Number Systems and Real Number Operations

Number Systems

• Integers: Whole numbers (positive, negative, and zero) e.g., ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠ 0. e.g., 1/2, 3, -5/4, 0.75
• Irrational Numbers: Numbers that cannot be expressed as a fraction. Their decimal representations are non-repeating and non-terminating. e.g., √2, π
• Real Numbers: The set of all rational and irrational numbers.
  • Every integer is a rational number.
  • Every rational number is a real number.
  • Every irrational number is a real number.

Operations with Real Numbers

• Addition: Combining numbers. Rules apply for integers, fractions, and decimals.
• Subtraction: Finding the difference between numbers. Rules for integers, fractions, and decimals apply.
  • Subtracting a negative number is the same as adding the corresponding positive number.
• Multiplication: Repeated addition. Rules for integer signs (positive, negative) are crucial.
• Division: Repeated subtraction. Rules for integer signs are important. Division by zero is undefined.

Exponents and Scientific Notation

• Exponent Rules: Understand rules for multiplication, division, powers of powers, and negative exponents.
• Scientific Notation: Expressing very large or very small numbers in a compact form using powers of 10. e.g., 3.2 x 10⁵ or 2.5 x 10^-3

Order of Operations (PEMDAS/BODMAS)

• Parentheses/Brackets: Evaluate expressions within grouped symbols first.
• Exponents/Orders: Calculate exponential expressions.
• Multiplication and Division: Perform these operations from left to right.
• Addition and Subtraction: Perform these operations from left to right.

Algebraic Expressions

• Variables: Letters representing unknown values.
• Constants: Numerical values.
• Terms: Parts of an expression separated by + or -. e.g., in 2x + 3y - 5, the terms are 2x, 3y, and -5.
• Combining Like Terms: Adding or subtracting terms with the same variables raised to the same powers. e.g., 2x + 3x = 5x
• Simplifying expressions

Linear Equations

• Solving linear equations in one variable: Isolate the variable using inverse operations (addition, subtraction, multiplication, division) to find the solution.
• Word problems: Translate word problems into algebraic equations
• Example: If 2x + 5 = 11, solve for x.

Linear Inequalities

• Solving linear inequalities: Similar to solving equations, but remember that multiplying or dividing by a negative number reverses the inequality sign.
• Graphing linear inequalities: Represent solutions on a number line.

Linear Equations in Two Variables

• Slope-intercept form: y = mx + b
• Calculating the slope from two points: (y₂ - y₁)/(x₂ - x₁)
• Graphing linear equations: Using the slope and y-intercept to plot the line.
• Systems of linear equations: Finding the intersection point of two or more linear graphs.

Polynomials

• Definitions: Expressions containing variables with whole number exponents.
• Addition, Subtraction, and Multiplication of Polynomials

Factoring

• Factoring: Techniques for breaking down polynomials into simpler expressions.

Geometry Review

• Basic Shapes: Identify and understand the properties of triangles, quadrilaterals, and other polygons (pentagons, hexagons).
• Perimeter: Calculate the distance around a polygon.
• Area: Measure the surface covered by a two-dimensional shape.

Measurement

• Units of Measurement: Know the standard units for length, area, volume, angles, and time.
• Converting between units.

Data Analysis

• Types of Data: Identify different data types (quantitative, qualitative).
• Measures of Central Tendency: Mean, median, mode.
• Measures of Dispersion: Range, standard deviation.
• Representing Data: Dot plots, histograms, box-and-whisker plots, stem-and-leaf plots, and scatter plots.

Probability

• Basic Probability Concepts: The likelihood of an event occurring.
• Outcomes, Events.