17

The internet has been so bored lately that it spent most of the week arguing about a simple math statement. That statement:

 8 ÷ 2(2+2) = ?

Mostly, people are arguing that the answer is either 16 or 1. Depending on the order you calculate the operations, you can see how both answers are attained.

This popular YouTube video explains the answer is 16. The part I cannot accept is that he claims the answer was 1 a hundred years ago. He claims a difference in understanding syntax due to limited type settings at the time are to blame. Now that we have more robust type settings, we understand the statement differently. Here's the quote starting from about 1 minute, with a few sentences skipped and my emphasis added:

This [16] is without a doubt the correct answer according to the modern interpretation of the order of operations. But let's do a little thought experiment. Let's travel back in time 100 years.

I actually found an academic paper discussing this very ambiguity in 1917. How would they solve this problem? ...

At that time mathematical typesetting was limited, so if you had something like X divided by the quantity 2Y, you would understand it as X over the quantity 2Y. ... That's because it would be hard to write this expression [x over 2y] in the limited type setting of the time. ...

This [Using the "old" understanding] would equal 1. This is not the correct answer today.

I have a very hard time believing that the standard has changed. Rather, I think there's a lot of people who don't know how to do it correctly. If you look around, you will find that even calculators don't agree. Common calculators seem to give 16, while more robust scientific calculators give 1. This leaves many people uncertain. It leaves me remembering from my school days that you often can't just put statements as written into the calculator, because calculators calculate, not solve.

But maybe I'm way off here and there is a "modern interpretation of the order of operations" that I am unaware of. The video cites a 1917 paper supporting the 1 answer. I want to see modern papers supporting the 16 answer and specifically arguing for a change.

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[I edited this answer in response to the helpful comments in the chats. Thanks to everyone for their feedback!]

0. Introduction

TLDR: No, none of the order of evaluation conventions changed since the 1600s. As detailed below, the only change during this period was to switch from "overbar" to parentheses to denote grouping. Writers sometimes deviated from or argued about the conventions. I don't see that the disagreements got more intense (the 1917 letter cited below is as intense as any Facebook flame war), or people who disagree with the convention lost any ground (no evidence that children were taught other conventions in the past any more often than now).

In the example "8÷2(2+2)", everyone seems to agree that the addition inside the parentheses is done first; and that a multiplication is implied between "2" and "(". But, which operation should we perform next

  • multuply 2 by 4, then divide 8 by their product, or

  • divide 8 by 2, then multiply their quotient by 4?

According to the algebraic conventions that most people learn in elementary school, we should perform the lefmost operation first, without ambiguity. But so many people are confused by or disagree with the conventions that it's really not clear what the writer intended, except to confuse the reader.

A good strategy to address this question and the surrounding confusion is to present a lot of historical context and links to primary sources.

Generally, Western Europeans agreed by the 17th century on the same conventions for precedence and associativity that we still use, although they have already been mostly using them for a couple of centuries prior.

For example, Google Books has Francisci Vietae Opera mathematica (collected works of François Viète printed in Latin by Frans van Schooten in 1646) and Teutsche Algebra by Johann H. Rahn (printed in 1659 in German). The language and the typeface may look unfamiliar, but the formulas look modern.

Europeans used to use "vinculum" - a horizontal bar above an expression - before they adopted parentheses to indicate grouping. (Some web pages say that it is a vertical bar, but it is not.) This was the only material change in the conventions since the 1600s. The vinculum still survives in the radical $\sqrt{.}$ ; in the horizontal bar in $\frac{.}{.}$ ; and in the obelus ÷.

The symbols ×, asterisk 2*x, raised/middle dot x·2, or juxtaposition 2x or 2 x denote the same operation: multiplication. The solidus 1/2, obelus 1÷2, fraction slash ¹⁄2, or fraction line $\frac{1}{2}$ all denote the same operation: division, multiplication's inverse: a÷b=ab^(-1). They have the same precedence and other semantics.

