Order of operations

For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.

[11] Beyond primary education, the symbol '÷' for division is seldom used, but is replaced by the use of algebraic fractions,[12] typically written vertically with the numerator stacked above the denominator – which makes grouping explicit and unambiguous – but sometimes written inline using the slash or solidus symbol '/'.

In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is conventionally interpreted as having higher precedence than division, so that e.g. 1 / 2n is interpreted to mean 1 / (2 · n) rather than (1 / 2) · n.[2][10][14][15] For instance, the manuscript submission instructions for the Physical Review journals directly state that multiplication has precedence over division,[16] and this is also the convention observed in physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz[c] and mathematics textbooks such as Concrete Mathematics by Graham, Knuth, and Patashnik.

[15][19] Mathematics education researcher Hung-Hsi Wu points out that "one never gets a computation of this type in real life", and calls such contrived examples "a kind of Gotcha!

parlor game designed to trap an unsuspecting person by phrasing it in terms of a set of unreasonably convoluted rules.

For example, Microsoft Excel and computation programming language MATLAB evaluate a^b^c as (ab)c, but Google Search and Wolfram Alpha as a(bc).

Mnemonic acronyms are often taught in primary schools to help students remember the order of operations.

[20][21] The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction,[22] is common in the United States[23] and France.

[24] Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally".

[25] The United Kingdom and other Commonwealth countries may use BODMAS (or sometimes BOMDAS), standing for Brackets, Of, Division/Multiplication, Addition/Subtraction, with "of" meaning fraction multiplication.

[26][27] Sometimes the O is instead expanded as Order, meaning exponent or root,[27][28] or replaced by I for Indices in the alternative mnemonic BIDMAS.

[25] For example, misinterpreting any of the above rules to mean "addition first, subtraction afterward" would incorrectly evaluate the expression[25]

Mnemonic acronyms have been criticized for not developing a conceptual understanding of the order of operations, and not addressing student questions about its purpose or flexibility.

[31][32] Students learning the order of operations via mnemonic acronyms routinely make mistakes,[33] as do some pre-service teachers.

[34] Even when students correctly learn the acronym, a disproportionate focus on memorization of trivia crowds out substantive mathematical content.

[25][22] Most programming languages use precedence levels that conform to the order commonly used in mathematics,[41] though others, such as APL, Smalltalk, Occam and Mary, have no operator precedence rules (in APL, evaluation is strictly right to left; in Smalltalk, it is strictly left to right).

The accuracy of software developer knowledge about binary operator precedence has been found to closely follow their frequency of occurrence in source code.

The rule that multiplication has precedence over addition was incorporated into the development of algebraic notation in the 1600s, since the distributive property implies this as a natural hierarchy.

As recently as the 1920s, the historian of mathematics Florian Cajori identifies disagreement about whether multiplication should have precedence over division, or whether they should be treated equally.

The term "order of operations" and the "PEMDAS/BEDMAS" mnemonics were formalized only in the late 19th or early 20th century, as demand for standardized textbooks grew.

However, while Chrystal's book initially establishes a rigid rule for evaluating expressions involving '÷' and '×' symbols, it later consistently gives implicit multiplication higher precedence than division when writing inline fractions, without ever explicitly discussing the discrepancy between formal rule and common practice.