Arithmetic Operations and Properties

Arithmetic Operations

• Addition (+) combines numbers to find a total. For example, 2 + 3 = 5.
• Subtraction (-) finds the difference between two numbers. For example, 5 - 2 = 3.
• Multiplication (× or *) combines equal groups. For example, 2 × 3 = 6 or 2 * 3 = 6.
• Division (÷ or /) splits a number into equal parts. For example, 6 ÷ 2 = 3 or 6 / 2 = 3.

Properties of Numbers

• Commutative property: The order of numbers in addition or multiplication doesn't change the result (e.g., 2 + 3 = 3 + 2).
• Associative property: The grouping of numbers in addition or multiplication doesn't change the result (e.g., (2 + 3) + 4 = 2 + (3 + 4)).
• Distributive property: Multiplying a sum or difference is the same as multiplying each of the numbers in the sum or difference by the multiplier and then adding or subtracting the products (e.g., 2 × (3 + 4) = (2 × 3) + (2 × 4)).
• Identity property of addition: Adding zero to any number results in the original number (e.g., 5 + 0 = 5).
• Identity property of multiplication: Multiplying any number by one results in the original number (e.g., 5 × 1 = 5).
• Inverse operations: Operations that undo each other (e.g., addition and subtraction, multiplication and division).

Number Systems

• Natural Numbers (or Counting Numbers): 1, 2, 3,... Used for counting things.
• Whole Numbers: 0, 1, 2, 3,... Include zero and all natural numbers.
• Integers: ..., -3, -2, -1, 0, 1, 2, 3,... Include both positive and negative whole numbers.
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Examples include fractions (like 1/2), decimals (like 0.25), and integers.
• Irrational Numbers: Numbers that cannot be expressed as a fraction of two integers. Examples include π (pi) and the square root of 2.
• Real Numbers: The set of all rational and irrational numbers.

Order of Operations (PEMDAS/BODMAS)

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

Fractions

• Numerator (top number): Represents the part of the whole.
• Denominator (bottom number): Represents the whole divided into parts.
• Equivalent fractions: Fractions that represent the same value.
• Simplifying fractions: Reducing a fraction to its lowest terms.

Decimals

• Decimal places: Positions to the right of the decimal point.
• Comparing decimals.

Exponents and Roots

• Exponents represent repeated multiplication. For example, 2³ = 2 × 2 × 2 = 8.
• Roots represent the inverse of exponents. For example, the square root of 9 (√9) is 3 because 3 × 3 = 9.

Geometry

• Basic shapes (e.g., squares, circles, triangles).
• Formulas for areas and perimeters of basic shapes.

Algebra

• Variables: Symbols (like x or y) that represent unknown quantities.
• Equations: Statements that show two expressions are equal.
• Solving equations: Finding the value of the variable that makes the equation true.
• Inequalities: Statements comparing two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to).