# Approximations of Irrational Numbers Worksheets

Irrational numbers are those values that you cannot write as a fraction (a/b) when both a and b are integers. A commonly used example is the world's most famous constant (pi or π). This is constant that express the ratio of any circle's circumference to its diameter. Regardless of the size of your circle, this value will always be the same. There are many times that we will need to work with constants like this and actually use them to solve problems. In order to do this, we need to learn how to estimate their value in a decimal form to make them workable. This entire section is dedicated to help you baby step your way to being very comfortable with this. These worksheets and lessons help students learn how to convert irrational numbers to the more understandable rational form.

### Aligned Standard: Grade 8 Number System - 8.NS.A.2

- Comparing Radicals Step-by-Step Lesson- Use a numbers line to compare radicals of five and seven.
- Guided Lesson - Approximate the value of radical numbers to the nearest tenth.
- Guided Lesson Explanation - We use numbers lines again to help us visualize where the values lie.
- Independent Practice - Approximate values for five problems and compare radicals for another five problems.
- Matching Worksheet - Sorry that we ended up with an odd number of problems on the this one. I know it makes it tougher to grade.
- Approximations of Irrational Numbers Five Pack - You are basically simplifying these irrational statements.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Point out the approximation of the value of irrational numbers.

- Homework 1 - Compare √10 and √12.
- Homework 2 - Because we only need to go to the tenths place, we could work from a list of the possible multiples.
- Homework 3 - Find the approximation of √150 to the nearest tenth.

### Practice Worksheets

More approximations. These make great in-class activities.

- Practice 1 - Compare √55 and √57.
- Practice 2 - How do these two stack up?
- Practice 3 - Wonderdog goes after it.

### Math Skill Quizzes

The quizzes also include operations with irrational integers.

- Quiz 1 - Solve to the nearest tenth.
- Quiz 2 - Solve to the nearest hundredth.
- Quiz 3 - Operations are included in this set of problems.

### How to Make Approximations of Irrational Numbers

Irrational numbers are real numbers that we cannot express as a simple fraction. These values possess decimals that repeat in fixed pattern forever. There will be plenty of times that we actually have to work with the math behind these values. In order to do this, we must estimate the approximate value of the irrational value that we are working with. More often than not, you will be working squares roots.

Here is a simple series of steps that you can apply to approximate these values. Start by getting an idea of where this value would fit on a number line. You can do this by identifying two perfect squares, one that results in a larger value and one that is smaller. The best place to start is by identifying the place value where square root would best fit. Once you identify this, look at the next value up and down and write that on your numbers line. From there plot your value on the number line based on carrying it out to the second and third place values. That should give you enough of an idea to create a nice line plot.

Decimal expansion of a rational number provides a similar sequence that comes through rational approximations. For example, the value of π is 3.14159…

The approximation of π can be carried out through:

r ^{0} = 3

r ^{1} = 3.1 = 31/10

r ^{2} = 3.14 = 314/100

r ^{3} = 3.141 = 3141/1000

These numbers give out a sequences and better approximation of the value of Pi. Similarly, √2 = 1.41421 which can be approximated by the rational number sequence:

r ^{0} = 1

r ^{1} = 1.4 = 14/10

r ^{2} = 1.41 = 141/100

r ^{3} = 1.414 = 1414/1000

This is will go on with the same frequency as the approximation of π.