Math Basics: Numbers and Operations

Numbers and Operations

• Number Systems:
  • Natural Numbers (1, 2, 3, ...)
  • Whole Numbers (0, 1, 2, 3, ...)
  • Integers (...,-3, -2, -1, 0, 1, 2, 3, ...)
  • Rational Numbers (fractions, decimals, percentages)
  • Irrational Numbers (non-repeating decimals)
  • Real Numbers (all rational and irrational numbers)
• Operations:
  • Addition (+)
  • Subtraction (-)
  • Multiplication (×)
  • Division (÷)
  • Exponents (^)
  • Roots (√)

Algebra

• Variables and Expressions:
  • Constants (numbers)
  • Variables (letters or symbols)
  • Algebraic Expressions (combinations of constants and variables)
• Equations and Inequalities:
  • Linear Equations (ax + by = c)
  • Quadratic Equations (ax^2 + bx + c = 0)
  • Inequalities (<, >, ≤, ≥)
• Functions:
  • Domain (input values)
  • Range (output values)
  • Function Notation (f(x))

Geometry

• Points, Lines, and Planes:
  • Points (locations in space)
  • Lines (sets of points extending infinitely)
  • Planes (flat surfaces)
• Angles and Measurements:
  • Degrees (angles)
  • Radians (alternative angle measurement)
  • Perimeter (distance around a shape)
  • Area (amount of space inside a shape)
• Shapes and Solids:
  • 2D Shapes (triangles, quadrilaterals, polygons, circles)
  • 3D Solids (prisms, pyramids, spheres, cones)

Data Analysis and Statistics

• Data Types:
  • Quantitative (numerical)
  • Qualitative ( categorical)
• Measures of Central Tendency:
  • Mean (average)
  • Median (middle value)
  • Mode (most frequent value)
• Data Visualization:
  • Charts (bar, pie, line)
  • Graphs (scatter plots, histograms)

Other Topics

• Trigonometry:
  • Triangles (right, oblique, similar)
  • Trigonometric Functions (sine, cosine, tangent)
• Calculus:
  • Limits (approaching values)
  • Derivatives (rates of change)
  • Integrals (accumulated quantities)

Numbers and Operations

• Natural Numbers start from 1 and are positive integers.
• Whole Numbers include 0 and all positive integers.
• Integers include all whole numbers and their negative counterparts.
• Rational Numbers include fractions, decimals, and percentages.
• Irrational Numbers are non-repeating decimals.
• Real Numbers combine all rational and irrational numbers.

Algebra

• Constants are numerical values.
• Variables are represented by letters or symbols.
• Algebraic Expressions combine constants and variables.
• Linear Equations are in the form of ax + by = c.
• Quadratic Equations are in the form of ax^2 + bx + c = 0.
• Inequalities use greater than, less than, or equal to symbols.
• Functions have a domain of input values and a range of output values.
• Function Notation is represented as f(x).

Geometry

• Points are locations in space.
• Lines are sets of points extending infinitely in two directions.
• Planes are flat surfaces.
• Degrees measure angles.
• Radians are an alternative angle measurement.
• Perimeter is the distance around a shape.
• Area is the amount of space inside a shape.
• 2D Shapes include triangles, quadrilaterals, polygons, and circles.
• 3D Solids include prisms, pyramids, spheres, and cones.

Data Analysis and Statistics

• Quantitative Data is numerical.
• Qualitative Data is categorical.
• Mean is a measure of central tendency.
• Median is the middle value in a dataset.
• Mode is the most frequent value in a dataset.
• Charts include bar, pie, and line charts.
• Graphs include scatter plots and histograms.

Other Topics

• Trigonometry deals with right, oblique, and similar triangles.
• Trigonometric Functions include sine, cosine, and tangent.
• Calculus deals with limits, derivatives, and integrals.
• Limits approach values.
• Derivatives measure rates of change.
• Integrals calculate accumulated quantities.

Multiplying Polynomials

Distributive Property

• Multiplying a single value to multiple terms inside a set of parentheses is possible using the distributive property.
• Formula: a(b + c) = ab + ac
• Example: 2(x + 3) = 2x + 6

Quadratic Expressions

• A quadratic expression is a polynomial of degree two, in the form ax^2 + bx + c
• Factoring quadratics involves expressing the expression as a product of two binomials
• Formula: ax^2 + bx + c = (px + q)(rx + s)
• Example: x^2 + 5x + 6 = (x + 3)(x + 2)

Binomial Expansion

• Expanding binomials involves multiplying two binomials to form a polynomial
• Formula: (a + b)(c + d) = ac + ad + bc + bd
• Example: (x + 2)(x + 3) = x^2 + 5x + 6

Complex Polynomial Multiplication

• Complex polynomial multiplication involves multiplying two polynomials with multiple terms
• Multiply each term in one polynomial by each term in the other polynomial
• Combine like terms to simplify the result
• Example: (x^2 + 3x + 2)(x + 1) = x^3 + 4x^2 + 5x + 2

FOIL Method

• FOIL stands for "First, Outer, Inner, Last" and is a method for multiplying two binomials
• Multiply the:
  • First terms: a*c
  • Outer terms: a*d
  • Inner terms: b*c
  • Last terms: b*d
• Combine the four terms to form the product
• Example: (x + 3)(x + 2) = x^2 + 5x + 6