Integers and Fractions in Mathematics

Definition: Whole numbers that can be positive, negative, or zero.
Examples: -3, -2, -1, 0, 1, 2, 3.
Properties:
- Closed under addition, subtraction, and multiplication.
- Not closed under division (e.g., -1 ÷ 2 is not an integer).
Operations:
- Addition: Combine values (e.g., 3 + (-2) = 1).
- Subtraction: Find the difference (e.g., 4 - 5 = -1).
- Multiplication: Product of values (e.g., -2 × 3 = -6).
- Division: Quotient of values (consider the sign).

Definition: A part of a whole, expressed as a ratio of two integers (numerator/denominator).
Types:
- Proper Fraction: Numerator < Denominator (e.g., 3/4).
- Improper Fraction: Numerator ≥ Denominator (e.g., 5/3).
- Mixed Number: Combination of whole number and proper fraction (e.g., 1 2/3).
Operations:
- Addition/Subtraction: Find a common denominator.
- Multiplication: Multiply numerators and denominators (e.g., 1/2 × 3/4 = 3/8).
- Division: Multiply by the reciprocal (e.g., 1/2 ÷ 3/4 = 1/2 × 4/3 = 2/3).
Simplification: Reduce fractions to lowest terms by dividing numerator and denominator by their greatest common divisor (GCD).

Definition: A way to represent fractions using a decimal point.
Types:
- Terminating Decimal: Ends after a finite number of digits (e.g., 0.75).
- Repeating Decimal: Has one or more digits that repeat indefinitely (e.g., 0.333...).
Conversion:
- From Fraction to Decimal: Divide numerator by denominator (e.g., 1/4 = 0.25).
- From Decimal to Fraction: Identify place value (e.g., 0.75 = 75/100 = 3/4).
Operations:
- Addition/Subtraction: Align decimal points.
- Multiplication: Multiply as whole numbers, count decimal places for placement.
- Division: Move decimal points to simplify before dividing.

Converting Between Types: Integers can be represented as fractions (e.g., 5 = 5/1) and as decimals (e.g., 5 = 5.0).
Order of Operations: Follow PEMDAS/BODMAS rules when performing calculations involving integers, fractions, and decimals.

Whole numbers include positive numbers, negative numbers, and zero.
Examples include -3, -2, -1, 0, 1, 2, 3.
Integers are closed under addition, subtraction, and multiplication, meaning the result of these operations will always yield an integer.
Division of integers is not closed; for instance, -1 divided by 2 results in a non-integer.
Key operations with integers:
- Addition combines values, such as 3 + (-2) resulting in 1.
- Subtraction determines the difference, such as 4 - 5 yielding -1.
- Multiplication gives the product, e.g., -2 × 3 results in -6.

A fraction represents a part of a whole, written as a ratio of two integers (numerator/denominator).
Types of fractions:
- Proper fractions have a numerator smaller than the denominator (e.g., 3/4).
- Improper fractions have a numerator equal to or greater than the denominator (e.g., 5/3).
- Mixed numbers consist of a whole number combined with a proper fraction (e.g., 1 2/3).
Operations involving fractions:
- Addition and subtraction require a common denominator for calculations.
- Multiplication involves multiplying top and bottom parts: e.g., 1/2 × 3/4 equals 3/8.
- To divide fractions, multiply by the reciprocal; e.g., 1/2 ÷ 3/4 can be simplified to 1/2 × 4/3 = 2/3.
Fraction simplification involves dividing the numerator and denominator by their greatest common divisor (GCD).

Decimals are a representation of fractions using a decimal point.
Types of decimals include:
- Terminating decimals which indicate a finite number of digits (e.g., 0.75).
- Repeating decimals where one or more digits repeat indefinitely (e.g., 0.333...).
Conversion methods:
- To convert fractions to decimals, divide the numerator by the denominator (e.g., 1/4 equals 0.25).
- To transform decimals into fractions, determine the place value, such as converting 0.75 to 75/100 which simplifies to 3/4.
Decimal operations:
- Addition and subtraction require alignment of decimal points.
- For multiplication, treat numbers as wholes then count decimal places for the final placement.
- In division, simplify by adjusting decimal points before dividing to ease calculations.

Integers can be expressed as fractions (e.g., the integer 5 is equivalent to 5/1) and decimals (e.g., 5 corresponds to 5.0).
Follow the order of operations by applying PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) or BODMAS for calculations involving integers, fractions, and decimals.