Problem 43 - Introductory Algebra for College Students

Question

Give an example of a number that is an irrational number and a real number.

Step-by-Step Solution

Answer

An example of a number that is an irrational number and also a real number is \( \sqrt{2} \). Another example would be Pi.

Step 1: Definition of irrational numbers

Understanding that an irrational number is a number that cannot be expressed as a ratio of two integers - it cannot be written as a simple fraction. In other words, it is a decimal that never ends and never settles into a repeating pattern.

Step 2: Examples

There are many examples of irrational numbers. One of the most well-known ones is \( \sqrt{2} \) which is approximately equal to 1.4142135, but the decimal form never ends or repeats. Another example is the number Pi. It begins with 3.14, but the decimal continues indefinitely without repeating.

Step 3: Irreational numbers are real numbers

By definition, all irrational numbers are also real numbers. The set of real numbers includes both rational numbers (which can be expressed as a ratio of two integers) and irrational numbers.