Key Concepts in Mathematics 1

Key Concepts in Mathematics 1

Number Systems

• Natural Numbers (N): Positive integers (1, 2, 3, ...).
• Whole Numbers (W): Natural numbers plus zero (0, 1, 2, ...).
• Integers (Z): Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers (Q): Numbers that can be expressed as a fraction (a/b, where b ≠ 0).
• Irrational Numbers: Numbers that cannot be expressed as fractions (e.g., √2, π).
• Real Numbers (R): All rational and irrational numbers.

Basic Arithmetic Operations

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

Properties of Operations

• Commutative Property:
  • Addition: a + b = b + a
  • Multiplication: a × b = b × a
• Associative Property:
  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)
• Distributive Property: a × (b + c) = (a × b) + (a × c)

Fractions

• Definition: A ratio of two integers (numerator/denominator).
• Simplifying: Dividing both numerator and denominator by their greatest common divisor (GCD).
• Addition/Subtraction: Find common denominator, adjust numerators, then operate.
• Multiplication: Multiply numerators together and denominators together.
• Division: Multiply by the reciprocal of the divisor.

Decimals

• Definition: A fractional number expressed in the base 10 system.
• Conversion:
  • Fraction to Decimal: Divide numerator by denominator.
  • Decimal to Fraction: Use place value to write as a fraction (e.g., 0.75 = 75/100 = 3/4).

Percentages

• Definition: A fraction expressed as a part of 100.
• Calculation:
  • To find percentage: (part/whole) × 100.
  • To find part: (percentage × whole) / 100.

Algebra

• Variables: Symbols representing numbers (e.g., x, y).
• Expressions: Combinations of variables and constants (e.g., 3x + 2).
• Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).
• Solving Equations: Isolate the variable using inverse operations.

Geometry

• Basic Shapes:
  • Triangle: 3 sides
  • Square: 4 equal sides
  • Rectangle: 4 sides, opposite sides equal
  • Circle: Round shape with a constant radius.
• Perimeter: Sum of all sides of a shape.
• Area: Surface covered by a shape.
  • Triangle: 1/2 × base × height.
  • Rectangle: length × width.
  • Circle: π × radius².

Data Handling

• Mean: Average of a set of numbers.
• Median: Middle value when data is organized in order.
• Mode: Most frequently occurring value.
• Range: Difference between the highest and lowest values in a data set.

Basic Probability

• Definition: Measure of the likelihood of an event occurring.
• Formula: Probability (P) = Number of favorable outcomes / Total number of outcomes.

Number Systems

• Natural Numbers (N): These are the counting numbers, starting from 1 and going up indefinitely (1, 2, 3, ...).
• Whole Numbers (W): These are the natural numbers plus zero (0, 1, 2, ...), encompassing all positive integers and the number zero.
• Integers (Z): Integers include all whole numbers but also their negative counterparts (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers (Q): Rational numbers are those that can be expressed as a fraction, where the numerator and denominator are integers, and the denominator is not zero (a/b, where b ≠ 0).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction are called irrational. Examples include the square root of 2 (√2) and pi (π).
• Real Numbers (R): The set of real numbers encompasses all rational and irrational numbers. This means almost any number you can think of is a real number.

Basic Arithmetic Operations

• Addition (+): Represents combining two numbers together.
• Subtraction (−): Represents finding the difference between two numbers.
• Multiplication (×): Repeated addition of the same number.
• Division (÷): Splitting a larger number into equal, smaller parts.

Properties of Operations

• Commutative Property: The order of operations doesn't matter for addition and multiplication.
  • Addition: a + b = b + a (e.g., 2 + 3 = 3 + 2)
  • Multiplication: a × b = b × a (e.g., 2 × 3 = 3 × 2)
• Associative Property: Grouping doesn't affect the outcome of addition and multiplication.
  • Addition: (a + b) + c = a + (b + c) (e.g., (2 + 3) + 4 = 2 + (3 + 4))
  • Multiplication: (a × b) × c = a × (b × c) (e.g., (2 × 3) × 4 = 2 × (3 × 4))
• Distributive Property: This allows you to multiply a number by a sum by multiplying each term in the sum separately and then adding the results.
  • a × (b + c) = (a × b) + (a × c) (e.g., 2 × (3 + 4) = (2 × 3) + (2 × 4))

Fractions

• Definition: A fraction represents a part of a whole and is written as a ratio of two integers (numerator/denominator).
• Simplifying: To simplify a fraction, divide both the numerator and denominator by their greatest common divisor (GCD).
• Addition/Subtraction: When adding or subtracting fractions, you need to find a common denominator for both fractions, adjust the numerators, and then perform the operation on the numerators.
• Multiplication: To multiply fractions, multiply the numerators together and the denominators together.
• Division: To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

Decimals

• Definition: Decimals are fractional numbers expressed using the base 10 system, where a dot (.) separates the whole number part from the fractional part.
• Conversion:
  • Fraction to Decimal: Divide the numerator of the fraction by the denominator.
  • Decimal to Fraction: Use place value to determine the fraction. For example, 0.75 is equivalent to 75/100, which can be simplified to 3/4.

Percentages

• Definition: A percentage represents a fraction expressed as a part of 100.
• Calculation:
  • To find the percentage: Divide the part by the whole and multiply by 100.
  • To find the part: Multiply the percentage by the whole and divide by 100.

Algebra

• Variables: Variables are symbols used to represent unknown numbers in equations.
• Expressions: Algebraic expressions combine variables and constants using arithmetic operations (e.g., 3x + 2).
• Equations: Equations express the equality between two algebraic expressions (e.g., 2x + 3 = 7).
• Solving Equations: To solve an equation for a variable, you use inverse operations to isolate the variable on one side of the equation.

Geometry

• Basic Shapes:
  • Triangle: A polygon with three sides and three angles.
  • Square: A quadrilateral with four equal sides and four right angles.
  • Rectangle: A quadrilateral with four sides, where opposite sides are equal and all angles are right angles.
  • Circle: A closed curve where all points are equidistant from a central point (the center).
• Perimeter: The total length of all the sides of a shape.
• Area: The amount of surface covered by a shape.
  • Triangle: Area = 1/2 × base × height
  • Rectangle: Area = length × width
  • Circle: Area = π × radius²

Data Handling

• Mean: The average of a set of numbers, calculated by summing all the numbers and dividing by the total number of values.
• Median: The middle value in a data set when it is arranged in order.
• Mode: The value that occurs most frequently in a data set.
• Range: The difference between the highest and lowest values in a data set.

Basic Probability

• Definition: Probability measures the likelihood of a particular event occurring.
• Formula: Probability (P) = Number of favorable outcomes / Total number of outcomes.