Understanding Integers

• Integers are whole numbers, which can be positive, negative, or zero
• They do not include fractions, decimals, or mixed numbers
• Examples of integers: -3, -2, -1, 0, 1, 2, 3

Number Line Representation

• Integers can be visually represented on a number line
• Zero is at the center, positive integers extend to the right, and negative integers extend to the left
• The number line helps to understand the order and relative values of integers

Basic Operations with Integers

• Addition, subtraction, multiplication, and division can be performed with integers
• Rules for these operations must be followed carefully to determine the correct sign of the result

Addition of Integers

• Adding two positive integers results in a positive integer
• Adding two negative integers results in a negative integer
• When adding a positive and a negative integer, subtract the smaller absolute value from the larger absolute value and use the sign of the integer with the larger absolute value

Subtraction of Integers

• Subtracting an integer is the same as adding its opposite
• To subtract b from a, change the sign of b and add it to a: a - b = a + (-b)

Multiplication of Integers

• Multiplying two positive integers results in a positive integer
• Multiplying two negative integers results in a positive integer
• Multiplying a positive integer and a negative integer results in a negative integer

Division of Integers

• Dividing two positive integers results in a positive integer
• Dividing two negative integers results in a positive integer
• Dividing a positive integer and a negative integer results in a negative integer
• Division by zero is undefined

Absolute Value

• The absolute value of an integer is its distance from zero on the number line
• The absolute value is always non-negative
• Notation: |a| represents the absolute value of a
• Example: |-5| = 5 and |5| = 5

Properties of Integer Operations

• Commutative Property: Applies to addition and multiplication, the order of the integers does not affect the result (a + b = b + a, a * b = b * a)
• Associative Property: Applies to addition and multiplication, the grouping of integers does not affect the result ((a + b) + c = a + (b + c), (a * b) * c = a * (b * c))
• Distributive Property: a * (b + c) = a * b + a * c
• Identity Property:
  • Addition: a + 0 = a (0 is the additive identity)
  • Multiplication: a * 1 = a (1 is the multiplicative identity)
• Inverse Property:
  • Addition: a + (-a) = 0 (-a is the additive inverse of a)

Ordering Integers

• Integers can be ordered from least to greatest
• On the number line, integers increase in value from left to right
• Any negative integer is less than any positive integer
• Zero is greater than any negative integer and less than any positive integer

Applications of Integers

• Integers are used in various real-world contexts
• Examples: representing temperatures (above or below zero), financial transactions (credits and debits), altitudes (above or below sea level)

Algebraic Expressions with Integers

• Integers are used in algebraic expressions and equations
• Variables can represent integers
• Solving equations may involve performing operations with integers

Integer Exponents

• Integers can be used as exponents
• A negative exponent indicates the reciprocal of the base raised to the positive exponent: a^(-n) = 1 / a^n

Integer Coordinates

• Integers can be used as coordinates in the Cartesian plane
• Points are represented as ordered pairs (x, y), where x and y are integers

Integer Sequences

• A sequence is an ordered list of numbers
• Integer sequences consist of integers
• Examples: arithmetic sequences, geometric sequences

Divisibility of Integers

• An integer a is divisible by an integer b if there exists an integer c such that a = b * c
• If a is divisible by b, then b is a factor of a
• Prime numbers are integers greater than 1 that have only two factors: 1 and themselves

Prime Factorization

• Prime factorization is the process of expressing an integer as a product of its prime factors
• Every integer greater than 1 can be uniquely expressed as a product of prime numbers

Greatest Common Divisor (GCD)

• The greatest common divisor (GCD) of two or more integers is the largest positive integer that divides all of the integers without leaving a remainder
• The Euclidean algorithm can be used to find the GCD of two integers

Least Common Multiple (LCM)

• The least common multiple (LCM) of two or more integers is the smallest positive integer that is a multiple of all of the integers
• The LCM can be found using the prime factorizations of the integers

Solving Linear Equations with Integer Coefficients

• Linear equations with integer coefficients can be solved using algebraic techniques
• These techniques include adding, subtracting, multiplying, and dividing both sides of the equation by integers

Inequalities with Integers

• Inequalities compare the relative values of integers
• Symbols used: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to)
• Solving inequalities involves finding the range of integers that satisfy the inequality

Absolute Value Equations and Inequalities

• Equations and inequalities involving absolute values require special techniques to solve
• Consider both positive and negative cases for the expression inside the absolute value

Word Problems Involving Integers

• Many word problems involve integers
• Careful reading and understanding of the problem are necessary to set up and solve the problem correctly

Advanced Topics

• Number Theory: Explores the properties and relationships of integers
• Cryptography: Uses integers and their properties to encrypt and decrypt messages