How to Approximate Irrational Numbers? (+FREE Worksheet!)

This article teaches you how to Approximate Irrational Numbers in a few simple steps.

Related Topics

Step by step guide to Approximate Irrational Numbers

Numbers that cannot be written as a fraction are called irrational. An irrational number is a non-repeating, non-terminating decimal and it does not have an exact place on the number line. Square roots of numbers that are not perfect squares are irrational.

We use approximations of irrational numbers to locate them approximately on a number line diagram.

Since the irrational numbers are radical numbers that are not a perfect square, to approximate them, follow the steps below

  • Step 1: First, we need to find the two consecutive perfect squares that the number is between. if  is our number, we can do this by writing this inequality: \(a^2< x <b^2\)
  • Step 2: Take the square root of each number:\(\sqrt{a^2}< \sqrt{x} <\sqrt{b^2}\)
  • Step 3: Simplify the square roots of perfect squares:\(a< \sqrt{x} <b\), then, \(\sqrt{x}\) is between \(a\) and \(b\).
  • Step 4: To find a better estimate, choose some numbers between \(a\) and \(b\).

Approximating Irrational Numbers Example 1:

Find the approximation of \(\sqrt{22}\)

Solution:

since \(\sqrt{22}\) is not a perfect square,  is irrational. To approximate \(\sqrt{22}\) first, we need to find the two consecutive perfect squares that \(22\) is between. We can do this by writing this inequality: \(16< 22 <25\). Now take the square root of each number: \(\sqrt{16}< \sqrt{22} <\sqrt{25}\). Simplify the square roots of perfect squares:

\(4< \sqrt{22} <5\), then, \(\sqrt{22}\) is between \(4\) and \(5\). To find a better estimate, choose some numbers between \(4\) and \(5\) Let’s choose \(4.6\), \(4.7\) and \(4.8\).

\(4.6^2=21.16\), \(4.7^2=22.09\), \(4.8^2=23.04\), \(4.7\) is closer to \(22\). Then: \(\sqrt{22}≈4.7\)

Approximating Irrational Numbers Example 2:

Find the approximation of \(\sqrt{74}\)

Solution:

since \(\sqrt{74}\) is not a perfect square,  is irrational. To approximate \(\sqrt{74}\) first, we need to find the two consecutive perfect squares that \(74\) is between. We can do this by writing this inequality: \(64< 74 <81\). Now take the square root of each number: \(\sqrt{64}< \sqrt{74} <\sqrt{81}\). Simplify the square roots of perfect squares:

\(8< \sqrt{74} <9\), then, \(\sqrt{74}\) is between \(8\) and \(9\). To find a better estimate, choose some numbers between \(8\) and \(9\) Let’s choose \(8.5\), \(8.6\) and \(8.7\).

\(8.5^2=72.25\), \(8.6^2=73.96\), \(8.7^2=75.69\), \(8.6\) is closer to \(74\). Then: \(\sqrt{74}≈8.6\)

Approximating Irrational Numbers Example 3:

Find the approximation of \(\sqrt{94}\)

Solution:

since \(\sqrt{94}\) is not a perfect square,  is irrational. To approximate \(\sqrt{94}\) first, we need to find the two consecutive perfect squares that \(94\) is between. We can do this by writing this inequality: \(81< 94 <100\). Now take the square root of each number: \(\sqrt{81}< \sqrt{94} <\sqrt{100}\). Simplify the square roots of perfect squares:

\(9< \sqrt{94} <10\), then, \(\sqrt{94}\) is between \(9\) and \(10\). To find a better estimate, choose some numbers between \(9\) and \(10\) Let’s choose \(9.6\), \(9.7\) and \(9.8\).

\(9.6^2=92.16\), \(9.7^2=94.09\), \(9.8^2=96.04\), \(9.7\) is closer to \(94\). Then: \(\sqrt{94}≈9.7\)

Approximating Irrational Numbers Example 4:

Find the approximation of \(\sqrt{26}\)

Solution:

since \(\sqrt{26}\) is not a perfect square,  is irrational. To approximate \(\sqrt{26}\) first, we need to find the two consecutive perfect squares that \(26\) is between. We can do this by writing this inequality: \(25< 26 <36\). Now take the square root of each number: \(\sqrt{25}< \sqrt{26} <\sqrt{36}\). Simplify the square roots of perfect squares:

\(5< \sqrt{26} <6\), then, \(\sqrt{26}\) is between \(5\) and \(6\). To find a better estimate, choose some numbers between \(5\) and \(6\) Let’s choose \(5.1\), \(5.2\) and \(5.3\).

\(5.1^2=26.01\), \(5.2^2=27.04\), \(5.3^2=28.09\), \(5.1\) is closer to \(26\). Then: \(\sqrt{26}≈5.1\)

Exercises for Approximating Irrational Numbers

Find the approximation of each.

  1. \(\color{blue}{\sqrt{41}}\)
  2. \(\color{blue}{\sqrt{52}}\)
  3. \(\color{blue}{\sqrt{59}}\)
  4. \(\color{blue}{\sqrt{72}}\)
  5. \(\color{blue}{\sqrt{17}}\)
  6. \(\color{blue}{\sqrt{10}}\)
  1. \(\color{blue}{6.4}\)
  2. \(\color{blue}{7.2}\)
  3. \(\color{blue}{7.7}\)
  4. \(\color{blue}{8.5}\)
  5. \(\color{blue}{4.1}\)
  6. \(\color{blue}{3.2}\)

Related to "How to Approximate Irrational Numbers? (+FREE Worksheet!)"

How to Determine Limits Using Algebraic Manipulation?How to Determine Limits Using Algebraic Manipulation?
How to Estimate Limit Values from the Graph?How to Estimate Limit Values from the Graph?
Properties of LimitsProperties of Limits
How to Find the Expected Value of a Random Variable?How to Find the Expected Value of a Random Variable?
How to Define Limits Analytically Using Correct Notation?How to Define Limits Analytically Using Correct Notation?
How to Solve Multiplication Rule for Probabilities?How to Solve Multiplication Rule for Probabilities?
How to Solve Venn Diagrams and the Addition Rule?How to Solve Venn Diagrams and the Addition Rule?
How to Find the Direction of Vectors?How to Find the Direction of Vectors?
Vectors IntroductionVectors Introduction
How to Find Addition and Subtraction of Vectors?How to Find Addition and Subtraction of Vectors?

What people say about "How to Approximate Irrational Numbers? (+FREE Worksheet!)"?

No one replied yet.

Leave a Reply