Basic Arithmetic Operations and Properties

Basic Arithmetic Operations

• Addition: Combining two or more numbers to find their total. The result is the sum.
• Subtraction: Finding the difference between two numbers. The result is the difference.
• Multiplication: Repeated addition of the same number. The result is the product.
• Division: Splitting a number into equal parts. The result is the quotient.

Properties of Numbers

• Commutative Property (Addition and Multiplication): Changing the order of numbers in an addition or multiplication problem does not change the outcome.
• Associative Property (Addition and Multiplication): Grouping numbers differently in an addition or multiplication problem does not change the outcome.
• Distributive Property: Multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products.
• Identity Property (Addition): Adding zero to any number results in the original number.
• Identity Property (Multiplication): Multiplying any number by one results in the original number.
• Zero Property (Multiplication): Multiplying any number by zero results in zero.

Place Value and Numeration

• Place value system: A system where the position of a digit determines its value.
• Expanded form: Writing a number as the sum of the values of its digits. Example: 123 = 100 + 20 + 3

Whole Numbers

• Counting numbers: 1, 2, 3, ... (and zero, if considered)
• Properties of whole numbers: Closure under addition and multiplication. (The sum or product of two whole numbers is also a whole number)

Fractions

• Fractions represent parts of a whole. A fraction has a numerator (top number) and a denominator (bottom number).
• Equivalent fractions: Fractions that represent the same part of a whole.
• Simplifying fractions: Reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common factor.
• Comparing fractions: Determining which fraction is larger or smaller, often by finding a common denominator.
• Adding and subtracting fractions: Requires a common denominator.

Decimals

• Decimal representation of fractions: Fractions can be converted to decimals and vice versa.
• Decimal place value: Each digit in a decimal has a specific place value that represents a power of ten. (e.g. tenths, hundredths, thousandths).
• Operations with decimals: Addition, subtraction, multiplication, and division are performed similarly to whole numbers, but with consideration of decimal points.

Integers

• Integers: Whole numbers and their opposites.
• Positive and negative integers: Positive numbers are greater than zero, negative numbers are less than zero.
• Ordering integers: Integers can be ordered on a number line.
• Operations with integers: Addition, subtraction, multiplication, and division rules apply, including rules for signs.

Exponents and Powers

• Exponents represent repeated multiplication of a base number.

Factors and Multiples

• Factors: Numbers that divide another number evenly.
• Multiples: Numbers that are the product of a given number and another whole number.
• Prime numbers: Whole numbers greater than 1 that have only two factors: 1 and themselves.
• Composite numbers: Whole numbers greater than 1 that have more than two factors.

Order of Operations

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

Real Numbers

• Real numbers include all rational and irrational numbers.
• Rational numbers can be expressed as a fraction, including integers and terminating or repeating decimals.
• Irrational numbers cannot be expressed as a fraction (non-repeating, non-terminating decimals).