Making the conventions more complicated, e.g. using different precedence for the same operation written differently, would only add confusion and no value. No one wants that except perhaps some math tutors (2.6 below).

Readers immediately noted (even the Teutsche Algebra alludes to it) that following an inline division with another division a/b/c or multiplication a/bc, which is the point of your absurdly contrived Facebook puzzle, may be hard to read and confusing. According to common conventions, "1 ÷ 2x" is not ambiguous. It can only mean (1÷2)×x" and not "1÷(2×x)". But students are taught, as a matter of good writing style, to use grouping (redundant parentheses in modern use) or display-style fraction bar:

1

—×x.

2

The stylistic guidance has been found literally for centuries in countless style guides and standards (SI, ISO, DIN...).

Students should also know that if they encounter 1÷2x, then it means (1÷2)×x according to the rules, but the writer might have intended 1÷(2×x).

I've seen three reasons offered as to why the multiplication might go before the division in your puzzle:

  • multiplication and division go from right to left, not from left to right;

  • Multiplication always goes before division;

  • Multiplication written as juxtaposition and/or division written as ÷ go before/after other multiplications/divisions.

These rules don't contradict each other. I will address each in turn.

1. Associativity

For operations such as subtraction and division, the associativity matters, and must be agreed on as a matter of convention. Because (a-b)-c is not the same as a-(b-c) for almost all a', b's, c's, as soon as people agreed that certain symbols would denote division and subtraction, they also were forced to agree on their associativity convention.

By convention, "-" and "÷" are left-associative: a-b-c means a+(-b)+(-c) which is (a-b)-c, and not a-(b-c) which is a-b+c. Likewise a÷b÷c means $(a÷b)÷c=ab^{-1}c^{-1}=a÷(bc)$ and not a÷bc which is (a÷b)c. In contrast, exponentiation is right-associative: 2^2^2 means 2^(2^2), not (2^2)^2.

Such conventions are arbitrary, somewhat accidental, are neither theories nor techniques, can be collectively invented, debated, revised, and agreed to, cannot be proven, debunked, or discovered. Base 10 (so 9+1 is 10) and writing the most significant digit on the left are similar conventions. It's not a math issue, but a language/notation issue.

There is nothing inherently correct about left-associativity. The Europeans who used the algebraic notation agreed that "a-b-c" would mean "(a-b)-c". We can hypothesize that they chose left associativity because Latin and Greek scripts are written left-to-right. Recall that medieval Europeans adopted Arabic/Indian numerals from the Islamic/Arabic civilization, whose script is written right-to-left, but who adopted the numerals in turn from India, whose Devanagari-based scripts are written left-to-right. So American, European, and Indian students, reading left-to-right, encounter the most significant digit first and the least significant digit last; but Arab (and Israeli) students, reading right-to-left, encounter the least significant digit first and the most significant digit last, which sometimes causes mild confusion. We can guess that if, in some alternative history, medieval Arabs got to choose the associativity convention for subtraction and division, they likely would have chosen right associativity (as well as flip the order of decimal digits).

In the computer language APL, all operators had the same precedence and were right-associative: a÷b-c meant a÷(b-c).

2. Precedence

2.1 PEMDAS mnemonic

For the precedence, most American middle school children (5th grade, according to the U.S. Common Core curriculum; earlier in other countries) are taught the mnemonic rule PEMDAS ("please excuse my dear aunt Sally" / "please eat mom's delicious apple strudels" / "people eat more donuts after school"), which stands for

  • Parentheses (called "brackets" in some other countries)

  • Exponents (powers and roots, also called "orders" or "indices" in some other countries)

  • Multiplication and (its inverse) division having the same precedence

  • Addition and (its inverse) subtraction having the same precedence

PEMDAS can be written as PEDMSA or PEDMAS. In UK and Canada, they sometimes call this rule BOMDAS or BIDMAS. In French-speaking countries, PEMDAS stands for parenthèses; exposants; multiplications et divisions; additions et soustractions.

Danica McKellar suggested the mnemonic "Pandas Eat: Mustard on Dumplings, and Apples with Spice" to emphasize that M and D have the same precedence, as do A and S.

