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Finding Approximate Value of an Irrational Number
A Complete Guide to Understanding Rational and Irrational Numbers with Practical Examples
Published: July 8, 2025
Finding Approximate Value of an Irrational Number
Any irrational number lies between two rational numbers, and its approximate position can be represented on the number line between these two rational numbers.
Example
Find two consecutive integers between which each of the following numbers lies, and determine its approximate position on the number line.
1. $\sqrt{11}$ 2. $\sqrt[3]{20}$
Solution
$\because$ Choose two consecutive perfect squares between which 11 lies.
$9 < 11 < 16$
$\therefore$ Apply the square root to all parts:
$\sqrt{9} < \sqrt{11} < \sqrt{16}$
$3 < \sqrt{11} < 4$
$\therefore$ The number $\sqrt{11}$ lies between the numbers 3 and 4.
$\because$ The number 11 is closer to 9 than to 16.
$\therefore$ The number $\sqrt{11}$ is closer to 3.
$\because$ Choose two consecutive perfect cubes between which 20 lies.
$8 < 20 < 27$
$\therefore$ Apply the cube root to all parts:
$\sqrt[3]{8} < \sqrt[3]{20} < \sqrt[3]{27}$
$2 < \sqrt[3]{20} < 3$
$\therefore$ The number $\sqrt[3]{20}$ lies between the numbers 2 and 3.
$\because$ The number 20 is closer to 27 than to 8.
$\therefore$ The number $\sqrt[3]{20}$ is closer to 3.
