Calculus I, II and III

College Algebra

Complex Analysis

An open circle on a number line is used to represent a boundary value that is not included in the interval. This is a graphical representation of what is known in mathematical notation as an open interval or a parenthesis. Let's go through the steps to understand this concept and how it is represented on a number line and in interval notation. ### Step 1: Understanding Boundary Values Boundary values are the endpoints of an interval. They define the smallest and largest values in the set. In the context of intervals, there are two types of boundary values: 1. **Inclusive Boundary**: This is when the endpoint is included in the interval. In mathematical notation, this is represented by a closed interval, which uses square brackets, `[` and `]`. 2. **Exclusive Boundary**: This is when the endpoint is not included in the interval. In mathematical notation, this is represented by an open interval, which uses parentheses, `(` and `)`. ### Step 2: Graphical Representation on a Number Line On a number line, these two types of boundaries are represented differently: 1. **Inclusive Boundary**: A closed circle or dot is drawn on the number line at the point corresponding to the boundary value. 2. **Exclusive Boundary**: An open circle is drawn on the number line at the point corresponding to the boundary value, indicating that this value is not part of the interval. ### Step 3: Interval Notation In interval notation, an interval is written using a pair of numbers that represent the lower and upper bounds, with the type of bracket indicating whether the bound is included or not. - For an interval including its endpoints, square brackets are used. For example, the interval from 1 to 3, including both 1 and 3, is written as `[1, 3]`. - For an interval not including its endpoints, parentheses are used. For example, the interval from 1 to 3, not including 1 and 3, is written as `(1, 3)`. ### Step 4: Example on a Number Line Let's consider an interval from 1 to 3 that does not include the number 1 but does include the number 3. On a number line, this would be represented by: - An open circle at the point corresponding to the number 1. - A closed circle at the point corresponding to the number 3. ### Step 5: Writing the Interval in Mathematical Notation Using interval notation, the interval described in Step 4 would be written as `(1, 3]`. This indicates that the interval starts just after 1 (since 1 is not included, as denoted by the open parenthesis) and goes up to and includes 3 (as denoted by the closed bracket). In summary, an open circle on a number line represents an exclusive boundary value, which corresponds to an open interval in mathematical notation. This indicates that the value at which the open circle is drawn is not included in the interval.

Question

What does an open circle on a number line represent in terms of intervals and boundary values in mat...

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