transform algebraic information from one form into another. For instance, if you want to find x and y such that x+y = a and x−y = b, you can add the two equations obtaining x+y+x−y = a+b, cancel the +y and the −y to get x+x = a+b, rewrite x+x as 2x to get 2x = a+b, and finally divide both sides by 2, obtaining x = (a+b)/2. If instead of adding the two original equations you subtract them, you end up with the equation y = (a−b)/2. Magic! Imagine trying to do this without symbols the way everyone did up until five hundred years ago.
What I’ve written above is not quite how Viète’s symbolic algebra worked. For one thing, he advocated a rule that the unknown quantities in a formula should be represented by vowels, reserving consonants for the known quantities. You don’t remember that rule? That’s not surprising; this convention lasted only a few decades before it was displaced by René Descartes’ counter-proposal that one should represent known quantities by letters near the beginning of the alphabet and unknowns by letters near the end, such as x. This convention is in force to this day; if someone asked you to solve ax2 + bx + c = 0 without providing further context, I bet it would never occur to you to solve for a in terms of b, c, and x. But even Descartes’ more modern version of symbolic algebra, expounded in his 1637 book La Gèometrie, differed from ours; Descartes wrote the square of x as xx. Back then it was a principle of good style that exponents should be used only for third powers and higher. European mathematicians followed that rule until they didn’t. Rules change.
THE DOWNSIDE
Not everyone approved of symbolic algebra; in 1654, the English minister John Webster launched what may be the very first salvo in the ongoing Four Hundred Years War over how math should be taught. Webster saw himself as a follower of Bacon, and for Webster symbolic methods in algebra achieved concision at the cost of comprehensibility. In his The Examination of Academies Webster singled out for special scorn what he saw as the overuse of symbols in math, comparing the practice to the use of hieroglyphics and alchemical symbols. His countryman Seth Ward replied with a forceful defense of the modern style, insisting that “the avoiding of confusion made by words was the end and motive of inventing mathematical symbols.” More famous than Webster or Ward was John Wallis, who asserted that through the use of symbols “the whole process of many operations is at once exposed to the eye in a short synopsis.” Math becomes (here again I quote Wallis) “intelligible, with much more ease than when involved in a multitude of words, and long periphrases of the several quantities and operations.”
I hasten to confess that a month ago I knew very little of this story; I learned about it from mathematical historian Helena Pycior’s writings. I’ve compressed some quotations (putting a mere space where a scholar would put “. . . ”) and modernized the Capitalization and spellynge, but aside from that I’ve distorted nothing intentionally (though if I’ve gotten anything else wrong, blame me, not her). Pycior has a vivid description of the revolution led by Viète and Descartes: “Symbols made algebra easier because they permitted mathematicians to paint, as it were, transparent thumbnail sketches of otherwise complicated algebraic situations.”
Webster was right when he compared symbolical algebra to alchemy (another word borrowed from Arabic, meaning “transmutation”), but algebra is the alchemy that works. Our success in transforming the pair of equations x+y = a and x−y = b into the pair of equations x = (a+b)/2 and y = (a−b)/2 wasn’t just a stroke of good luck; it’s part of a general methodology for transmuting algebraic questions into algebraic answers by successively pouring mathematical relationships from one vessel to another, artful adding just the right algebraic reagents at each step of the way. One can understand why Viète hoped that his methods would allow humanity to solve all algebraic problems!3
Webster was right in a different way: if you’re not tutored in the ways of symbolic algebra, the cryptic symbols lock you out of algebraic thinking. The revolution of Viète and Descartes divided society into a clergy of algebra-adepts and a complementary laity consisting of everybody else. During my own lifetime, the American civil-rights-worker turned education-reformer Robert Moses noticed that lack of knowledge of symbolic algebra was a key obstacle in young Black people’s access to other forms of knowledge, and so created the Algebra Project to address the problem – to restore one part of what has been broken at a societal level.
I’m not saying symbolic algebra was a step in the wrong direction! The science of the past two centuries would be impossible without the concision that symbols afford, and algebra wouldn’t have become such a universal language if its symbols were replaced by words. I personally have loved algebra from the memorable day a teacher showed me how it explained and unified numerical patterns I’d noticed (see my essay “Thoughts from the Outfield”), and much of my research would be unthinkable (or at least not thinkable by me) if it didn’t use symbols. But we should also take a moment to note in passing that without symbolic algebra, there’d have been be no need for an Algebra Project; a purely rhetorical algebra takes longer to read and write but it’s more accessible to those who haven’t been mathematically tutored.
