In mathematics, operations refer to actions like addition, subtraction, multiplication, division, and exponentiation. For consistent results, a standard order is followed to evaluate expressions. Without this, expressions could be interpreted differently, causing confusion.

The order of operations is summarized by the acronym PEMDAS ¹ 

1. Parentheses
2. Exponentiation
3. Multiplication and Division (left to right)
4. Addition and Subtraction (left to right)

For example, the expression \(1 + 2 \times 3\) evaluates as \(1 + (2 \times 3) = 7\) , not \((1 + 2) \times 3 = 9\) , because multiplication takes precedence over addition.

These conventions simplify notation and ensure clarity, formalized in textbooks by the late 19th century.

Although using mnemonic acronyms like PEMDAS is helpful for remembering the order of operations, relying on them without understanding the underlying concepts can lead to mistakes. For example, since "A" for addition comes before "S" for subtraction in PEMDAS, one might incorrectly assume that addition always precedes subtraction. This could lead to evaluating \(2 - 3 + 4\) as \(2 - (3 + 4) = 2 - 7 = -5\) , whereas the correct result is \((2 - 3) + 4 = -1 + 4 = 3\) . Therefore, it is important to understand what happens with the order of operations rather than simply memorizing the sequence or mnemonics.

Special Cases and Exceptions

1. Division and Multiplication : Division and multiplication are evaluated left to right unless specified otherwise. For example, \(6 \div 2 \times 3\) evaluates as \((6 \div 2) \times 3 = 9\) .
2. Serial Exponentiation : When exponentiation is indicated by stacked symbols in superscript notation, the standard rule is to work from the top down: \[a^{b^c} = a^{\left(b^c\right)}\] which typically does not equal \((a^b)^c\) . This convention is practical because exponentiation has the property that \({\left(a^b\right)}^c = a^{(b\cdot c)}\) , making serial exponentiation unnecessary. So, for example, we have \[3^{2^4}=3^{16} \neq (3^2)^4=3^8=9^4.\]
3. Implied Multiplication : Multiplication indicated by juxtaposition (written without a multiplication symbol) forms a visual unit and takes precedence over most other operations. In academic literature, when inline fractions are combined with implied multiplication without explicit parentheses, the multiplication is typically given higher precedence than division. For example, \(1/2n\) is interpreted as \(1/(2\cdot n)\) rather than \((1/2)·n\) . However, some authors advise against using expressions like \(a/bc\) , recommending the explicit use of parentheses as in \(a/(bc)\) .