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. Proportions/Equivalent Fractions

Study Guide

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Worksheet/Answer key

Proportions/Equivalent Fractions

Create And Print more Fractions worksheets with Comparing Fractions, Equivalent Fractions, and Simplifying Fractions

The resources above cover the following skills:

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Grade 6 Curriculum Focal Points (NCTM)

Number and Operations: Connecting ratio and rate to multiplication and division

Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., 'If 5 items cost $3.75 and all items are the same price, then I can find the cost of 12 items by first dividing $3.75 by 5 to find out how much one item costs and then multiplying the cost of a single item by 12'). By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students solve a wide variety of problems involving ratios and rates.