I cannot find the earliest reference to PEMDAS, but here is a 1945 book that cites "the rule of signs (BODMAS)".

2.2 PEMDAS alternatives

Many web pages argue against PEMDAS. For example, an Excel trainer writes:

PEMDAS and the Mathematical Order of Operations with Microsoft Excel... Example of PEMDAS: = 2 + 3 * 4 Most people would read that formula from left to right and calculate 2+3 equals 5 and 5 times 4 equals 20, and it wouldn’t be entirely wrong. However, because multiplication has a higher order of precedence than addition, Excel will first calculate 3*4 equals 12 and 12+2 equals 14.

I hope that not many people think that 2+3×4 means (2+3)×4.

Some assert that "MD" means that all multiplications must be done before all divisions. For example, in a similar discussion on a physics forum, someone wrote:

I have a friend who argues for CPMD (commutative property of multiplication over division) when using PEMDAS. Basically, left to right, you solve for multiplication first, then go back and do division. PEMDAS came into being around 1986 and argues that multiplication and division hold the same priority, so for the famous equation 6/2(2+1), pre-1986 as he was taught you'd get 1; post 1986 using PEMDAS as I was taught, you'd get 9.

No, before 1986 students were not commonly taught that multiplication had higher precedence than division. The "famous equation" is not an equation (contains no = sign). There's no "CPMD".

2.3 Lennes 1917 Leter to American Math Monthly

I found a letter entitled Discussions: Relating to the Order of Operations in Algebra by Prof. N.J.Lennes from University of Montana in the Questions and Discussions section of the February 1917 issue of American Mathematical Monthly (probably still © Mathematical Association of America). (Lennes was also the second author of the 1907 Slaught algebra textbook, cited below.) Lennes is forceful and uses a lot of italics for emphasis:

Following this theoretical development most of the current text-books give a rule like the following:

A series of operations involving multiplication and dimsion alone shall be performed in the order in which they occur from left to right.

§ 2. Actual Usage.

The Above Rule Contrary to Actual Usage. The rule stated above is agreed to by practically all those writers on algebra who make any mention of the matter at all. Chrystal gives a detailed development and writers on elementary algebra have in general followed him. It would, however, follow from this rule for carrying out multiplications and divisions in order from left to right, that

$$9a^2 ÷ 3a = (9a^2 ÷ 3) × a = 3a^3.$$

But I have not been able to find a single instance where this is so interpreted. The fact is that the rule requiring the operations of multiplication and division to be carried out from left to right in all cases, is not followed by anyone. For example, in case an indicated product follows the sign ÷ the whole product is always used as divisor, except in the theoretical statement of the case.

Writers meet the situation in different ways:

(a) Some always use the fractional form to indicate division, this being equivalent to a symbol of aggregation. Thus, $ab/cd = (ab) ÷ (cd)$.

(b) Some write out the words in full, thus: “divide this expression by that expression.”

(c) Some use the sign ÷ to mean that the whole product, following the sign ÷, shall be the divisor.

The most important exception to (c) occurs in the development of the theory in such a text as Chrystal. In Chrystal, $÷ u × v$ is sometimes used to mean $(÷ u) v$. In such cases, however, the notation $÷ u × v$ and not $÷ uv$ is used.

Chrystal in one case writes $2^2 ÷ 3^2 × 5^2 = (2^2 ÷ 3^2) × 5^2$. (Note the sign × to indicate multiplication.) He also writes $pa/pb = pa ÷ (pb)$. (Note the parenthesis.) This comes nearer consistency than is usually the case. However, in no case does Chrystal write $9a^2 ÷ 3a$ as the equivalent of $(9a^2 ÷ 3) × a$. He overcomes the difficulty by never using the sign ÷ with a product after it.