And even those who have been mathematically tutored tend to find symbolic notation tricky and come to rely on rules like “FOIL” to prevent blunders.
TRAINING WHEELS
Viète and Descartes and their successors didn’t just sever algebra from words; they severed algebra from pictorial thinking, at least in terms of how the subjects are now taught in school. (It’s no small irony that Descartes made this severing possible in a work called La Gèometrie.) In the 1500s and 1600s, students wouldn’t have made the nowadays-common mistake of equating the square of a sum with the sum of the squares, not just because they didn’t have symbols that encouraged the mistake of confusing (a+b)2 with a2+b2, but also because they would picture actual geometrical squares:
The left half of the picture shows a square with sides of length a+b; the right half shows two smaller squares, one with sides of length a and one with sides of length b, superimposed with the outline of the big square. Since the two small squares on the right can fit inside the big square on the left at the same time with room to spare, it’s visually clear that (a+b)2 is bigger than (not equal to!) a2 plus b2. Likewise, in thinking about the product of a+b and c+d, students of an earlier era would be likely to picture a rectangle with sides of length a+b and c+d (as shown on the left) and mentally split up that rectangle into four smaller rectangles (as shown on the right):
With this picture it’s easy to see that a+b × c+d, the area of the big rectangle, is equal to the sum of the four products a×c, a×d, b×c, and b×d, since these are the areas of the four small rectangles. It’s common nowadays to write these products as ac, ad, bc, and bd, using the convention that juxtaposed quantities with no operation-sign between them are to be multiplied. (This convention appears in Europe mathematical literature in 1544 but seems to have been used in India five hundred years earlier; see the St. Andrews Maths History article listed in the References.)
When algebra became more abstract and multiplication was construed not as an operation on physical magnitudes but on pure numbers, then those geometrical images faded from view and students lost much of their ability to apply visual insight to algebra. Instead students relied on rules such as this one: When you want to multiply together two expressions, each of which is a sum of two terms, multiply the First terms of the expressions together, multiply the Outer terms together, multiply the Inner terms together, multiply the Last terms together, and finally add those four products, where the meanings of “First”, “Last”, “Outer”, and “Inner” are clarified by this diagram:
The mnemonic for the First-Outer-Inner-Last rule is FOIL.
Is FOIL an immutable rule? Certainly the order of the terms doesn’t matter; FILO would be just as good a rule as FOIL (if less memorable for non-pastry-lovers). What matters is that each of the four products needs to be included in the sum exactly once: FIL and FOLIO would be bad rules, the former because it leaves out ad, the latter because it includes it twice. What too many students miss out on is that what underlies and justifies the FOIL procedure is the distributive law:
(a+b)(c+d) = ac + ad + bc + bd
(There are actually two distributive laws, called the left-distributive law and the right-distributive law; I’ve combined them here in part because I can never remember which is which.) The distributive law is the immutable bedrock on which the mutable and somewhat arbitrary rule FOIL rests. Rules are consequences of underlying laws, and when you understand the law that underlies a rule, the rule ceases to be perceived as “This is what you do (because the textbook says so)” and is seen as just How Things Are.
I typically use First-Outer-Inner-Last ordering, or occasionally First-Inner-Outer-Last, but I never use a mnemonic; instead, when I have to multiply a sum of m terms by a sum of n terms (where m and n could be 2, but one or both of them could be larger) I just make sure that I paired each term of the first sum with each terms of the second sum in all mn possible ways (or, as Francis Bacon would have said, “in all ways possible”) and take the sum of all those products. If I’m worried I might forget some products or include some twice, I draw arcs as I do the working (paying special attention to signs):
The last line is the desired expansion of (a−b)(c−d).
The problem with FOIL is that it can do an all-too-good job of sparing the student from the task of thinking about why algebra works. But if through the use of FOIL a student comes to understand that FOIL is just a way of ensuring that they’ve paired the terms in all possible ways, and if they transfer that understanding to cases where the mnemonic doesn’t apply (when m or n is bigger than 2), then for that student, FOIL did what training wheels are supposed to do: enable a person to function without training wheels.
And if you still use FOIL, I don’t judge you. Even when I’m going shopping for just four things, I like to use a shopping list to ensure that I don’t mess up.
