Learn on PengiReveal Math, AcceleratedUnit 13: Irrational Numbers, Exponents, and Scientific Notation

Lesson 13-3: Understand Irrational Numbers

In Lesson 13-3 of Reveal Math, Accelerated, Grade 7 students learn to identify and approximate irrational numbers, including non-perfect square roots such as √19 and √44, by locating them between consecutive integers and testing decimal values on a number line. Students distinguish irrational numbers from rational numbers by understanding that irrational numbers are nonrepeating, nonterminating decimals that cannot be expressed as a ratio of two integers. The lesson applies these skills to real-world contexts, such as estimating perimeter and radius using square roots.

Section 1

Defining Rational and Irrational Numbers

Property

Together, rational and irrational numbers make up the real numbers. A rational number can be written as the ratio of two integers, $\frac{a}{b}$ where $b \neq 0$ , and its decimal form either stops or repeats. An irrational number cannot be written as a ratio of two integers, and its decimal form never stops and never repeats. When a positive integer is not a perfect square, its square root is an irrational number.

Examples

The numbers $5$ , $-\frac{3}{8}$ , and $0.333...$ are rational because they can be written as fractions ( $\frac{5}{1}$ , $-\frac{3}{8}$ , $\frac{1}{3}$ ) and their decimals terminate or repeat. $\sqrt{81}$ is also rational because $9^2 = 81$ , so $\sqrt{81} = 9$ .
The number $\sqrt{50}$ is irrational because $50$ is not a perfect square, so its decimal form goes on forever without repeating.
$\sqrt{2}$ , $\sqrt{7}$ , and $\sqrt{15}$ are all irrational numbers because the numbers under the radical are not perfect squares.

Explanation

Section 2

Estimating Square Roots with Perfect Squares

Property

To approximate an irrational square root $\sqrt{n}$ to the nearest integer, find the two consecutive perfect squares that $n$ is between. If $a^2 < n < (a+1)^2$ , then the value of $\sqrt{n}$ is strictly between the integers $a$ and $a+1$ :

a < \sqrt{n} < a+1

Examples

Approximate $\sqrt{30}$ to the nearest integer.

The perfect squares closest to $30$ are $25$ and $36$ .

25 < 30 < 36

\sqrt{25} < \sqrt{30} < \sqrt{36}

5 < \sqrt{30} < 6

Since $30$ is closer to $25$ than to $36$ , the best integer approximation is $5$ . So, $\sqrt{30} \approx 5$ .

Approximate $\sqrt{85}$ to the nearest integer.

The perfect squares closest to $85$ are $81$ and $100$ .

81 < 85 < 100

\sqrt{81} < \sqrt{85} < \sqrt{100}

9 < \sqrt{85} < 10

Since $85$ is closer to $81$ than to $100$ , the best integer approximation is $9$ . So, $\sqrt{85} \approx 9$ .

Section 3

Refining Square Root Approximations

Property

To get a more precise approximation of an irrational square root, use trial and error. After finding the two consecutive integers the square root is between, test decimal values within that range by squaring them. Continue to test values with more decimal places to squeeze the gap and narrow down the range.

Examples

To approximate $\sqrt{30}$ to one decimal place: We know it is between $5$ and $6$ . Let's test decimals between $5$ and $6$ :

$5.4^2 = 29.16$
$5.5^2 = 30.25$
Since $30$ is much closer to $30.25$ than $29.16$ , $\sqrt{30}$ is approximately $5.5$ .

To compare $\sqrt{5}$ and $2.3$ , we square both numbers to see their true size:

$(\sqrt{5})^2 = 5$
$2.3^2 = 5.29$
Since $5 < 5.29$ , we know that $\sqrt{5} < 2.3$ .

Explanation

Since you cannot write down the exact decimal value of an irrational number, you have to trap it! By testing decimals and squaring them, you are squeezing the gap between rational numbers to find an approximation that is as accurate as you need it to be.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Rational and Irrational Numbers

Property

Examples

The numbers $5$ , $-\frac{3}{8}$ , and $0.333...$ are rational because they can be written as fractions ( $\frac{5}{1}$ , $-\frac{3}{8}$ , $\frac{1}{3}$ ) and their decimals terminate or repeat. $\sqrt{81}$ is also rational because $9^2 = 81$ , so $\sqrt{81} = 9$ .
The number $\sqrt{50}$ is irrational because $50$ is not a perfect square, so its decimal form goes on forever without repeating.
$\sqrt{2}$ , $\sqrt{7}$ , and $\sqrt{15}$ are all irrational numbers because the numbers under the radical are not perfect squares.