The followers of Chrystal have too often blindly copied his theory, but have not taken pains, as he did, to avoid inconsistency in usage. Examples of such inconsistency in theory and usage could be multiplied ad infinitum. One text, which has been in very wide use, states (in developing the theory)

$$60 - 40 ÷ 5 × 3 - 20 = 60 - \frac{40}{5} - 20,$$

but on the next page we read:

$$10bc ÷ 12a = \frac{10bc}{12a}.$$

The Established Usage. When an indicated product follows the sign ÷ the whole product is, by overwhelming preponderance of actual usage, to be regarded as the divisor. Hence, thé true rule as to the order of operations when both multiplications and divisions are involved is not the one stated above, but the following:

All multiplications are to be performed first and the divisions next.

That is, $9a^2 ÷ 3a = 3a$ and not $3a^3$.

The multiplications may be taken in any order, but the divisions are to be taken an the order in which they occur from left to right.

That is, the associative law holds for the former but not for the latter.

Thus, $3 × 5 × 2= (3 × 5) × 2$ or $= 3 × (5 × 2)$; but, $16 ÷ 4 ÷ 2 = (16 ÷ 4) ÷ 2$ and does not $= 16 ÷ (4 ÷ 2)$.

Compare the corresponding rules for addition and subtraction in § 1.

Mathematical Idioms. It might be agreed that, for the sake of simplicity and logical coherence, the past tense of the verb to drink should be drinked, but even so, English speaking people would continue to say drank, and not drinked. Precisely, for the same reason, all who know anything about the language of algebra regard $9a^2 ÷ 3a$ as equal to $3a$ and not $3a^3$, and, therefore, the rule just given is the correct one as determined by actual usage. When a mode of expression has become wide-spread, one may not change it at will. It is the business of the lexicographer and grammarian to record, not what he may think an expression should mean (no matter how far-fetched the usual or idiomatic usage may seem), but what it is actually understood to mean by those who use it. The language of algebra contains certain idioms and in formulating the grammar of this language we must note them. For example, that $9a^2 ÷ 3a$ is understood to mean $3a$ and not $3a^3$ is such an idiom. The matter is not logical but historical.

Lennes proposes to give multiplication higher precedence than division always, even when explicitly indicated, so 2÷2×2 would mean 2÷(2×2) rather than (2÷2)×2, because he thinks that 2-2+2 means 2-(2+2) rather than (2-2)+2: corresponding rules for addition and subtraction - he writes. Not surprisingly, the proposed rule did not make it into any books. Lennes is mistaken about "the established usage" (except perhaps among the unlucky students that he himself taught). He cites one source - Chrystal, which, Lennes admits, follows the centuries-old conventions, rather than the new rule proposed by Lennes. What "one text, which has been in very wide use" is Lennes criticizing?

I found Chrystal's text on Google books: Introduction to Algebra: For the Use of Secondary Schools and Technical Colleges by George Chrystal, London, 1898. On page 6, Chrystal mentions the vinculum (overbar) as a viable alternative to parentheses, somewhat anachronistic for a 19th century text. This book is very confusingly written (I fully agree with Lennes' criticisms). On page 7, Chrystal seems to suggest that a÷bc might be mis-interpreted as a÷(bc) but is not clear. Lennes asserts in his point (c) that some other writers (whom he does not name) interpret a÷bc as a÷(bc) but that Chrystal interprets a÷bc as (a÷b)×c.

2.5 Florian Cajori

Florian Cajori wrote several books about the history of math teaching. In A History of Mathematical Notations, 1928, v. 1, p. 274, Cajori writes:

Order of operations in terms contains both ÷ and ×.-If an arithmetical or algebraical term contains ÷ and ×, there is at present no agreement as to which sign shall be used first. "It is best to avoid such expressions" [and he cites a 1833 French book as the source of this excellent advice]. For instance, if in 24 ÷ 4 × 2 the signs are used as they occur in the order from left to right, the answer is 12; if the sign × is used first, the answer is 3.

Some authors follow the rule that the multiplication and division shall be taken in the order in which they occur [citing Luby and Touton, 1910, reference below]. Other textbook writers direct that multiplications in any order be performed first, then divisions as they occur from left to right [citing Slaught and Lennes, 1907, reference below]. The term a ÷ b × b is interpreted by Fisher and Schwatt [reference below] as (a ÷ b) × b. An English committee recommends the use of brackets to avoid ambiguity in such cases.