SORRY, AUNT SALLY, YOU’RE NOT EXCUSED
Now we come to one of my least favorite topics of mathematical discussion: the PEMDAS rule. I kind of gave PEMDAS the finger a few paragraphs back when I wrote a+b × c+d (raise your hand if you noticed this), because I used spacing to make my meaning clear; if I’d written a + b×c + d you’d have interpreted it differently. Since a founding purpose of symbolic algebra was “the avoiding of confusion”, and since using spacing to resolve ambiguity is a dangerous game (if there’s anything more mutable than a symbol it’s the size of the absence of a symbol), mathematicians had to come up with a convention for what a + b × c + d should mean. They agreed (and later on we’ll discuss why nobody disagreed) that in an expression like 1 + 2 × 3 + 4, you perform the multiplication first, and that if you really want to perform the additions first, you have to override the default interpretation by writing (1 + 2) × (3 + 4).
Those parentheses are the “P” in PEMDAS, and the fact that the “M” comes before the “A” reminds us that we do multiplications before additions unless there are parentheses telling us to do otherwise. (The parentheses had better come first, because if parentheses don’t take precedence over other operations, then they can’t override the other conventions, so what good are they?) So instead of a+b × c+d, I should have written (a+b)×(c+d) to be PEMDAS-compliant. If your hand was up, you can lower it now.
PEMDAS was latent in something else I wrote earlier: “If instead of adding the two original equations you subtract them, you end up with the equation y = (a−b)/2.” Let’s check this. If I take the equation x+y = a and subtract the equation x−y = b from it, I get x+y−x−y = a−b, and if I cancel the x and the −x and I cancel the y and the −y, I get … wait, what? 0 = a−b?!? Where did I go wrong? My error was leaving out some much-needed parentheses to remind myself that I’m subtracting x−y from x+y, and I forgot to remember that (x+y)−(x−y) is equal to x+y−x+y (because, as commercials keep reminding us, when you pay less you save more). Part of the fine print in PEMDAS is that you do addition and subtraction from left to right unless parentheses intervene.
Oh, did I forget to state the PEMDAS rule? It says that in expressions involving more than one operation, then except where Parentheses dictate otherwise, the operations are to be performed in the following order: Exponentiation, Multiplication, Division, Addition, and Subtraction. (A once-common mnemonic for this was “Please Excuse My Dear Aunt Sally”, but that was considered dorky even in the century I grew up in.)
One reason I dislike discussions of PEMDAS is that they’re too often based on the unstated assumption that PEMDAS is holy writ. It’s just a convention, folks! But what I really hate are the memes that suggest that if you think 1 plus 2 times 3 is 9 rather than 7, you don’t know math. Those memes are mean-spirited; they make some people think that they’re bad at math when they’ve merely failed to instantly solve a trickily-worded problem that was designed to trip people up.
I also don’t like the fact that this kind of social media content displaces content about actual mathematics, and leaves too many people with the impression that math is about conventions like PEMDAS rather than deep mysteries like the Riemann Hypothesis.
But what I really really hate is when PEMDAS gets applied to oral mathematics (a kind of mathematics that existed many centuries before PEMDAS existed). If you’re a competitor on a quiz show and the host asks “What is the value of one plus two . . . times three?”, leering at you in a mean-spirited way during the pause, the mathematically correct answer is “You’re being a jerk.”
But PEMDAS is a pretty good rule, because it reduces the need for lots of infernal stupid parentheses.4 To see why it’s good to have multiplication and division take precedence over addition and subtraction, or at least see why it’s good to have multiplication take precedence over addition and subtraction, it’s helpful to think about polynomials (a topic I wrote about in “Let x Equal x” and “What Lovelace Did: From Bombelli to Bernoulli to Babbage”) and the way they’re naturally expressed as sums of products (but not always expressible as products of sums).
