Numbers, Data, and Problem Solving

Sets of Numbers

Understanding the classification of numbers is foundational in mathematics. Numbers are grouped into sets based on their properties and uses.

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).

• Whole Numbers: Natural numbers plus zero (0, 1, 2, 3, ...).

• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...).

• Rational Numbers: Numbers that can be written as a fraction , where and are integers and .

• Irrational Numbers: Numbers that cannot be written as a fraction, such as or .

• Real Numbers: All rational and irrational numbers.

Example: Classify , , , and into their respective sets.

Additional info: The relationship among these sets is often shown as nested circles, with real numbers encompassing all others.

Order of Operations

To evaluate mathematical expressions correctly, follow the order of operations, often remembered by the acronym PEMDAS:

• P: Parentheses

• E: Exponents

• M: Multiplication

• D: Division

• A: Addition

• S: Subtraction

Negation and exponents require careful attention, especially when negative signs are involved.

Example: Evaluate and .

Additional info: ; .

Scientific and Decimal Notation

Scientific notation is used to express very large or very small numbers concisely.

• Definition: , where is a number between 1 and 10, and is an integer.

• Conversion: Move the decimal point to create a coefficient between 1 and 10, adjusting the exponent accordingly.

Example: in scientific notation is .

Operations with Scientific Notation

Arithmetic operations can be performed directly in scientific notation, often using calculators with "E" notation.

• Multiplication: Multiply coefficients and add exponents: .

• Division: Divide coefficients and subtract exponents: .

Example: .

Percent Change

Percent change measures the relative increase or decrease between two values.

• Formula:

• Interpretation: Positive result indicates increase; negative indicates decrease.

Example: If a price rises from \frac{60-50}{50} \times 100 = 20\%$.

Applications: Percent Change and Scientific Notation

Real-world problems often require combining percent change and scientific notation, such as calculating speed or volume.

• Example: Finding the speed of Earth or the volume of a soda can using formulas and scientific notation.

Additional info: Practice with textbook exercises and videos enhances understanding.

Visualizing and Graphing Data

One-Variable Data Analysis

Analyzing data involves summarizing and interpreting values from a single variable.

• Mean: The average value, calculated as .

• Median: The middle value when data is ordered.

Example: For temperatures 70, 72, 68, 75, 71, mean is .

Domain and Range of a Relation

The domain and range describe the set of possible input and output values for a relation.

• Domain: All possible input (x) values.

• Range: All possible output (y) values.

Example: For the relation {(1,2), (2,3), (3,4)}, domain is {1,2,3}, range is {2,3,4}.

Additional info: Implied domain excludes values that make expressions undefined (e.g., division by zero).

Distance Formula

The distance between two points in the plane is calculated using the distance formula.

• Formula:

Example: Between (1,2) and (4,6): .

Midpoint Formula

The midpoint between two points is the average of their coordinates.

• Formula:

Example: Between (1,2) and (4,6): .

Center and Radius of a Circle

The standard equation of a circle allows identification of its center and radius.

• Standard Equation:

• Center:

• Radius:

Example: has center (3, -2) and radius 4.

Finding the Standard Equation of a Circle

To write the equation of a circle, use its center and radius, or complete the square for general equations.

• General Equation:

• Completing the Square: Rearranging terms to match the standard form.

Example: can be rewritten by completing the square.

Graphing Circles and Other Equations with a Calculator

Graphing calculators are used to visualize equations, set viewing windows, and create scatterplots or line graphs.

• Viewing Rectangle: Adjust axes to display relevant data.

• Scatterplot: Plotting data points to observe relationships.

• Line Graph: Connecting points to show trends.

Example: Graph or a circle equation using calculator functions.

Quick Summary Table

The following table summarizes exam objectives, textbook examples, and key videos for Sections 1.1–1.2.