The old textbooks can be found in Google Books:

But Cajori is mistaken: Slaught and Lennes actually wrote on page 30:

In operations involving parentheses the expressions within the parentheses should be performed first if possible. Then perform the indicated multiplications and divisions, and finally the remaining additions and subtractions.

Lennes proposed this rule change in the 1917 letter, but I don't see the proposed rule in the Slaught 1907 textbook (10 years earlier).

The "English committee" is a reference to a Mahematical Gazette article that I could not find online.

2.5 Jeff Miller's web page

I found a mostly excellent page by Jeff Miller, which says:

The convention that multiplication precedes addition and subtraction was in use in the earliest books employing symbolic algebra in the 16th century. The convention that exponentiation precedes multiplication was used in the earliest books in which exponents appeared...

In 1898 in Text-Book of Algebra by G. E. Fisher and I. J. Schwatt, a÷b×b is interpreted as (a÷b)×b.

In 1907 in High School Algebra, Elementary Course by Slaught and Lennes, it is recommended that multiplications in any order be performed first, then divisions as they occur from left to right.

In 1910 in First Course of Algebra by Hawkes, Luby, and Touton, the authors write that ÷ and × should be taken in the order in which they occur.

In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: "Indicated operations are to be performed in the following order: first, all multiplications and divisions in their order from left to right; then all additions and subtractions from left to right."

In 1913, Second Course in Algebra by Webster Wells and Walter W. Hart has: "Order of operations. In a sequence of the fundamental operations on numbers, it is agreed that operations under radical signs or within symbols of grouping shall be performed before all others; that, otherwise, all multiplications and divisions shall be performed first, proceeding from left to right, and afterwards all additions and subtractions, proceeding again from left to right."

Jeff Miller cites the same textbooks as Cajori and also Wells and Hart; and repeats Cajori error regarding Slaught and Lennes.

Several Web pages point to Jeff Miller's page and claim that he found an old book that said that multiplications had higher precedence than divisions, but the Slaught book does not say that, and they cannot point to any other books that say this.

2.6 Tutors and Unaries

PEMDAS is often called "wrong", "misleading", "oversimplified", "fuzzy", and "inconsistent" in online ads by math tutors seeking to scare parents to hire someone to "teach" kids altertative rules, not found in textbooks, "more sophisticated" than PEMDAS.

Indeed, PEMDAS does not specify associativity or the precedence of functions (like trig) or unary negation. E.g., does "-1^n" mean or "-(1^n)" or "(-1)^n"? By convention -1^n means -(1^n); so you must write (-1)^n if that's what you really mean. sin x^2 usually means sin (x^2), while sin^2 x means (sin x)^2, but these are not usually taught right away. (sin ax/2 or sin x(y-pi)+z really need parentheses.) Children do learn additional algebra conventions (none of which contradict PEDMAS) in later grades, just as they learn first to subtract for non-negative results, and later how to subtract a larger number from a smaller one; learn square roots of non-negative numbers first, and square roots of negative numbers later.

3. Juxtaposition

Finally, we come to the juxtaposition part. We recall that some 8 years earlier someone asked on Math exchange: What is “multiplication by juxtaposition”? and quoted:

The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations

To which the moderator The Chaz 2.0 replied:

This question is more about how we deal with trolling and nuisances.

Read this page: http://knowyourmeme.com/memes/48293

Then maybe we can start a meta article about identifying and dealing with these threads.

Between the math forums that I moderate and otherwise frequent, and dozens of other forums

(a short list is here: http://www.mymathforum.com/viewtopic.php?f=13&t=20148&p=79150#p79150),

I'd guess that thousands of hours have been wasted on this garbage.

and closed the question. Perhaps he didn't like the blatantly false "general consensus" assertion.

