Number Systems

Number Systems

• Real Numbers: A combination of rational and irrational numbers.
• Rational Numbers: Can be expressed in the form p/q, where p and q are integers and q ≠ 0.
  • Properties:
    • Closure: The result of adding, subtracting, multiplying, or dividing two rational numbers is always a rational number.
    • Commutativity: The order of rational numbers does not change the result of addition and multiplication.
    • Associativity: The order in which rational numbers are added or multiplied does not change the result.
    • Distributivity: a(b + c) = ab + ac
• Irrational Numbers: Cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
  • Examples: π, e, √2
• Recurring Decimals: Decimals that have a repeating pattern of digits.
  • Example: 0.12341234...
• Non-Recurring Decimals: Decimals that do not have a repeating pattern of digits.
  • Example: π

Polynomials

• Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
  • Example: 2x^2 + 3x - 4
• Degree of a Polynomial: The highest power of the variable in the polynomial.
  • Example: The degree of 2x^2 + 3x - 4 is 2.
• Terms of a Polynomial: The individual parts of the polynomial separated by + or - signs.
  • Example: The terms of 2x^2 + 3x - 4 are 2x^2, 3x, and -4.
• Types of Polynomials:
  • Monomial: A polynomial with only one term.
    • Example: 2x^2
  • Binomial: A polynomial with two terms.
    • Example: x^2 + 3x
  • Trinomial: A polynomial with three terms.
    • Example: x^2 + 3x - 4
• Addition and Subtraction of Polynomials:
  • To add or subtract polynomials, combine like terms.
  • Example: (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x

Coordinate Geometry

• Cartesian Plane: A two-dimensional plane with a horizontal x-axis and a vertical y-axis.
• Coordinates: A pair of numbers (x, y) that represent a point on the Cartesian plane.
• Plotting Points: To mark a point on the Cartesian plane using its coordinates.
• Distance Formula: The distance between two points (x1, y1) and (x2, y2) is given by √((x2 - x1)^2 + (y2 - y1)^2).
• Section Formula: The coordinates of a point that divides a line segment in the ratio m:n are given by ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)).
• Midpoint Formula: The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by ((x1 + x2) / 2, (y1 + y2) / 2).

Number Systems

• Real numbers combine rational and irrational numbers.
• Rational numbers can be expressed as p/q, where p and q are integers and q ≠ 0, and have properties of closure, commutativity, associativity, and distributivity.
• Irrational numbers cannot be expressed as p/q, and examples include π, e, and √2.
• Recurring decimals have a repeating pattern of digits, such as 0.12341234..., while non-recurring decimals do not, like π.

Polynomials

• A polynomial is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication.
• The degree of a polynomial is the highest power of the variable, such as 2 in 2x^2 + 3x - 4.
• Terms of a polynomial are individual parts separated by + or - signs, like 2x^2, 3x, and -4 in 2x^2 + 3x - 4.
• Types of polynomials include monomials (one term), binomials (two terms), and trinomials (three terms).
• To add or subtract polynomials, combine like terms, such as (2x^2 + 3x) + (x^2 - 2x) = 3x^2 + x.

Coordinate Geometry

• The Cartesian plane is a two-dimensional plane with a horizontal x-axis and a vertical y-axis.
• Coordinates are a pair of numbers (x, y) representing a point on the Cartesian plane.
• Plotting points involves marking a point on the plane using its coordinates.
• The distance formula calculates the distance between two points (x1, y1) and (x2, y2) as √((x2 - x1)^2 + (y2 - y1)^2).
• The section formula gives the coordinates of a point dividing a line segment in the ratio m:n as ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n)).
• The midpoint formula calculates the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) as ((x1 + x2) / 2, (y1 + y2) / 2).

Explore the properties and definitions of real and rational numbers, including their properties and forms. Learn about closure, commutativity, and associativity of rational numbers.