Section 1.1 - Real Numbers and the Rectangular Coordinate System

Sets of Real Numbers

The real numbers are a fundamental concept in algebra, encompassing several important subsets. Understanding these sets is essential for working with equations, graphs, and mathematical models.

• Natural Numbers: The set of counting numbers, starting from 1 and increasing by 1 each time. Definition:

• Whole Numbers: The set of natural numbers including zero. Definition:

• Integers: The set of whole numbers and their negatives. Definition:

• Rational Numbers: Numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Definition:

• Irrational Numbers: Numbers that cannot be written as a simple fraction; their decimal expansions are non-repeating and non-terminating. Definition:

• Real Numbers: The set of all rational and irrational numbers. Definition:

The Set of Real Numbers and the Number Line

Real numbers can be represented visually on a number line, which is a straight line where each point corresponds to a real number. The point corresponding to zero is called the origin.

• Number Line: Used to graphically represent real numbers, showing their order and relative position.

• Origin: The point on the number line corresponding to $0$.

Example: The graph below shows the set of real numbers from to $5$:

The Rectangular Coordinate System (Cartesian Plane)

The rectangular coordinate system, also known as the Cartesian coordinate system, is formed by two number lines intersecting at right angles at their origins. This system allows us to graph points, lines, and curves in two dimensions.

• Axes: The horizontal axis is called the x-axis, and the vertical axis is called the y-axis.

• Origin: The point where the axes intersect.

• Quadrants: The plane is divided into four regions called quadrants, labeled I, II, III, and IV.

• Point Representation: Each point in the plane is represented by an ordered pair , where is the x-coordinate and is the y-coordinate.

Example: The point is located in one of the four quadrants depending on the signs of and .

Viewing Windows (Graphing Calculators and Desmos)

When using graphing calculators or graphing software, the viewing window determines which portion of the coordinate plane is displayed. Standard abbreviations are used to describe the window settings.

• Xmin: Minimum value of

• Xmax: Maximum value of

• Ymin: Minimum value of

• Ymax: Maximum value of

• Xsc: Scale (distance between tick marks on the x-axis)

• Ysc: Scale (distance between tick marks on the y-axis)

Application: Adjusting these settings allows for better visualization of graphs and data.

Rounding Numbers: Place Value Table

Rounding is the process of approximating a number to a specified place value. The table below shows how to round numbers to the nearest tenth, hundredth, and thousandth.

Example: Finding Roots on a Calculator

Roots can be approximated using a calculator and rounded to a specified decimal place.

• Find to the nearest thousandth.

• Find to the nearest thousandth.

• Find to the nearest thousandth.

Method: Enter the expression into the calculator and round the result appropriately.

Example: Approximating Expressions with a Calculator

Complex expressions can be evaluated and rounded to a specified decimal place using a calculator.

• Approximate to the nearest hundredth.

• Approximate to the nearest hundredth.

• Approximate to the nearest hundredth.

Method: Enter the expression into the calculator and round the result to the required place value.