The Lennes letter proposes always giving multiplication higher precedence than division, even when explicitly specified, not only when implied by juxtaposition.

When Microsoft Excel encounters juxtaposition in a formula, it says: "Microsoft Excel found an error in your formula you entered. Do you want to accept the correction proposed below?" and proposes to insert a multiplication symbol "*".

Wolfram's Mathematica is one of the few computer systems that can assume that juxtaposition means multiplication. We observe that http://www.wolframalpha.com/input/?i=1%2F2x outputs x/2, but, for some examples, also warns: "Assuming multiplication". Mathematica correctly rewrites "a/b c" and "a/b×c" into "ac/b". Some people get confused by "a/bc", because "bc" is a single token, not product b×c.

WeBWorK interprets juxtaposition ("e.g. 2x, 2 x or 2*x, also 2(3+4)") as multiplication, with the same precedence.

In the derivative dy/dx (a quotient of two differentials), "dx" is not the product "d×x", but (more or less) unary d (or ∂) applied to x.

An expression like $\Bbb Z/n\Bbb Z$ can stupefy someone ingorant of the math, because the symbols look like division and multiplication of numbers, but are not. "Blackboard bold" $\Bbb Z$ denotes a group (of integers under addition), not a number. $n\Bbb Z$ is neither n×Z nor n·Z, but its normal subgroup consisting of multiples of n. The / (pronounced "mod") denotes the "factor" (or "quotient") group (not denoted ÷), here the cyclic group of order n. (And $\Bbb Z×\bbbZ$ and $\Bbb Z*\bbbZ$ are not products of two numbers either.)

3.1 American Mathematical Society

The source of the American Mathematical Society urban legend appears to be this example from an early version of the Guide to Reviewers, archived here, that says:

Formulas. You can help us to reduce production and printing costs by avoiding excessive or unnecessary quotation of complicated formulas. We linearize simple formulas, using the rule that multiplication indicated by juxtaposition is carried out before division. For example, your TeX-coded display

$${1\over{2\pi i}}\int_\Gamma {f(t)\over (t-z)}dt$$

is likely to be converted to

$(1/2\pi i)\int_\Gamma f(t)(t-z)^{-1}dt$

in our production process.

This was written by a confused typesetting person, not by a math person. Replacing display

1

——

by inline 1/2π would not be acceptable to most AMS authors (bad style and changes the meaning). AMS fixed this in later versions. I find nothing on the current version of the AMS web site to endorse such rewriting.

3.2 Physics idiom

Some physics people write the energy kT or RT as an inseparable term, no longer a product of the Bolzmann's constant $k_B$ (or ideal gas constant R = k × Avogadro's constant $N_A$) and the Kelvin temperature T. For example, this Wikipedia page https://en.wikipedia.org/wiki/Chemistry#Energy (3rd paragraph) uses "E/kT". This Wikipedia page https://en.wikipedia.org/wiki/Activation_energy#Relationship_with_Gibbs_energy_of_activation uses "/RT" a few times. The Feynman Lectures on Physics uses "/kT". Course of Theoretical Physics by Landau & Lifshitz omits the k (the reader has to know that T really means kT) but does write ")/2mT". Some write c/λT, where λ is wavelength. This means to multiply by the temperature but (some physics people say smugly) this makes no physical sense, so the writer meant "/(kT)".

Some non-math statistics books write the exponent in standard deviation as -(x-μ)^2/2σ^2, meaning they multiply by σ^2.

3.3 Stanford book

I'm aware of only one book that endorses the confusing notation, and it is a computer science book, not a math book: Concrete Mathematics by Graham, Knuth, and Patashnik says at the beginning of "a note on notation": "some of the symbolism in this book has not (yet?) become standard" (that's an understatement) and later says at the very bottom of "a note on notation" on page xi: "An expression of the form 'a/bc' means the same as 'a/(bc)'".

Knuth, Landau, or Feynman are no fools, but they would have had points taken off in an elementary school algebra class. I hope someone corrects future editions of their books.

