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3.3 Order of Operations

3 min read•june 18, 2024

is your math GPS, guiding you through calculations. It's a simple acronym that helps you navigate complex equations by telling you which operations to do first. Without it, you'd be lost in a sea of numbers.

Mastering PEMDAS is crucial for solving math problems correctly. It's not just about memorizing an order; it's about understanding how different operations interact. This skill is essential for everything from basic arithmetic to advanced algebra.

Order of Operations (PEMDAS)

Order of operations in arithmetic

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  • PEMDAS acronym represents the order to perform arithmetic operations
    • : Evaluate expressions inside parentheses first (, )
    • : Calculate , , and other exponents
    • and : Perform from left to right
    • and : Perform from left to right
  • Applies the correct sequence of operations for accurate results
    • Ignoring the order leads to incorrect answers
  • Example: 3[+](https://www.fiveableKeyTerm:+)4×2−13 [+](https://www.fiveableKeyTerm:+) 4 \times 2 - 1
    • Multiply 4×2=84 \times 2 = 8 first according to PEMDAS
    • Then add and subtract from left to right: 3+8−1=103 + 8 - 1 = 10
  • Operations with the same precedence (multiplication/division, addition/subtraction) are performed from left to right (following the )
    • Example: 24÷4×224 \div 4 \times 2
      1. Divide 24÷4=624 \div 4 = 6 first since it appears on the left
      2. Then multiply 6×2=126 \times 2 = 12 to get the final result

Expressions with grouping symbols

  • Parentheses and other grouping symbols have the highest precedence in PEMDAS
    • Perform operations inside the innermost parentheses first
    • Work outward to the next set of parentheses until all are resolved
  • Grouping symbols override the standard order of operations
    • Ensures parts of the expression are evaluated together
  • Example: 2×(3+4)−12 \times (3 + 4) - 1
    1. Add inside parentheses: 2×7−12 \times 7 - 1
    2. Multiply: 14−114 - 1
    3. Subtract to get the final result: 1313
  • Nested parentheses are evaluated from the innermost pair outward
    • Example: 2×(3+(4−1))2 \times (3 + (4 - 1))
      1. Innermost parentheses: 2×(3+3)2 \times (3 + 3)
      2. Outer parentheses: 2×62 \times 6
      3. Multiply to get the final result: 1212

Simplification of complex expressions

  • Break down complex expressions into smaller parts
    • Apply PEMDAS rules to each part separately
    • Simplify innermost parentheses, exponents, multiplication/division, addition/subtraction
  • Example: 3+2×(4−1)[](https://www.fiveableKeyTerm:)2÷33 + 2 \times (4 - 1)[^](https://www.fiveableKeyTerm:^)2 \div 3
    1. Parentheses: 3+2×32÷33 + 2 \times 3^2 \div 3
    2. Exponents: 3+2×9÷33 + 2 \times 9 \div 3
    3. Multiplication and division from left to right:
      • 2×9=182 \times 9 = 18
      • 18÷3=618 \div 3 = 6
      • Simplified: 3+63 + 6
    4. Addition: 99
  • Remember equal precedence of multiplication/division and addition/subtraction
    • Perform left to right when at the same level
  • Use parentheses to group parts of the expression for clarity
    • Example: (3+2)×(4−1)=15(3 + 2) \times (4 - 1) = 15 is different from 3+2×(4−1)=93 + 2 \times (4 - 1) = 9
    • Grouping affects the order and the final result

Algebraic Expressions and Properties

  • combine numbers, variables, and operations
  • The allows regrouping of terms without changing the result
    • Example: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
  • involves substituting specific values for variables in algebraic expressions

Key Terms to Review (30)