One way to think about polynomials is as a way of bringing order to the confusion that ensues when you allow x to enter arithmetic on the same footing as ordinary numbers and you create new entities from old using addition, subtraction, and multiplication. Suddenly you’ve got things like (x + 3) × (x + 7) to contend with. Oh, if only there were a one-size-fits-all form in which to write such expressions! The good news is that there is such a standard form: any expression that you can obtain by combining numbers with the variable x using addition and multiplication (and what the heck, let’s throw in subtraction too) can be written in the form ax+b or ax2+bx+c or ax3+bx2+cx+d or ax4+bx3+cx2+dx+e or … well, you get the idea. Such expressions are called polynomials in the variable x.5 Now, you may choose to write ax2+bx+c as axx+bx+c like Descartes, or even put in those tacit multiplication signs and write a×x×x+b×x+c,6 but you won’t need any parentheses when you write such a polynomial, because it’s already expressed as a sum of products (along with things that are even simpler than products, such as numbers and x itself). But how would you write x2+1 as a product of sums? Or x5+x+1?7
Putting it aphoristically (hello, Francis Bacon): products are good building blocks when the cement is addition but sums are bad building blocks when the cement is multiplication. And if you trace this asymmetry between addition and multiplication back to its source, you’ll find that it comes down to the fact that, even though multiplication is distributive over addition, addition is not distributive over multiplication. That is, it is a truth universally acknowledged that (a+b)×(c+d) equals (a×c)+(a×d)+(b×c)+(b×d), but if you replace × by + and vice versa and try to assert that that (a×b)+(c×d) equals (a+c)×(a+d)×(b+c)×(b+d), then you’ve left the golden realm of the always-true and stumbled into the swamplands of the nearly-always-false.
IMMUTABLE LAWS?
The asymmetry between multiplication and addition (the former is distributive over the latter but the latter doesn’t return the compliment) is the immutable bedrock on which PEMDAS rests, and explains why, even though the rule is called BOMDAS in England8 and undoubtedly has other names in other places, nobody gives addition default-precedence over multiplication. That’s the thing about conventions: they may be partly arbitrary but they’re also grounded in necessity. (When it comes to two-way two-lane roads, there are countries where cars and trucks drive on the right side and countries where cars and trucks drive on the left side, but no countries where cars drive on one side and trucks drive on the other.)
Is the distributive law an immutable rule or a human convention? It’s a bit of both. In the context of the counting numbers, the distributive property is a theorem, provable from more basic axioms about how counting numbers behave. In the context of other, more capacious number systems, the distributive law is technically a theorem because it can be derived from the axioms and definitions that govern those number systems, but it’s equally true to say that those axioms and definitions were chosen in part because they played nicely with the distributive property. Even before math had much use for multiplying one negative number by another, preference was given to the convention that the product of two negative numbers is a positive number, in part because it’s the only choice that’s fully consistent with the distributive law (if we want a+b times c+d to equal ac+ad+bc+bd without exception, then we need −1 times −1 to equal +1).9 Hamilton was inventing the quaternions (see “Hamilton’s Quaternions, or, The Trouble with Triples”), and he saw that the distributive law, the commutative law for multiplication, and the law of moduli10 were inconsistent with one another, he chose to scrap the commutative law. This is an instance of what 19th century mathematician George Peacock dubbed the Principle of Permanence of Form, which essentially says that when you’re designing a new number system, you should retain as many of the algebraic properties of the old system as you can.
But some rules get jettisoned, and these are typically rules that decree that a certain expression is nonsensical or that a certain procedure is impossible. Nonsense may become sense if one moves to an appropriate context, and the impossible may become possible if one relaxes an unstated assumption. You say I can’t subtract a greater number from a lesser? Watch me invent negative numbers. You say I can’t divide by 0? Hold my beer while I invent the projectively extended real numbers (see my essay “Dividing by Zero”). You say it just doesn’t make sense to sum the geometric series 1+2+4+8+…? I’ll hold my own beer in one hand while while I invent p-adic numbers with the other and then I’ll shout “It does now!” (see my essay “Marvelous Arithmetics of Distance”). I don’t mean to suggest that these advances were made with transgressive intent; the pioneers who gave us these new systems weren’t rebels. But they also weren’t the kind of people who unthinkingly obeyed DO NOT TRESPASS signs in the mathematical realm.11
So my parting advice to you (but don’t call it a rule!) is, Don’t give undue reverence to school-rules. Some are mere conventions. Others are sneakily context-dependent. And often a rule that begins with “You can’t” turns out to be an oversimplified statement of a theorem that says “You can’t . . . unless . . . ”, where figuring out what comes after “unless” is at the heart of mathematics as a creative enterprise.
Thanks to Jeremy Cote, Sandi Gubin, David Jacobi, Joseph Malkevitch, and Evan Romer.
This essay is a supplement to chapter 6 of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds cool and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!
