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Order of Operations ○｜Definition｜1st｜20251119205401-00-⌔
Order of operations - Wikipedia
Order of operations
In mathematics and computer programming, the order of operations is a collection of conventions about which arithmetic operations to perform first in order to evaluate a given mathematical expression.
These conventions are formalized with a ranking of the operations. The rank of an operation is called its precedence, and an operation with a higher precedence is performed before operations with lower precedence. Calculators generally perform operations with the same precedence from left to right,1 but some programming languages and calculators adopt different conventions.
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation.23 Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base.2 Thus 3 + 5 = 28 and 3 × 5 = 75.
These conventions exist to avoid notational ambiguity while allowing notation to remain brief.4 Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses () can be used. For example, (2 + 3) × 4 = 20 forces addition to precede multiplication, while (3 + 5) = 64 forces addition to precede exponentiation. If multiple pairs of parentheses are required in a mathematical expression (such as in the case of nested parentheses), the parentheses may be replaced by other types of brackets to avoid confusion, as in [2 × (3 + 4)] − 5 = 9.
These conventions are meaningful only when the usual notation (called infix notation) is used. When functional or Polish notation is used for all operations, the order of operations results from the notation itself.
Printed 2026-06-28.
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Link to original Footnotes
“Calculation operators and precedence: Excel”. Microsoft Support. Microsoft. 2023. Retrieved 2023-09-17. ↩
Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. “2.4.1.1. Definition arithmetischer Ausdrücke” [Definition of arithmetic expressions]. In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik [Pocketbook of mathematics] (in German). Vol. 1. Translated by Ziegler, Viktor (23rd ed.). Thun, Switzerland: Harri Deutsch. pp. 115–120, 802. ISBN 3-87144-492-8. Regel 7: Ist F(A) Teilzeichenreihe eines arithmetischen Ausdrucks oder einer seiner Abkürzungen und F eine Funktionenkonstante und A eine Zahlenvariable oder Zahlenkonstante, so darf F A dafür geschrieben werden. [Darüber hinaus ist noch die Abkürzung F(A) für (F(A)) üblich. Dabei kann F sowohl Funktionenkonstante als auch Funktionenvariable sein.] ↩ ↩2
Peterson, Dave (Sep–Oct 2019). The Math Doctors (blog). Order of Operations: “Why?”; “Why These Rules?”; “Subtle Distinctions”; “Fractions, Evaluating, and Simplifying”; “Implicit Multiplication?”; “Historical Caveats”. Retrieved 2024-02-11.
Peterson, Dave (Aug–Sep 2023). The Math Doctors (blog). Implied Multiplication: “Not as Bad as You Think”; “Is There a Standard?”; “You Can’t Prove It”. Retrieved 2024-02-11. ↩Swokowski, Earl William (1978). Fundamentals of Algebra and Trigonometry (4 ed.). Boston: Prindle, Weber & Schmidt. ISBN 0-87150-252-6. p. 1: The language of algebra […] may be used as shorthand, to abbreviate and simplify long or complicated statements. ↩
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