^: The caret symbol (^) is used to represent exponentiation, which is the operation of raising a number to the power of an exponent. This symbol helps simplify mathematical expressions by indicating how many times a base number is multiplied by itself. Understanding exponentiation is crucial for performing calculations accurately and efficiently, especially when following the order of operations.
+: 'The plus sign (+) is a mathematical symbol used to denote addition, which is one of the four basic arithmetic operations. It signifies the process of combining two or more numbers or quantities to get a total. Understanding its role is essential, as it directly influences how calculations are performed, especially when following the order of operations in mathematics.'
÷: The symbol '÷' represents division, which is one of the four basic arithmetic operations. It indicates the process of distributing a quantity into equal parts or finding out how many times one number is contained within another. Understanding how to use this symbol is crucial in mathematical calculations, especially when following the correct order of operations to ensure accurate results.
×: The symbol '×' represents multiplication, a fundamental arithmetic operation used to combine groups of equal size. It serves as a method to calculate the total amount when a number is added to itself multiple times, making it essential for understanding various mathematical concepts. This symbol plays a crucial role in factorization processes, particularly when determining whether a number is prime or composite, and it is also vital in correctly applying the order of operations in more complex mathematical expressions.
Addition: Addition is the mathematical operation of combining two or more numbers to find their total. This basic arithmetic function forms the foundation for more complex calculations, including those involving rational numbers, and is crucial for establishing order in mathematical expressions through operations.
Algebraic expressions: Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They can represent a wide range of values and relationships, allowing for the manipulation and simplification of equations. Understanding algebraic expressions is essential for performing calculations using the order of operations, which dictates the sequence in which mathematical operations should be carried out to arrive at the correct answer.
Associative property: The associative property is a fundamental principle in mathematics that states the way numbers are grouped in addition or multiplication does not affect the final result. This means that when adding or multiplying three or more numbers, you can change the grouping of the numbers without changing the sum or product. This property emphasizes the flexibility in computations and helps simplify expressions, making it crucial in various mathematical contexts.
BODMAS: BODMAS is an acronym that stands for Brackets, Orders, Division and Multiplication, Addition and Subtraction. It is a crucial guideline that helps determine the order in which operations should be performed in mathematical expressions to ensure correct results. Understanding BODMAS allows for clarity in calculations and prevents ambiguity in complex equations, making it essential for problem-solving in mathematics.
Braces: Braces are a pair of curly brackets used in mathematical expressions to group numbers, variables, or operations together. This grouping helps clarify the order of operations and indicates that the calculations within the braces should be performed first before moving on to other parts of the expression.
Brackets: Brackets are symbols used in mathematics to indicate the order of operations in an expression. They help clarify which calculations should be performed first, ensuring that mathematical expressions are evaluated correctly. Different types of brackets, such as parentheses '()', square brackets '[]', and curly braces '{}', can be used interchangeably in many cases, but their consistent use can help avoid confusion in complex expressions.
Commutative Property: The commutative property is a fundamental mathematical principle stating that the order in which two numbers are added or multiplied does not change the result. This property highlights that rearranging the elements in addition or multiplication still yields the same outcome, making calculations more flexible and efficient. It plays a crucial role in simplifying expressions and solving equations across various mathematical contexts.
Decimals: Decimals are a way of expressing numbers that are not whole, using a decimal point to separate the whole number part from the fractional part. This notation allows for representing values between integers, enabling precision in measurements and calculations. Decimals play a crucial role in operations involving fractions, percentages, and real-world applications like finance and science.
Distributive Property: The distributive property is a fundamental algebraic principle that states that multiplying a number by a sum is the same as multiplying each addend separately and then adding the results. This property is essential for simplifying expressions and solving equations, and it connects to various mathematical concepts such as logic, operations with real numbers, manipulation of exponents, and algebraic expressions.
Division: Division is a mathematical operation that represents the process of splitting a quantity into equal parts or determining how many times one number is contained within another. This operation plays a critical role in various mathematical concepts, providing a foundation for understanding relationships between numbers and forming the basis for operations involving fractions, ratios, and proportions. Division interacts with other operations like multiplication and is essential in contexts like calculating averages and understanding rational numbers.
Division in a base: Division in a base is the process of dividing numbers expressed in a specific numeral system, such as binary, octal, or hexadecimal. It follows similar principles to division in the decimal system but requires adjustments based on the base.