ENDNOTES
#1. In between rhetorical algebra and symbolic algebra came “syncopated algebra”, which was less jazzy than it sounds.
#2. Bacon’s famous aphorism “Knowledge itself is power”, usually abridged to “Knowledge is power”, is clear enough, but some of his other terse sentences are more enigmatic. I can’t resist pointing the reader to a fun story of how the terse spoken utterance “‘Knowledge is power.’ – Francis Bacon” misled young John Barber: https://thehabit.co/knowledge-is-power-france-is-bacon/
#3. It emerged in later centuries that to solve solve many (indeed most) algebraic equations you need tools from outside algebra, but that’s another story.
#4. The bizarre ending of that sentence was a shout-out to the LISP programming language which has so many parentheses in its structure that an old joke held that the name of the language was an acronym for Lots of Infernal Stupid Parentheses.
#5. If we allow division as well, then the one-size-fits-all form for everything you can build from x is P/Q where P and Q are polynomials in x. Such expressions are called rational functions of x.
#6. It was precisely because of the possibility of confusing x with × that Leibniz suggested early on that “×” was not such a good choice for representing multiplication; he preferred “·”, which you’ll sometimes see even today. I’m not sure who first pointed out the problem with using superscripts to denote both exponents and footnotes, though.
#7. If you’re willing to use complex numbers you can write x2+1 as (x+i)(x−i), and you can write x5+x+1 as (x+a)(x+b)(x+c)(x+d)(x+e) but a, b, c, d, and e will be nasty numbers that you can’t express using the ordinary operations of algebra even if you allow taking square roots, cube roots, etc. This is related to what I alluded to but declined to discuss further in footnote 3. And if you throw in a second variable y, then even complex numbers won’t save you; most two-variable polynomials simply cannot be factored (that is, written as products of sums) in any sense. On the other hand, the two-variable expression x2y2 + 2x2y + 3xy2 + 5xy can be written as a sum of products, because hey, it’s already written that way.
#8. B is for Brackets, but I haven’t been able to find out how the O got in there. Some say it stands for Order, but I prefer to think that it stands for Of, just as the A and I in TARDIS stand for And and In.
#9. If we plug a = 1, b = −1, c = 1, and d = −1 into (a + b)(c + d) = ac + ad + bc + bd we get (1 + −1)(1 + −1) = (1)(1) + (1)(−1) + (−1)(1) + (−1)(−1). The left hand side is (0)(0), which is 0, so we want the right hand side to be zero as well; and if we’ve already accepted that (1)(1) = 1 and (1)(−1) = −1 and (−1)(1) = 1 we have no choice but to set (−1)(−1) equal to 1. For more on the issue than you’ll probably want to read, see my essays Going Negative, part 1, Going Negative, part 2, Going Negative, part 3, and Going Negative, part 4.
#10. The law of moduli was the assertion that the modulus of the product of two quaternions should equal the product of the moduli of the two quaternions considered separately. Hamilton wisely recognized that the law of moduli for complex numbers was one of the most profound properties complex numbers satisfied, so he was determined to have his new hypercomplex numbers satisfy a law of moduli as well, even before he knew what sort of thing his new numbers were or gave them their name.
#11. Georg Cantor created the most radical mathematics of the late nineteenth century, but he wasn’t a rebel by temperament; at the start of his career he just wanted to understand subsets of the real numbers. When his work led him to infinite ordinals that broke the commutative law for addition, the commutative law for multiplication, and the distributive law, he accepted what the math was whispering to him.
REFERENCES
John McWhorter, Lets chill out about apostrophes (New York Times, May 16, 2024). https://www.nytimes.com/2024/05/16/opinion/lets-chill-out-about-apostrophes.html
Robert Moses, Radical Equations: Civil Rights from Mississippi to the Algebra Project, 2001.
Helena Pycior, “George Peacock and the British Origins of Symbolical Algebra”, Historia Mathematica 8 (1981), pp. 23–45, esp. 27–31.
Helena Pycior, “Internalism, Externalism, and Beyond: 19th-Century British Algebra”, Historia Mathematica 11 (1984), pp. 424–441, esp. 430.
Helena Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra through the Commentaries on Newton’s Universal Arithmetick, 1997.
St. Andrews Maths History, Earliest Uses of Symbols of Operation. https://mathshistory.st-andrews.ac.uk/Miller/mathsym/operation/
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