Arithmetic Operations and Properties

Study Notes

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).

Description

This quiz covers fundamental arithmetic operations including addition, subtraction, multiplication, and division. It also explores key properties of numbers such as commutative, associative, distributive, and identity properties. Test your understanding of these essential mathematical concepts.