These worksheets will give your students practice in expressing irrational numbers to a given degree.

To put it in the most simplistic form, if you can’t write a value as a simple fraction, it is not rational. They describe that value as an irrational number. The most notorious irrational number is Pi. That grand old 3.14 and a ton of numbers after it. You'll find a set of worksheets here that center around algebraic expressions with irrational solutions. You will learn how to approximate the value of irrational numbers that are the result of these problems. Each set of answers is to be notated to a given degree (tenths, hundredths, etc.) This set contains all introductory material, practice questions, reviews, longer exercise sheets, and quizzes. These worksheets will help your students to learn how to approximate irrational numbers and understand the nature of them.



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Approximating Irrational Numbers Lesson

Approximating Irrational Numbers Lesson

Learn how to solve problems with irrational numbers: √7 to the nearest tenth.

Approximations Worksheet 1

Approximations Worksheet 1

This will give you a series of ten problems to work on. For each problem below, solve to the nearest tenth. Example: 3/7 √8

Worksheet 2

Worksheet 2

We follow the same form of problems here, but the values can be confusing for some students. For each problem below, solve to the nearest tenth. Example: 11/7 √7

Approximating Irrational Numbers Review Sheet

Approximating Irrational Numbers Review Sheet

You will work on problems like this: √53 to the nearest hundredth. There are many different ways to approach these types of problems.

Skill Quiz

Skill Quiz

You will approximate the end value of wild values. Example: 1/7 √2

Do Now Worksheet

Do Now

On this worksheet you will have your class do some problems on their own and then regroup as a whole. Here is an example problem for you: 12/17 √7

How to Approximate Irrational Numbers

We all know that some numbers are rational, meaning they can be expressed as a fraction, while others are irrational, meaning they cannot. Most people believe that all irrational numbers are infinitely long and impossible to understand fully. However, this isn't necessarily true - while we may not be able to write down an irrational number in its entirety, we can still approximate it to whatever level of accuracy we need.

Continued Fractions

Continued fractions are a way of representing a number as a sequence of fractions. For example, the number 3 can be represented as a continued fraction like this:

3 = 1 +1/2+1/3+ 1/4 ...

This might not look like much, but it's a compelling way of approximating numbers. Let's see how it works. Suppose we want to approximate the number ℼ. We can start by representing it as a continued fraction:

ℼ = 3 +1/2+1/5+1/8+ 1/11 + ...

Now, if we take the first few terms of this continued fraction, we get an approximation for ℼ:

3 + 1/2 ≈ 3.5

3 +1/2+ 1/5 ≈ 3.6

3 +1/2+1/5+ 1/8 ≈ 3.625

As you can see, the more terms we include in the continued fraction, the more accurate our approximation becomes. So how do we use this to approximate an irrational number like the square root of 2? We can start by representing it as a continued fraction:

√2= 1 +1/2+1/3+ 1/4 + ...

Now, if we take the first few terms of this continued fraction, we get an approximation for the square root of 2:

1 + 1/2≈ 1.5

1 +1/2+ (1 )/3≈ 1.583

1 +1/2+1/3+ 1/4 ≈ 1.615

As you can see, the more terms we include in the continued fraction, the more accurate our approximation becomes.

Things to Keep in Mind While Using Continued Fractions

There are a few things to remember when using continued fractions to approximate numbers.

First, the more terms you include in the continued fraction, your approximation will be more accurate.

Second, if you're trying to approximate an irrational number like ℼ or the square root of 2, you'll never be able to get an exact answer, but you can get arbitrarily close.

Babylonian Method

Another method to approximate irrational numbers is by using the Babylonian method. The Babylonian method is a way of approximating the square root of a number by using successive guesses. It gets its name from the ancient Babylonians, who were some of the first people to use it. Here's how it works:

Suppose we want to find the square root of 10. We can start by making an educated guess, like so:

√10 ≈ 3

Now, we can use this guess to make a better one:

√10≈ ((3 +10/3))/2 = 7/3

And we can use this new guess to make an even better one:

√10 ≈ ((7/3 + 10/7))/2 = 17/7

We can keep doing this as many times as we want, and each time we'll get a more accurate approximation for the square root of 10.

Some Additional Methods to Approximate Square Roots

The Babylonian method is an excellent way to approximate the square root of a number, but it's not the only way. In fact, there are an infinite number of ways to approximate the square root of a number. Here are a few other methods:

  • The Trial and Error Method: This is probably the most intuitive method. You keep guessing until you find the correct answer.
  • The Newton-Raphson Method: This is more mathematically sophisticated but is not too difficult to understand. Essentially, you start with a guess and then use calculus to find a better guess.
  • The Bisection Method: This is a method that's used in computer programming. It's a bit beyond the scope of this blog post, but it is a useful and strong computational method.

Wrapping Up

No matter which method you use, approximating the irrational number will always be inexact. However, with a little practice, you can get pretty close!