Basic Arithmetic Rules
Arithmetic is the bedrock of mathematics. Before diving into algebra or calculus, it’s essential to understand the basic rules governing numbers. This guide explains the key arithmetic properties and helps you build a solid foundation—perfect for students who want clarity and confidence.
1. Order of Operations (PEMDAS)
To solve any numerical expression correctly, follow this order:
Parentheses → Exponents → Multiplication & Division → Addition & Subtraction
Example:
\(8 + 2 \times (3^2 – 1) = 8 + 2 \times (9 – 1) = 8 + 2 \times 8 = 8 + 16 = 24\)
2. Commutative Property
This applies only to addition and multiplication. You can change the order of numbers without changing the result.
\(a + b = b + a\)
\(a \times b = b \times a\)
Example:
\(4 + 7 = 7 + 4 = 11\)
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3. Associative Property
Changing the grouping of numbers does not affect the result (addition and multiplication only):
\((a + b) + c = a + (b + c)\)
\((a \times b) \times c = a \times (b \times c)\)
Example:
\((2 + 3) + 4 = 2 + (3 + 4) = 9\)
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4. Distributive Property
This rule connects multiplication with addition or subtraction:
\(a \times (b + c) = a \times b + a \times c\)
Example:
\(3 \times (4 + 5) = 3 \times 4 + 3 \times 5 = 12 + 15 = 27\)
5. Identity and Zero Rules
- Additive identity: \(a + 0 = a\)
- Multiplicative identity: \(a \times 1 = a\)
- Multiplying by zero: \(a \times 0 = 0\)
Why It Matters
These rules help you simplify and solve problems confidently. Once mastered, you’ll easily transition into algebra, expressions, and equations.
For example, understanding these lets you simplify expressions like:
\(\frac{a^2 – b^2}{a – b} = a + b\)
and recognize deeper structures in calculus such as:
\(\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}\)
Use our solver after each rule to reinforce what you’ve learned. Mastery begins with repetition and clarity.