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Chapter 1: Problem 4

Fill in the blank. An interval that involves the infinity symbol is an _____ interval.

Short Answer

Expert verified
infinite

Step by step solution

01

Understand the Concept of Intervals

Intervals in mathematics define a range of numbers. They can be finite or extend to infinity.
02

Define Infinity

Infinity is a concept representing an unbounded quantity that is greater than any real number.
03

Identify the Type of Interval

An interval that includes the infinity symbol (鈭�) indicates that it extends without bound in at least one direction.
04

Fill in the Blank

Recognize that when an interval extends infinitely, it is known as an 'infinite interval'.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

interval notation
Interval notation is a system used to describe subsets of the real number line. It tells us where the interval starts and ends.
It also includes if these start and end points are included or excluded.
The notation typically uses parentheses ( ) and brackets [ ].
  • [a, b] represents all numbers between a and b, including a and b.
  • (a, b) represents all numbers between a and b but does not include a and b themselves.
  • [a, b) includes a but not b, while (a, b] includes b but not a.
  • When dealing with infinity, you usually use parentheses since infinity isn't a specific, reachable value.
For example, (-鈭�, 3] would mean all numbers less than or equal to 3 but no lower bound.
infinity
Infinity is a concept in mathematics that describes something without any bound or limit. More specifically, it is used to represent values that grow beyond any finite measure.
In interval notation, we use the infinity symbol 鈭� to show this unbounded growth or extension.Some important points about infinity:
  • Infinity is not a number but a concept. You can't treat it like a normal number.
  • The symbol for positive infinity is \( +\resizebox{!}{12pt}\infty\) and for negative infinity, it is \( -\resizebox{!}{12pt}\infty).\)
  • In interval notation, when you use 鈭�, always use open parentheses because you can't actually 'reach' infinity. For example, (3, 鈭�) means all numbers greater than 3.
Understanding infinity helps in grasping the concept of infinite intervals, as these intervals extend without any limit in at least one direction.
types of intervals
Intervals can be categorized into different types based on their bounds. Here's a simple breakdown of these types:
  • Finite intervals: These intervals have both upper and lower bounds within real numbers. Examples include [1, 5] or (2, 6).
  • Infinite intervals: These are intervals that extend infinitely in at least one direction. Examples include (-鈭�, 4] or [3, 鈭�).
  • Open intervals: Intervals that do not include their end points, like (2, 5).
  • Closed intervals: Intervals that do include their end points, like [3, 6].
  • Half-open (or half-closed) intervals: These include only one of their end points. Examples are [4, 8) and (1, 9].
Each type of interval is useful in different mathematical contexts. Understanding them is essential to solving problems like the one in the exercise.

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