3.4 Buggy calculators

Some electronic calculators are known to use the "juxtaposition" logic. For example, this photo: x

shows two Casio calculators. The one on the right is unambiguously right; the one on the left (fx-570MS) incorrectly gives higher precedence to "multiplication implied by juxtaposition" or "abbreviated" and is just buggy. Some old Texas Instruments models had this bug, but it was corrected in all models since 1996 (TI FAQ). Some (not all) old Casio models used extremely complicated rules (Casio User's Guide, pages 34-35).

Likewise, some buggy old calculators implemented exponentiation as left-associative instead of right-associative: a^b^c=(wrong!)(a^b)^c=a^(b*c).

These calculator bugs were never intentional. The early solid-state calculator engineers (late 1950s) simply did not know yet how to parse algebraic formulas into Polish notation / trees according to common mathematical conventions. Similar problems vexed the developers of the early FORTRAN compilers (also late 1950s) and were soon mostly resolved (e.g., Backus 1959), but the new techniques did not trickle down to calculators for years.

4. Conclusions

To summarize, I've encountered many people whose math teachers who did not do a good job - who held believes about math that were factually wrong. For example, a while back I wrote on Quora Why is American math education so abysmal? and mentioned how children were taught that it isn't a set unless it's surrounded by {}s. Another example is writing an equal sign for "therefore" or "if and only if": many American children were taught to write expressions like "x-1 = 2 = x = 3" and let the reader figure out what the middle "=" means, rather than "x-1 = 2 $\leadsto$ x = 3" or, if they really must, "(x-1 = 2) = (x = 3)". I'm not aware of any books that endorse any of this.

The phrase "American history teacher" is ambiguous because "American" could be modifying "history" (a teacher of American history) or "teacher" (an American teacher of history). "American math teacher/tutor" might mean someone teaching unsound unwritten rules not found in textbooks - buyer beware.

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    Comments are not for extended discussion; this conversation has been moved to chat. – Sklivvz Aug 6 at 10:55
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    Please take comments to chat. – Oddthinking Sep 7 at 2:29
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    You seem to contradict yourself. You argue that 9a^2÷3a=3a, but then say that 6÷2(1+2)=9. In fact it equals 1 by actual usage, except for the fact that no mathematician would ever write that. If they did, they would mean 1. – Matt Samuel Sep 10 at 20:59
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    "So we're down to evaluating "8÷2×4"." No, the whole argument is whether or not 8÷2a is the same as 8÷2×a. – swbarnes2 Sep 11 at 22:17
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    This board doesn't set up interpret math formatting like others do,,, people.reed.edu/~iswanson/abstractalgebra.pdf page 3 shows how it's supposed to look like. – Dimitri Vulis Oct 9 at 20:17
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The answer to the title question is no for the simple reason that the (fuzzy, inconsistent) rules mentioned in Lennes's 1917 note are still in wide use today.

I think it's important to distinguish between elementary-school arithmetic notation and the notation used by real mathematicians, which is also used in grade-school algebra and calculus and which I'll call algebraic notation.

In elementary-school arithmetic notation, multiplication and division are always written × and ÷. In algebraic notation, multiplication is indicated by juxtaposition or · or × and division is indicated by a line (ideally a horizontal line, since that's unambiguous, but often a diagonal line). I have never seen ÷ used for division in modern algebraic notation, and since juxtaposition isn't used in elementary-school arithmetic, there is no modern notation in which you would ever see a string like 8 ÷ 2(2+2). The only place you see such things is in questions that are intended to provoke arguments on the Internet. But anyway...

There are more subtle, but important, differences between real mathematics and what's taught in elementary school.

First, algebra has operator precedence rules, but it doesn't have an order of operations. The precedence rules mean that 5 + 6 + 7 · 8 is equivalent to 5 + 6 + (7 · 8), but if you're evaluating that expression, it's perfectly fine to add 5 and 6 before you multiply 7 and 8. It's not fine to add 6 and 7 first, but that isn't because multiplications have to happen before additions; it's because 6 and 7 are not operands of the same operator, even though it sort of looks like they are when the parentheses are omitted.

