Problem 9 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show the interval on a number line. $$ (0, \infty) $$

Step-by-Step Solution

Answer

Draw a straight horizontal line to represent the number line. Mark the number 0 on it. At 0, draw an open circle to show that it is not included in the interval. Then, draw an arrow starting from the open circle at 0 and extending to the right, representing an interval that goes towards positive infinity.

Step 1: Draw a straight horizontal line

Draw a straight horizontal line to represent the number line.

Step 2: Mark the endpoint

Mark the number 0 (the endpoint of the interval) on the number line.

Step 3: Represent the open interval

At the number 0, draw an open circle to indicate that the endpoint is not included in the interval.

Step 4: Indicate the direction of the interval

Draw an arrow starting from the open circle at 0 and extending to the right, representing an interval that goes towards positive infinity.

Key Concepts

Understanding Open Interval Notation

Open interval notation is a way of describing a range of numbers where the endpoints are not included in the set. It is denoted by parentheses \( (a, b) \) where \( a \) and \( b \) are the lower and upper bounds, respectively.

For example, the interval \( (0, \infty) \) represents all positive numbers greater than 0 but not including 0 itself. The symbol \( \infty \) stands for infinity, indicating that there is no upper limit to the values in the interval. When graphing this type of interval on a number line, you draw an open circle at the endpoint 0, which visually conveys that 0 is not a part of the interval. This is critical for students to understand because it distinguishes open intervals from closed intervals, where the endpoints are included and are represented by square brackets.

Infinite Intervals on Number Lines

Infinite intervals are used to represent sets of numbers that have no bound in one or both directions. These are often indicated by the symbol \( \infty \) for positive infinity and \( -\infty \) for negative infinity.

The interval \( (0, \infty) \) is an example of an infinite interval, where numbers start from just above 0 and continue without end. To properly graph this on a number line, after marking an open circle at 0, one should draw a line or arrow extending towards the right direction to indicate the set of all positive numbers. This visual representation helps students understand the concept of infinite growth in a particular direction. It's also essential to distinguish between different types of infinite intervals, such as \( (a, \infty) \) where \( a \) is a finite number, and \( (-\infty, b) \) where \( b \) is a finite number, and each has a different graphic representation.

Graphing on Number Lines

Graphing on number lines is a fundamental visual tool that helps illustrate various mathematical concepts, including intervals. A number line is a straight line with numbers placed at intervals along its length, usually with equal spacing.

When graphing intervals, such as \( (0, \infty) \) on a number line, specific symbols and directions are used to convey the characteristics of the interval. Open circles are used to denote that an endpoint is not included (open interval), while closed circles indicate that an endpoint is included (closed interval). Arrows are used to represent the direction and continuity of the interval. For instance, an arrow pointing rightwards from an open circle at 0, as in our example, shows that the interval includes all numbers greater than 0, heading towards infinity. This graphical representation is particularly useful in visualizing concepts like inequality solutions, ranges of functions, and understanding sets of real numbers. By mastering the art of graphing on number lines, students can easily interpret and solve problems involving intervals.