Order of operations - Definition and Overview

In arithmetic and elementary algebra, certain rules are used for the order in which the operations in algebraic expressions are to be evaluated. These precedence rules (which are mere notational conventions, not mathematical facts) are also used in many programming languages and by most modern calculators. In computing the standard algebraic notation is known as infix notation. This article assumes the reader is familiar with addition, division, exponential powers, multiplication, and subtraction.

The standard order of operations

1. Evaluate subexpressions contained within parentheses, starting with the innermost expressions. (Brackets [ ] are used here to indicate what is evaluated next.)

<math>(4+[10/2])/9=[4+5]/9=1 \,<math>

2. Evaluate exponential powers; for iterated powers, start from the right:

<math>2^{[3^2]}=[2^9]=512 \,<math>

3. Evaluate multiplications and divisions, starting from the left:

<math>[8/2]\times3=[4\times3]=12 \,<math>

4. Evaluate additions and subtractions, starting from the left:

<math>[7-2]-4+1=[5-4]+1=[1+1]=2 \,<math>

The expression: 2 + 3 × 4 is evaluated to 14, and not 20, because multiplication precedes addition. If the intention is to perform the addition first, parentheses must be used: (2 + 3) × 4 = 20.

In the UK, the acronym BODMAS is used for Brackets, raise to the power Of, Division, Multiplication, Addition, Subtraction. The "O" may also stand for "or", or may by replaced with an "I", for Indices.

In the US, the acronym PEMDAS (for Parentheses, Exponentation, Multiplication/Division, Addition/Subtraction) is used instead, sometimes expressed as the mnemonic "Please Excuse My Dear Aunt Sally".

Example

• Given:

<math>3-(5-(7+1))^2\times(-5)+2<math>

• Evaluate the innermost subexpression (7 + 1):

<math>3-(5-8)^2\times(-5)+2<math>

• Evaluate the subexpression within the remaining parentheses (5 − 8):

<math>3-(-3)^2\times(-5)+2<math>

• Evaluate the power of (−3)²:

<math>3-9\times(-5)+2<math>

• Evaluate the multiplication 9 × (−5):

<math>3-(-45)+2<math>

• Evaluate the subtraction 3 − (−45):

<math>48+2<math>

• Evaluate the addition 48 + 2:

<math>48+2=50<math>

Special cases

In the case that a factorial is in an expression, it is evaluated first. When new operations are defined they are generally presumed to take precedence over other operations unless defined otherwise. In the case where repeated operators of the same type are used, such as in

<math>a/b/c<math>

the expression is evaluated from left to right, as

<math>((a/b)/c)<math>

See also

• Common operator notation (for a more formal description)
• associativity
• commutativity
• distributivity

External link

• Order of operations (http://planetmath.org/?op=getobj&from=objects&id=3951) on PlanetMath.