Exponents: Exponents are a mathematical notation that represents the number of times a base is multiplied by itself. This concept is crucial in simplifying expressions and performing calculations efficiently, as it allows for the representation of large numbers in a compact form. Understanding exponents is essential for mastering the order of operations, since operations involving exponents must be performed before multiplication and division, following the established hierarchy of calculations.
Fractions: Fractions represent a part of a whole and are expressed as the ratio of two integers, with the top number called the numerator and the bottom number called the denominator. They play a critical role in arithmetic and algebra, helping to illustrate division and proportions. Understanding fractions is essential for working with real numbers, which include both rational and irrational values.
Integers: Integers are whole numbers that can be positive, negative, or zero, encompassing all the numbers without fractional or decimal components. They play a crucial role in various mathematical concepts, including arithmetic operations, number theory, and the structure of real numbers. Integers are fundamental for understanding prime and composite numbers, as these classifications are specific to the set of positive integers.
Multiplication: Multiplication is a mathematical operation that combines groups of equal sizes to find the total quantity. It serves as a fundamental building block in mathematics, allowing us to simplify and solve problems involving repeated addition, scaling quantities, and working with rational numbers. This operation is also essential for understanding various mathematical concepts, including the order of operations, base systems, and financial calculations such as interest.
Multiplication and division in bases: Multiplication and division in bases involve performing these arithmetic operations within non-decimal numeral systems, such as binary (base-2) or hexadecimal (base-16). These operations follow similar principles to base-10 but require adjustments for different digits and place values.
Numerical evaluation: Numerical evaluation refers to the process of calculating the value of an expression using specific numerical values for its variables. This involves substituting values into mathematical expressions and performing operations according to established rules, particularly the order of operations. Understanding numerical evaluation is crucial for accurately solving mathematical problems and interpreting results in various contexts.
Operand: An operand is a quantity on which an operation is performed in mathematical expressions. In the context of calculations, operands can be numbers, variables, or even complex expressions, and they work together with operators to form equations that convey mathematical relationships. Understanding operands is crucial for mastering how to structure and evaluate expressions correctly, especially when applying the order of operations.
Operator: An operator is a symbol or function that indicates a specific mathematical operation to be performed on one or more operands. In the context of mathematical expressions, operators dictate the order in which calculations are carried out, impacting the final result. Understanding how operators work is essential for evaluating expressions correctly and following established rules for computation.
Order of precedence: Order of precedence refers to the rules that dictate the sequence in which operations are performed in mathematical expressions. These rules ensure consistency in the interpretation of calculations, allowing for accurate results regardless of how complex the expression may be. By establishing a clear hierarchy for operations, such as addition, subtraction, multiplication, and division, the order of precedence helps prevent ambiguity in calculations.
Parentheses: Parentheses are symbols used in mathematics to indicate that the operations contained within them should be performed before any outside operations. They play a critical role in establishing order and clarity in mathematical expressions, especially when multiple operations are involved. By grouping numbers or expressions with parentheses, one can alter the sequence of calculations, ensuring the intended interpretation of an expression is achieved.
PEMDAS: PEMDAS is an acronym that represents the order of operations used in mathematics to solve expressions correctly. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding this order is essential when dealing with calculations involving rational numbers and is foundational in systems like the Hindu-Arabic positional system, ensuring that calculations are performed systematically and accurately.
Powers: In mathematics, powers refer to the operation of raising a number, known as the base, to an exponent, which indicates how many times the base is multiplied by itself. This concept is crucial when evaluating expressions and performing calculations, especially when adhering to the order of operations. Understanding how to manipulate and simplify expressions involving powers allows for more efficient problem-solving and accurate results in mathematical computations.
Roots: Roots refer to the values that satisfy an equation, particularly when it involves polynomial equations. They are crucial in understanding how functions behave and can indicate where a function crosses the x-axis. In various mathematical contexts, finding roots often involves operations, manipulations with exponents, and solving equations, highlighting their importance in diverse mathematical applications.
Subtraction: Subtraction is the mathematical operation of finding the difference between two numbers, where one number is taken away from another. It is often represented by the minus sign (-) and plays a crucial role in various mathematical processes, including simplifying expressions, solving equations, and managing real-world situations like budgeting and finance.
Subtraction in bases: Subtraction in bases involves performing arithmetic subtraction in number systems other than base 10, such as binary (base 2), octal (base 8), or hexadecimal (base 16). It requires understanding how to borrow and subtract digits within those specific bases.
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3.4 Rational Numbers