Operations

PEMDAS: The Order of Operations

When evaluating mathematical expressions with multiple operations, we follow a specific order to ensure consistent results. This order is often remembered using the acronym PEMDAS:

1. Parentheses (or brackets)
2. Exponents (powers, roots)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Some students remember this as “Please Excuse My Dear Aunt Sally.”

Addition Concepts

Addition is the operation of combining quantities. It is indicated by the + symbol.

• Terms: The numbers being added are called addends, and the result is called the sum.
• Properties:
  • Commutative Property: a + b = b + a (the order doesn’t matter)
  • Associative Property: (a + b) + c = a + (b + c) (grouping doesn’t matter)
  • Identity Property: a + 0 = a (adding zero doesn’t change the value)

Subtraction Concepts

Subtraction is the operation of finding the difference between quantities. It is indicated by the – symbol.

• Terms: In a – b, a is the minuend, b is the subtrahend, and the result is the difference.
• Properties:
  • Non-Commutative: a – b ≠ b – a (order matters)
  • Non-Associative: (a – b) – c ≠ a – (b – c) (grouping matters)
  • Identity Property: a – 0 = a (subtracting zero doesn’t change the value)
• Relationship to addition: a – b = c is equivalent to a = b + c

Multiplication Concepts

Multiplication is the operation of repeated addition. It is indicated by the × or * symbol.

• Terms: In a × b, a and b are factors, and the result is the product.
• Properties:
  • Commutative Property: a × b = b × a (order doesn’t matter)
  • Associative Property: (a × b) × c = a × (b × c) (grouping doesn’t matter)
  • Identity Property: a × 1 = a (multiplying by 1 doesn’t change the value)
  • Zero Property: a × 0 = 0 (any number multiplied by 0 equals 0)
  • Distributive Property: a × (b + c) = (a × b) + (a × c)

Division Concepts

Division is the operation of splitting into equal parts or finding how many groups. It is indicated by the ÷, / or ) symbols.

• Terms: In a ÷ b, a is the dividend, b is the divisor, and the result is the quotient. Any amount left over is the remainder.
• Properties:
  • Non-Commutative: a ÷ b ≠ b ÷ a (order matters)
  • Non-Associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) (grouping matters)
  • Identity Property: a ÷ 1 = a (dividing by 1 doesn’t change the value)
  • Zero Property: 0 ÷ a = 0 (zero divided by any non-zero number equals 0)
  • Undefined: a ÷ 0 is undefined (division by zero is not defined)
• Relationship to multiplication: a ÷ b = c is equivalent to a = b × c

Operations with Integers

Integers are whole numbers and their negatives, including zero: {…, -3, -2, -1, 0, 1, 2, 3, …}

Addition of Integers

• Same signs: Add absolute values and keep the sign
• Different signs: Subtract the smaller absolute value from the larger, and use the sign of the larger absolute value

Subtraction of Integers

• Change the operation to addition and change the sign of the second number: a – b = a + (-b)
• Then follow the rules for addition of integers

Multiplication of Integers

• Same signs (positive × positive or negative × negative): Result is positive
• Different signs (positive × negative or negative × positive): Result is negative

Division of Integers

• Same signs (positive ÷ positive or negative ÷ negative): Result is positive
• Different signs (positive ÷ negative or negative ÷ positive): Result is negative

Exponents

An exponent indicates how many times a number (the base) is multiplied by itself. For example, 2³ = 2 × 2 × 2 = 8.

Rules of Exponents

• Product rule: x^a × x^b = x^a+b
• Quotient rule: x^a ÷ x^b = x^a-b
• Power rule: (x^a)^b = x^a×b
• Zero exponent: x⁰ = 1 (for any non-zero x)
• Negative exponent: x^-a = 1/x^a

Roots

A root is the inverse operation of an exponent. For example, the square root of 9, written as √9, is 3 because 3² = 9.

Types of Roots

• Square root: √x (or x^1/2)
• Cube root: ∛x (or x^1/3)
• Fourth root: ∜x (or x^1/4)
• General: n^th root of x is x^1/n

Basic Algebraic Concepts

Algebra uses letters (variables) to represent unknown values. Algebraic expressions are combinations of variables, numbers, and operations.

Simplifying Expressions

• Combine like terms: Terms with the same variables raised to the same powers
• Apply the distributive property: a(b + c) = ab + ac
• Apply rules of exponents

Evaluating Expressions

To evaluate an expression, substitute values for variables and calculate the result.

Basic Equation-Solving Principles

• Equality principle: Performing the same operation on both sides of an equation maintains the equality.
• Isolation goal: Solve for the variable by isolating it on one side of the equation.

Steps for Solving Linear Equations

1. Simplify both sides of the equation (combine like terms)
2. Move all variable terms to one side and all constant terms to the other side
3. Divide both sides by the coefficient of the variable

Solving Word Problems

Word problems require translating verbal descriptions into mathematical operations. Follow these general steps:

1. Read the problem carefully to understand what is being asked
2. Identify the known information and what you need to find
3. Assign variables for unknown quantities if needed
4. Write equations or set up operations to represent the relationships in the problem
5. Solve the equations or perform the operations
6. Check your answer to ensure it makes sense in the context of the problem