There is likewise no rule that operations happen left to right. Algebraic fields have two associative binary operations, addition and multiplication, and two unary operations, additive inverse and multiplicative inverse. You'll notice that binary subtraction isn't on the list. The expression 10 - 9 - 8 is just shorthand for 10 + -9 + -8, and because addition is associative, you're free to start by adding -9 and -8 (or even 10 and -8, since it's also commutative). Mathematicians don't talk about properties of subtraction, such as non-associativity or "left-associativity", because subtraction just isn't a thing in mathematics. The real operations are addition and additive inverse.

Binary - is at least an infix operator like + following well defined rules. But the division slash / (which could also be written ÷, though it isn't these days) is not an infix operator at all. It is just a sort of typographically rotated version of the horizontal division line, for cases where vertical space is at a premium. If you're lucky, the typesetter will put the numerator and denominator at different heights so there is no ambiguity – for example, 1/2·x can only mean (1/2)·x, and 1/2·x can only mean 1/(2·x) – or they will make liberal use of parentheses, as I just did. But often enough they don't do that, and you just have to guess, or know from experience, what the intended meaning is.

It's hard to find examples because you can't really google for them. I arbitrarily decided to look at some papers (co)written by Terence Tao, because he's one of the most respected and lauded active mathematicians so he makes a good authority.

  • In The primes contain arbitrarily long arithmetic progressions (with Ben Green; a famous paper proving what's often called the Green-Tao theorem), the notation ℤ/Nℤ appears. This is a very common notation; it means ℤ/(Nℤ), but you'll never see it with parentheses because anyone who knows enough math to understand it with parentheses also knows what it means without. This may not be a great example though because the / isn't ordinary division and it's never (in my experience) written as a horizontal line.

  • In the same paper, expressions like N−1/2+ε appear; this means N(−1/2)+ε.

  • In the same paper, on page 36, the expression 1/2k(k+4)! appears. I think this means 1/(2k((k+4)!)).

  • In the same paper, on page 46, the expression 1/6m appears. I'm not certain what this means but my guess is that it's 1/(6m).

  • In The asymptotic distribution of a single eigenvalue gap of a Wigner matrix, the formula log n/2π² appears a few times; I think this means log (n/(2π²)).

I would have liked to find examples where 1/2x means (1/2)x and where 1/1+x means 1/(1+x), because the real point I wanted to make is that the notation is ambiguous. But that's what I found in the amount of time I felt like spending on it.

What I found does support that what Lennes wrote about the real-world meaning of ÷ in 1917 is still true (of /) today. Nothing has fundamentally changed. This is also supported by my decades of experience reading mathematics papers, but I can't really cite that as a reference.

For some reason, the other answer to this question, by Dimitri Vulis, has taken a hardline position that a/bc means (a/b)c always and anyone who disagrees is a fool. That just isn't true. The reality is that a/bc usually, in papers by professional mathematicians, means a/(bc); while in elementary school arithmetic it means nothing at all.

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    Welcome to Skeptics! Your first seven paragraphs need some references to support your claims. – Oddthinking Sep 4 at 8:06
  • I strongly disapprove of Landau writing "T" whenever he means "kT"; or "1/2mT" when he might mean "1/(2mAT)". Such imperfect notation doesn't make Landau a fool, though. – Dimitri Vulis Sep 4 at 9:49
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    "The reality is that a/bc usually, in papers by professional mathematicians, means a/(bc); while in elementary school arithmetic it means nothing at all." But what would it mean in high school algebra? My memory says it's also a/(bc). But what does that have to do with something like 2/3(4) or 2/3×4, when the numbers are already given, like in the question? – fredsbend Sep 7 at 2:12
  • Great answer. The third paragraph should alone be able to settle the debate. – Cedric H. Sep 12 at 8:46
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    Just said it's a great answer then read the last paragraph again. Don't you think you're confusing "professional mathematicians" with "old papers with very poor typesetting"? – Cedric H. Sep 12 at 8:52

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