Basic Mathematics Study Notes

Basic Mathematics Study Notes

1. Fundamental Operations

• Addition (+): Combining two or more numbers to get a sum.
• Subtraction (−): Finding the difference between two numbers.
• Multiplication (×): Repeated addition of a number.
• Division (÷): Splitting a number into equal parts.

2. Number Types

• Natural Numbers: Counting numbers (1, 2, 3, ...).
• Whole Numbers: Natural numbers including zero (0, 1, 2, ...).
• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers: Numbers that can be expressed as a fraction (p/q, where p and q are integers).
• Irrational Numbers: Numbers that cannot be expressed as a fraction (e.g., √2, π).

3. Order of Operations

• Use the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

4. Fractions

• Numerator: The top part of a fraction.
• Denominator: The bottom part of a fraction.
• Simplifying Fractions: Dividing the numerator and denominator by their greatest common divisor (GCD).
• Converting Mixed Numbers: Changing a mixed number to an improper fraction (e.g., 2 1/3 to 7/3).

5. Decimals

• Place Value: Each digit's value based on its position (tenths, hundredths, etc.).
• Conversion: Changing fractions to decimals and vice versa.
• Rounding: Adjusting a number to a specified place value.

6. Percentages

• Definition: A fraction out of 100.
• Conversion: Changing a fraction or decimal to a percentage (multiply by 100).
• Calculating Percentages: Finding a percentage of a number (e.g., 20% of 50 = 0.2 × 50).

7. Ratios and Proportions

• Ratio: A comparison of two quantities (e.g., 3:2).
• Proportion: An equation that states two ratios are equal (e.g., 3/4 = 6/8).

8. Basic Geometry

• Shapes:
  • Triangle: 3 sides
  • Square: 4 equal sides
  • Rectangle: Opposite sides are equal
• Perimeter: The total distance around a shape.
• Area: The space within a shape (e.g., for rectangles A = length × width).

9. Algebra Basics

• Variables: Symbols representing unknown values (e.g., x, y).
• Expressions: Combinations of numbers, variables, and operations (e.g., 3x + 2).
• Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).

10. Measurement

• Units: Standard quantities used to measure (length, area, volume).
• Conversions: Changing from one unit to another (e.g., inches to centimeters).

11. Data and Statistics

• Mean: Average of a data set.
• Median: The middle value when data is ordered.
• Mode: The value that appears most frequently.

Key Tips

• Practice basic operations regularly to improve speed and accuracy.
• Familiarize yourself with different number types and their properties.
• Use visual aids (like number lines or graphs) for better understanding.
• Apply mathematical concepts to real-life situations for practical understanding.

Fundamental Operations

• Addition (+): Combines values to obtain a total (sum).
• Subtraction (−): Calculates the difference between two numbers.
• Multiplication (×): Represents repeated addition of a number.
• Division (÷): Divides a number into specified equal parts.

Number Types

• Natural Numbers: Series of positive counting numbers starting from 1.
• Whole Numbers: Includes all natural numbers and zero.
• Integers: Comprises whole numbers and their negative counterparts.
• Rational Numbers: Can be expressed as a fraction of two integers (p/q).
• Irrational Numbers: Cannot be expressed as a fraction (examples include √2 and π).

Order of Operations

• Follow PEMDAS for solving equations:
  • Parentheses first
  • Exponents second
  • Multiplication and Division from left to right
  • Addition and Subtraction from left to right

Fractions

• Numerator: Indicates how many parts are taken.
• Denominator: Indicates the total number of equal parts.
• Simplifying: Reduces fractions by dividing the numerator and denominator by their GCD.
• Mixed Numbers: Can be converted into improper fractions (e.g., 2 1/3 becomes 7/3).

Decimals

• Place Value: Each digit's significance changes based on its position (e.g., tenths, hundredths).
• Conversions: Fractions can be turned into decimals and vice versa.
• Rounding: Adjusts values to a designated place value (e.g., nearest whole number).

Percentages

• Definition: Represents a ratio or fraction as parts of 100.
• Conversion: To express a fraction or decimal as a percentage, multiply by 100.
• Calculating Percentages: To find a percentage of a value, multiply the percentage (as a decimal) by the number.

Ratios and Proportions

• Ratio: Compares two quantities (expressed like 3:2).
• Proportion: States that two ratios are equivalent (e.g., 3/4 = 6/8).

Basic Geometry

• Shapes:
  • Triangle: Has three sides.
  • Square: Contains four equal sides.
  • Rectangle: Opposite sides are equal in length.
• Perimeter: Total distance around a geometric figure.
• Area: Measurement of the space inside a shape, calculated for rectangles as A = length × width.

Algebra Basics

• Variables: Symbols (like x or y) that stand for unknown values.
• Expressions: Combinations of numbers, variables, and operations (like 3x + 2).
• Equations: Statements asserting that two expressions hold the same value (e.g., 2x + 3 = 7).

Measurement

• Units: Standard measurements for assessing dimensions like length and volume.
• Conversions: Process of changing a measurement from one unit to another (e.g., from inches to centimeters).

Data and Statistics

• Mean: Average value calculated by summing data points and dividing by their count.
• Median: Middle value derived from ordered datasets.
• Mode: The value that occurs most frequently in a dataset.

Key Tips

• Regular practice of fundamental operations enhances speed and accuracy.
• Understand various number types and their individual characteristics.
• Utilize visual tools, such as number lines and graphs, for improved comprehension.
• Link mathematical concepts to everyday instances for practical application.