Order of Operations with Negative Numbers Calculator

What is Order of Operations?

Order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed in an expression. These rules ensure that everyone calculates expressions consistently, regardless of how they might be written.

The most commonly used acronym for these rules is PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

These rules apply to all mathematical expressions, including those with negative numbers.

PEMDAS Rules

The PEMDAS acronym provides a clear hierarchy for evaluating mathematical expressions:

1. Parentheses: Solve expressions inside parentheses first. This includes brackets [] and braces {}.
2. Exponents: Next, calculate any exponents or powers in the expression.
3. Multiplication and Division: Perform multiplication and division from left to right as they appear in the expression.
4. Addition and Subtraction: Finally, complete addition and subtraction from left to right.

Negative Numbers in Order of Operations

Negative numbers follow the same order of operations rules as positive numbers. The key is to carefully track the signs during each step of the calculation.

When dealing with negative numbers, remember that:

• Subtracting a negative number is the same as adding its absolute value.
• Adding a negative number is the same as subtracting its absolute value.
• Multiplying or dividing by a negative number changes the sign of the result.

Worked Examples

Example 1: Simple Expression

Evaluate 10 - 5 × 2 + 3

1. First, perform multiplication: 5 × 2 = 10
2. Then, perform subtraction and addition from left to right: 10 - 10 + 3 = 3

Example 2: Expression with Parentheses

Evaluate (8 - 3) × (4 + 2)

1. First, solve the parentheses: (8 - 3) = 5 and (4 + 2) = 6
2. Then, perform multiplication: 5 × 6 = 30

Example 3: Expression with Negative Numbers

Evaluate -4 × 3 + 6 ÷ 2 - 5

1. First, perform multiplication and division from left to right: -4 × 3 = -12 and 6 ÷ 2 = 3
2. Then, perform addition and subtraction from left to right: -12 + 3 - 5 = -14

Common Mistakes

When working with order of operations, especially with negative numbers, several common mistakes can occur:

• Ignoring parentheses: Forgetting to solve expressions inside parentheses first.
• Incorrect order of operations: Performing addition before multiplication or vice versa.
• Sign errors: Forgetting to change the sign when multiplying or dividing by a negative number.
• Left-to-right confusion: Not performing operations from left to right when they have equal precedence.