Basic Math Rules: Complete Guide

1. Commutative Rule

Addition: a + b = b + a

Multiplication: a × b = b × a

2. Associative Rule

Addition: (a + b) + c = a + (b + c)

Multiplication: (a × b) × c = a × (b × c)

3. Distributive Rule

a × (b + c) = (a × b) + (a × c)

4. Identity Rule

Addition: a + 0 = a

Multiplication: a × 1 = a

5. Inverse Rule

Addition: a + (-a) = 0

Multiplication: a × (1/a) = 1

1. Order of Operations (PEMDAS/BODMAS)

• Brackets/Parentheses
• Orders (Exponents/Powers)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

2. The Equals Rule

Things equal to the same thing are equal to each other

3. Properties of Zero

• Any number plus zero equals the number
• Any number times zero equals zero

4. Properties of One

• Any number times one equals the number
• Any number divided by one equals the number

5. Sign Rules

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

Consistency Rule

Mathematical statements must be consistent across all operations

Substitution Rule

Equal values can be substituted for each other in any expression

Addition Rule

• When adding positive numbers, the sum increases
• When adding negative numbers, the sum decreases

Multiplication Rule

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative

The standard rule refers to the order of operations:

1. Brackets/Parentheses
2. Orders (Exponents/Powers)
3. Division and Multiplication (left to right)
4. Addition and Subtraction (left to right)

Rule 5 refers to the fifth step in problem-solving:

1. Read the problem
2. Understand the given information
3. Determine what's being asked
4. Plan the solution
5. Execute the plan and check the answer

Rule 3 refers to the distributive property:

a(b + c) = ab + ac

Reflexive Property

a = a

Symmetric Property

If a = b, then b = a

Transitive Property

If a = b and b = c, then a = c

Addition Property

If a = b, then a + c = b + c

Multiplication Property

If a = b, then a × c = b × c

Counting Rules

• Numbers go from left to right
• Each number has its own symbol
• Zero means nothing

Addition Rules

• Adding makes numbers bigger
• Order doesn't matter (2+3 = 3+2)

Subtraction Rules

• Subtraction makes numbers smaller
• Order matters (5-3 ≠ 3-5)

What you do to one side of an equation, you must do to the other side to maintain equality.

Two quantities that are equal remain equal when the same operation is performed on both.