Irrational Numbers Study Guide

An irrational number is a real number that cannot be expressed as a simple fraction. In other words, it cannot be written as a fraction p/q where p and q are integers, and q is not equal to zero.

Characteristics of Irrational Numbers:

• Cannot be expressed as a simple fraction
• Have an infinite, non-repeating decimal expansion
• Exist on the number line as non-terminating and non-repeating decimals

Examples of Irrational Numbers:

Some common examples of irrational numbers include:

• π (pi) = 3.14159265359...
• e (Euler's number) = 2.71828182845...
• √2 (square root of 2) = 1.41421356237...

Identifying Irrational Numbers:

To determine if a number is irrational, you can use the following methods:

1. Prime Factorization: If the square root of a number does not simplify to an integer, then the number is irrational. For example, the square root of 5 (√5) cannot be simplified to an integer, so 5 is irrational.
2. Decimal Expansion: If a number has a non-repeating and non-terminating decimal expansion, then it is irrational. For example, the decimal expansion of π is non-repeating and non-terminating, making it irrational.

Operations with Irrational Numbers:

When performing operations with irrational numbers, the result can also be irrational. For example, adding or multiplying two irrational numbers may result in another irrational number.

Real-Life Examples:

Irrational numbers are encountered in various real-life situations, such as calculating the circumference of a circle, solving certain mathematical equations, and understanding patterns in nature.

Study Tips:

When studying irrational numbers, it's helpful to:

• Practice identifying irrational numbers using prime factorization and decimal expansion methods
• Explore real-life examples of irrational numbers to understand their significance
• Work on solving problems involving irrational numbers to improve your understanding

Remember, irrational numbers are an important concept in mathematics and have fascinating properties that make them worth exploring!