Irrational Numbers

An irrational number is a real number that cannot be expressed as a fraction of two integers and does not terminate or repeat. In other words, it cannot be written as a simple fraction.

Properties of Irrational Numbers

Cannot be expressed as a simple fraction
Non-terminating and non-repeating decimal expansion
Existence of irrational numbers can be proved through the method of contradiction
When added to a rational number, the result is always irrational
When multiplied by a non-zero rational number, the result is always irrational

Examples of Irrational Numbers

Some common examples of irrational numbers include:

√2 (square root of 2)
π (pi)
e (Euler's number)
√3 (square root of 3)

Study Tips

When studying irrational numbers, it's important to:

Understand the definition and properties of irrational numbers
Practice identifying irrational numbers and distinguishing them from rational numbers
Learn how to approximate irrational numbers using decimal expansions
Use visual aids such as number lines and geometric representations to understand irrational numbers
Explore real-world applications of irrational numbers, such as in geometry and trigonometry

Conclusion

Understanding irrational numbers is an important concept in mathematics, and it is essential for various fields including geometry, algebra, and calculus. By grasping the properties and examples of irrational numbers, you can develop a solid foundation for further mathematical studies.

◂Math Worksheets and Study Guides Eighth Grade. Real numbers

The resources above cover the following skills:

The Number System

Know that there are numbers that are not rational, and approximate them by rational numbers.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. [8-NS1]

Expressions and Equations

Work with radicals and integer exponents.

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. [8-EE2]