Problem 46 - College Algebra

Question

Write the set in interval notation. $$ \\{x \mid x \text { is all real numbers }\\} $$

Step-by-Step Solution

Answer

\((- fty, fty)\)

Step 1: Identify the Set

The set \( \{x \mid x \text{ is all real numbers }\} \) represents the collection of every real number. This means that every number that can be found on the number line is included in the set.

Step 2: Determine the Interval

Since the set includes all real numbers, this implies that the interval encompasses every possible number from negative infinity to positive infinity.

Step 3: Express in Interval Notation

The interval notation for all real numbers is denoted by \((-fty, fty)\). In interval notation, \(-fty\) signifies no lower bound and \(fty\) signifies no upper bound.

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics, making up what we consider the number line. These include all the rational and irrational numbers. Rational numbers, like 3, -7, and 1/2, can be expressed as a ratio of two integers. Irrational numbers, like \( \sqrt{2} \) and \( \pi \), cannot be expressed as simple fractions. However, both types of numbers, rational and irrational, make up the real numbers. Real numbers are crucial for representing measurements, distances, and many mathematical concepts in the real world. They cover all numbers you can find on a number line, except for imaginary numbers, which do not fit on this line.

Number Line

The number line is a visual way to represent real numbers in a straight, horizontal line. The center point is zero, with positive real numbers extending to the right and negative real numbers to the left. This line helps to easily visualize not only simple numbers but also their relations to one another, such as ordering and size comparison. By marking points on the line, we can translate algebraic expressions into visual formats. The number line offers a simple way to understand complex expressions and operations like addition and subtraction, wherein we "move" along the line to reach results.

Set Notation

Set notation is a formal way to specify a collection of elements, like numbers. For example, the set of all real numbers can be written as \( \{x \mid x \text{ is a real number} \} \). This notation means "the set of all x such that x is a real number." Set notation is useful because it cleanly expresses the inclusion of elements in a set without listing them individually. It allows us to express the concept of a set—like a group of numbers sharing a property—in a concise mathematical way. Mathematicians use set notation extensively to define various groups of numbers or objects.

Infinity in Interval Notation

In interval notation, infinity symbols are used to express unbounded limits. For real numbers, you can use \( -\infty \) and \( \infty \) to indicate an interval that continues indefinitely. The interval notation \((-\infty, \infty)\) is used to represent all real numbers, stretching from negative to positive infinity. Here, the parentheses indicate that infinity is a concept, not a number, so it can't be "included" or reached. Interval notation is a neat and concise way to show an entire set of real numbers, making it essential for students to understand how and why we use these infinity signs in mathematical expressions.