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

This article teaches you how to Approximate Irrational Numbers in a few simple steps.
Related Topics
- How to Simplify Radical Expressions
- How to Solve Radicals
- How to Solve Radical Equations
- How to Rationalize Radical Expressions
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.
- \(\color{blue}{\sqrt{41}}\)
- \(\color{blue}{\sqrt{52}}\)
- \(\color{blue}{\sqrt{59}}\)
- \(\color{blue}{\sqrt{72}}\)
- \(\color{blue}{\sqrt{17}}\)
- \(\color{blue}{\sqrt{10}}\)

- \(\color{blue}{6.4}\)
- \(\color{blue}{7.2}\)
- \(\color{blue}{7.7}\)
- \(\color{blue}{8.5}\)
- \(\color{blue}{4.1}\)
- \(\color{blue}{3.2}\)
Related to This Article
More math articles
- The Ultimate STAAR Algebra 1 Course (+FREE Worksheets)
- The Ultimate 7th Grade SOL Math Course (+FREE Worksheets)
- 8th Grade NYSE Math FREE Sample Practice Questions
- GED Math Study guide: 11 Steps to pass the GED Test in 2023!
- 8th Grade LEAP Math Worksheets: FREE & Printable
- FREE 6th Grade Common Core Math Practice Test
- The Ultimate 6th Grade NDSA Math Course (+FREE Worksheets)
- Completing the Puzzle: How to Finishing Equations when Multiplying Fractions by Whole Numbers Using Models
- 10 Most Common 3rd Grade ACT Aspire Math Questions
- How to Evaluate Recursive Formulas for Sequences
What people say about "How to Approximate Irrational Numbers? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.