Number System and Fractions

Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).
Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).
Integers: Whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3 ...).
Rational Numbers: Numbers that can be expressed as a fraction (p/q, where q ≠ 0).
Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).
Real Numbers: All rational and irrational numbers.

Definition: A fraction represents a part of a whole (numerator/denominator).
Types of Fractions:
- Proper Fractions: Numerator < denominator (e.g., 3/4).
- Improper Fractions: Numerator ≥ denominator (e.g., 5/4).
- Mixed Numbers: A whole number combined with a proper fraction (e.g., 1 1/4).
Addition/Subtraction:
- Same denominator: Add/Subtract numerators.
- Different denominators: Find LCM, adjust fractions, then add/subtract.
Multiplication: Multiply numerators and denominators (a/b) × (c/d) = (a×c)/(b×d).
Division: Multiply by the reciprocal (a/b) ÷ (c/d) = (a/b) × (d/c).

Definition: A way of representing fractions using a decimal point (e.g., 0.75).
Types of Decimals:
- Terminating Decimals: Decimal that ends (e.g., 0.5, 0.75).
- Recurring Decimals: Decimal that repeats (e.g., 0.333...).
Conversion:
- Fraction to Decimal: Divide numerator by denominator.
- Decimal to Fraction: Write decimal as fraction and simplify.
Operations:
- Addition/Subtraction: Align decimal points.
- Multiplication: Multiply like whole numbers, count decimal places.
- Division: Adjust to eliminate decimals in the divisor.

Basic Shapes:
- Circle: Round shape with a center point; radius and diameter.
- Triangle: Three sides; classified as scalene, isosceles, or equilateral.
- Quadrilaterals: Four sides; types include squares, rectangles, rhombuses, trapezoids.
Angles:
- Acute: Less than 90 degrees.
- Right: Exactly 90 degrees.
- Obtuse: Greater than 90 but less than 180 degrees.
Perimeter: Total distance around a shape (sum of all sides).
Area: Space within a shape.
- Rectangle: Length × Width.
- Triangle: (Base × Height) / 2.
- Circle: π × (radius)^2.

Definition: A way to express a number as a fraction of 100.
Conversion:
- Fraction to Percentage: Multiply by 100.
- Decimal to Percentage: Multiply by 100.
Calculations:
- Finding Percentage: (Part/Whole) × 100 = Percentage.
- Increase/Decrease: New Value = Original Value ± (Percentage × Original Value).
Applications:
- Used in financial calculations (discounts, interest rates).
- Expressing statistics and data analysis.

Natural Numbers: The set of counting numbers starting from 1 and continuing indefinitely (1, 2, 3,...).
Whole Numbers: Includes all natural numbers plus zero, represented as (0, 1, 2, 3,...).
Integers: Comprises all whole numbers and their negative counterparts, ranging from -∞ to +∞ (..., -3, -2, -1, 0, 1, 2, 3...).
Rational Numbers: Any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
Irrational Numbers: Numbers that cannot be expressed as a fraction, including non-repeating and non-terminating decimals (e.g., √2, π).
Real Numbers: Encompasses all rational and irrational numbers, forming the complete number line.

Definition: A fraction denotes a part of a whole, expressed as a numerator over a denominator (numerator/denominator).
Types of Fractions:
- Proper Fractions: The numerator is less than the denominator (e.g., 3/4).
- Improper Fractions: The numerator is equal to or greater than the denominator (e.g., 5/4).
- Mixed Numbers: A combination of a whole number and a proper fraction (e.g., 1 1/4).
Addition/Subtraction:
- Same Denominator: Simply add or subtract the numerators while keeping the denominator the same.
- Different Denominators: Find the Least Common Multiple (LCM), adjust fractions, and then perform the operation.
Multiplication: Multiply the numerators together and multiply the denominators together (a/b) × (c/d) = (a×c)/(b×d).
Division: Involves multiplying by the reciprocal of the divisor (a/b) ÷ (c/d) = (a/b) × (d/c).

Definition: A method to represent fractions using a decimal point, such as 0.75.
Types of Decimals:
- Terminating Decimals: Decimals that have a finite number of digits (e.g., 0.5, 0.75).
- Recurring Decimals: Decimals that have digits that repeat infinitely (e.g., 0.333...).
Conversion:
- Fraction to Decimal: Divide the numerator by the denominator.
- Decimal to Fraction: Express the decimal as a fraction and simplify.
Operations:
- Addition/Subtraction: Align the decimal points before performing the operation.
- Multiplication: Multiply as with whole numbers, then count and adjust for decimal places.
- Division: Adjust the divisor to eliminate decimals before performing the division.

Basic Shapes:
- Circle: Defined by a round shape with a center point, characterized by radius and diameter.
- Triangle: Composed of three sides and can be classified as scalene, isosceles, or equilateral based on side lengths.
- Quadrilaterals: Four-sided shapes, which include squares, rectangles, rhombuses, and trapezoids.
Angles:
- Acute Angle: Measures less than 90 degrees.
- Right Angle: Measures exactly 90 degrees.
- Obtuse Angle: Measures greater than 90 but less than 180 degrees.
Perimeter: The total distance around a shape, calculated by summing all sides.
Area: The amount of space enclosed within a shape.
- Rectangle: Area is calculated as Length × Width.
- Triangle: Area is calculated using (Base × Height) / 2.
- Circle: Area is determined by π × (radius)^2.

Definition: A method to express a number as a fraction of 100, commonly denoted with the symbol %.
Conversion:
- Fraction to Percentage: Multiply the fraction by 100.
- Decimal to Percentage: Multiply the decimal by 100.
Calculations:
- Finding Percentage: The formula (Part/Whole) × 100 yields the percentage.
- Increase/Decrease Calculations: New Value is obtained using Original Value ± (Percentage × Original Value).
Applications: Widely used in financial contexts such as calculating discounts and interest rates, as well as in statistics and data analysis.