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:

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.
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!

◂Math Worksheets and Study Guides Sixth Grade. Plotting Points

Vocabulary/Answer key

Plotting Points

The resources above cover the following skills:

The Number System

Apply and extend previous understandings of numbers to the system of rational numbers.

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. [6-NS6]

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. [6-NS6b]

Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. [6-NS6c]

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. [6-NS8]