Problem 65 - Algebra for College Students

Question

Give, in interval notation, the unknown numbers in each description. A number is between 0 and 1

Step-by-Step Solution

Answer

(0, 1)

Step 1: Understand the Problem

Identify the given conditions for the number. A number is said to be between 0 and 1.

Step 2: Identify the Endpoints

The number must be greater than 0 and less than 1. Hence the endpoints are 0 and 1.

Step 3: Determine Types of Intervals

Intervals can be open or closed, but for 'between 0 and 1,' excludes the endpoints. This means an open interval is needed.

Step 4: Write Interval Notation

Express the interval without including the endpoints as an open interval. Thus, the interval notation is \( (0, 1) \).

Key Concepts

open interval

An open interval is a range of numbers where the endpoints are not included. It is written using parentheses. For example, the open interval between 0 and 1 is written as \( (0, 1) \). The numbers within this interval are always \textbf{greater than} 0 and \textbf{less than} 1. Open intervals allow values to get very close to the endpoints but never actually reach them. This is very useful in real-world scenarios where extremes are not practical or possible.

endpoints

Endpoints are the values that mark the beginning and end of an interval. In the exercise mentioned, the endpoints are 0 and 1. Because the interval is open, these endpoints are not included in the final range. This differentiation is key in interval notation. There are two main types of endpoints in interval notation:

• Open endpoints which use parentheses \((a, b)\)
• Closed endpoints which use brackets \([a, b]\)

Open intervals exclude the endpoints, while closed intervals include them.

greater than and less than

When working with inequalities, 'greater than' and 'less than' help define the range of values. In the given example, the number must be greater than 0 and less than 1. This can be mathematically written as:

• 0 < x < 1

This means that the number can take any value between 0 and 1, but not actually 0 or 1. Through interval notation, this relationship translates to \( (0, 1) \). Understanding 'greater than' and 'less than' is essential for correct interval notation, as they guide the inclusion or exclusion of endpoints.