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

• 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:

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:

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\(8< \sqrt{74} <9\), 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:

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\(5< \sqrt{26} <6\), 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. \(\color{blue}{3.2}\)

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