Explanation

Section 2

Estimating Square Roots with Perfect Squares

Property

a < \sqrt{n} < a+1

Examples

Approximate $\sqrt{30}$ to the nearest integer.

The perfect squares closest to $30$ are $25$ and $36$ .

25 < 30 < 36

\sqrt{25} < \sqrt{30} < \sqrt{36}

5 < \sqrt{30} < 6

Since $30$ is closer to $25$ than to $36$ , the best integer approximation is $5$ . So, $\sqrt{30} \approx 5$ .

Approximate $\sqrt{85}$ to the nearest integer.

The perfect squares closest to $85$ are $81$ and $100$ .

81 < 85 < 100

\sqrt{81} < \sqrt{85} < \sqrt{100}

9 < \sqrt{85} < 10

Since $85$ is closer to $81$ than to $100$ , the best integer approximation is $9$ . So, $\sqrt{85} \approx 9$ .

Section 3

Refining Square Root Approximations

Property

Examples

To approximate $\sqrt{30}$ to one decimal place: We know it is between $5$ and $6$ . Let's test decimals between $5$ and $6$ :

$5.4^2 = 29.16$
$5.5^2 = 30.25$
Since $30$ is much closer to $30.25$ than $29.16$ , $\sqrt{30}$ is approximately $5.5$ .

To compare $\sqrt{5}$ and $2.3$ , we square both numbers to see their true size:

$(\sqrt{5})^2 = 5$
$2.3^2 = 5.29$
Since $5 < 5.29$ , we know that $\sqrt{5} < 2.3$ .

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Defining Rational and Irrational Numbers

Property

Examples

The numbers $5$ , $-\frac{3}{8}$ , and $0.333...$ are rational because they can be written as fractions ( $\frac{5}{1}$ , $-\frac{3}{8}$ , $\frac{1}{3}$ ) and their decimals terminate or repeat. $\sqrt{81}$ is also rational because $9^2 = 81$ , so $\sqrt{81} = 9$ .
The number $\sqrt{50}$ is irrational because $50$ is not a perfect square, so its decimal form goes on forever without repeating.
$\sqrt{2}$ , $\sqrt{7}$ , and $\sqrt{15}$ are all irrational numbers because the numbers under the radical are not perfect squares.

Explanation

Section 2

Estimating Square Roots with Perfect Squares

Property

a < \sqrt{n} < a+1

Examples

Approximate $\sqrt{30}$ to the nearest integer.

The perfect squares closest to $30$ are $25$ and $36$ .

25 < 30 < 36

\sqrt{25} < \sqrt{30} < \sqrt{36}

5 < \sqrt{30} < 6

Since $30$ is closer to $25$ than to $36$ , the best integer approximation is $5$ . So, $\sqrt{30} \approx 5$ .

Approximate $\sqrt{85}$ to the nearest integer.

The perfect squares closest to $85$ are $81$ and $100$ .

81 < 85 < 100

\sqrt{81} < \sqrt{85} < \sqrt{100}

9 < \sqrt{85} < 10

Since $85$ is closer to $81$ than to $100$ , the best integer approximation is $9$ . So, $\sqrt{85} \approx 9$ .

Section 3

Refining Square Root Approximations

Property

Examples

To approximate $\sqrt{30}$ to one decimal place: We know it is between $5$ and $6$ . Let's test decimals between $5$ and $6$ :

$5.4^2 = 29.16$
$5.5^2 = 30.25$
Since $30$ is much closer to $30.25$ than $29.16$ , $\sqrt{30}$ is approximately $5.5$ .

To compare $\sqrt{5}$ and $2.3$ , we square both numbers to see their true size:

$(\sqrt{5})^2 = 5$
$2.3^2 = 5.29$
Since $5 < 5.29$ , we know that $\sqrt{5} < 2.3$ .