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)
  • Irrational Numbers (non-repeating decimals)
  • Real Numbers (all rational and irrational numbers)
  • Complex Numbers (with imaginary parts)

• Operations:

  • Addition (+)
  • Subtraction (-)
  • Multiplication (×)
  • Division (÷)
  • Exponents ( powers)
  • Roots (square, cube, etc.)

Algebra

• Equations and Inequalities:

  • Linear Equations (e.g., 2x + 3 = 5)
  • Quadratic Equations (e.g., x^2 + 4x + 4 = 0)
  • Inequalities (e.g., 2x - 3 > 5)
  • Systems of Equations (e.g., 2x + y = 4, x - 2y = -3)

• Functions:

  • Domain and Range
  • Linear Functions (e.g., f(x) = 2x + 1)
  • Quadratic Functions (e.g., f(x) = x^2 + 2x + 1)
  • Exponential Functions (e.g., f(x) = 2^x)

Geometry

• Points, Lines, and Planes:

  • Midpoint Formula
  • Distance Formula
  • Slope Formula
  • Angle Properties (e.g., complementary, supplementary, vertical)

• Shapes:

  • Properties of Triangles (e.g., congruent, similar, right)
  • Properties of Quadrilaterals (e.g., rectangle, square, trapezoid)
  • Properties of Circles (e.g., center, radius, circumference)

Calculus

• Limits:

  • Basic Limit Properties (e.g., sum, product, chain rule)
  • One-Sided Limits
  • Infinite Limits

• Derivatives:

  • Rules of Differentiation (e.g., power rule, product rule, quotient rule)
  • Geometric Interpretation of Derivatives
  • Higher-Order Derivatives

• Integrals:

  • Basic Integration Rules (e.g., power rule, substitution method)
  • Area Between Curves
  • Volume of Solids

Numbers and Operations

• Natural Numbers are positive integers starting from 1, with no upper bound (e.g., 1, 2, 3, ...).
• Whole Numbers are non-negative integers, including 0 (e.g., 0, 1, 2, 3, ...).
• Integers include all whole numbers and their negative counterparts (e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers are fractions that can be expressed as a finite decimal or ratio of integers (e.g., 3/4, 22/7).
• Irrational Numbers are non-repeating decimals that cannot be expressed as a finite ratio of integers (e.g., π, e).
• Real Numbers comprise all rational and irrational numbers.
• Complex Numbers have imaginary parts, expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit.
• Addition (+) is the operation of combining two or more numbers to get a total or a sum.
• Subtraction (-) is the operation of finding the difference between two numbers.
• Multiplication (×) is the operation of repeating a number a certain number of times.
• Division (÷) is the operation of finding how many times one number fits into another.
• Exponents (powers) are used to represent repeated multiplication.
• Roots (square, cube, etc.) are used to find the value of a number that, when multiplied by itself, gives a specified value.

Algebra

• Linear Equations have a degree of 1 and can be written in the form ax + by = c, where a, b, and c are constants.
• Quadratic Equations have a degree of 2 and can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.
• Inequalities are statements that compare two values using greater than, less than, or equal to.
• Systems of Equations are sets of two or more equations that must be solved simultaneously.
• Functions have a domain (set of input values) and a range (set of output values).
• Linear Functions have a constant rate of change and can be written in the form f(x) = mx + b, where m and b are constants.
• Quadratic Functions have a parabolic shape and can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Exponential Functions have a rate of change that is proportional to the input value and can be written in the form f(x) = a^x, where a is a constant.

Geometry

• The Midpoint Formula calculates the midpoint of a line segment, given the coordinates of the endpoints.
• The Distance Formula calculates the distance between two points, given their coordinates.
• The Slope Formula calculates the slope of a line, given the coordinates of two points.
• Angle Properties include complementary, supplementary, and vertical angles.
• Triangles have properties such as congruency, similarity, and right triangles.
• Quadrilaterals have properties such as rectangles, squares, and trapezoids.
• Circles have properties such as center, radius, and circumference.

Calculus

• Limits are used to study the behavior of functions as the input values approach a certain point.
• Basic Limit Properties include the sum, product, and chain rule.
• One-Sided Limits are used to study the behavior of functions as the input values approach a certain point from one side.
• Infinite Limits are used to study the behavior of functions as the input values approach infinity.
• Derivatives measure the rate of change of a function with respect to its input.
• Rules of Differentiation include the power rule, product rule, and quotient rule.
• Geometric Interpretation of Derivatives includes the concept of tangent lines.
• Higher-Order Derivatives are used to study the rate of change of a function multiple times.
• Integrals are used to find the accumulation of a function over a given interval.
• Basic Integration Rules include the power rule and substitution method.
• Area Between Curves is used to find the area between two curves.
• Volume of Solids is used to find the volume of a solid object.