Topics
1. Real Numbers
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United States of America
Grade 8

1.05 Estimate irrational numbers

Lesson
Practice

Introduction

In past lessons, we have looked at the difference between  rational and irrational numbers  . We have also looked at evaluating  square roots of perfect squares  and  cube roots of perfect cubes  . In this lesson, we will learn how to estimate the values of irrational numbers.

Estimate irrational numbers

The value of \pi, square roots of non-perfect squares, and cube roots of non-perfect cubes are examples of irrational numbers. Recall the decimal expansion of an irrational number is a non-terminating, non-repeating decimal.

So far, we have estimated the values of irrational numbers using a calculator. However, there are ways we can approximate the values of irrational numbers without using a calculator.

Exploration

Consider the following fact:\begin{aligned}\text{If } a &\lt b \text{,}\\\text{then } \sqrt{a} &\lt \sqrt{b}\end{aligned}For example, 4\lt9, so \sqrt{4}\lt\sqrt{9}. We know this is true because \sqrt{4}=2 and \sqrt{9}=3, and 2 \lt 3.

Consider the expression \sqrt{40}.

  1. What are the closest square numbers that are smaller and bigger than 40?
  2. What does this tell us about the integers \sqrt{40} lies between?
  3. Which integer is \sqrt{40} closest to? Explain your reasoning.
  4. Explain why \sqrt{40} lies between 6.3 and 6.4.
  5. Is \sqrt{40} closer to 6.3 or 6.4? Explain your reasoning.
  6. How can you estimate the value of \sqrt{40} correct to two decimal places?

To estimate the value of a square root expression by hand, we can square integers or decimal numbers to determine what rational numbers the given expression lies between. We can use the same process for cube roots.

Recall there are infinitely many decimal places between the integers on a number line. Between the integers are tenths, between the tenths are hundredths, and so on.

00.10.20.30.40.50.60.70.80.91

When estimating the value of square root and cube root expressions, it is important to consider which squares the number underneath the given square root or cube root lies closet to for the best approximation and placement on the number line.

Examples

Example 1

Represent the following values on the same number line:

A
\pi^2
B
\sqrt{50}
C
2\sqrt{17}
Worked Solution
Create a strategy

Estimate the values of the numbers using decimals and plot the values on the same number line.

Apply the idea

Recall that \pi estimated to two decimals is 3.14.

\displaystyle \pi^2\displaystyle \approx\displaystyle \left(3.14\right)^2Square 3.14
\displaystyle =\displaystyle 9.8596Evaluate the multiplication

This means \pi ^2 will be close to 10.

Estimate the value of 2\sqrt {17} by comparing to the closest square numbers that are bigger and smaller than \sqrt {17}, making an estimate based on which integer \sqrt{17} is closer to, then multiplying our estimate by 2.

\displaystyle \sqrt{16}<\sqrt{17}\displaystyle <\displaystyle \sqrt{25}
\displaystyle 4<\sqrt{17}\displaystyle <\displaystyle 5

The value of \sqrt {17} is closer to \sqrt {16} than \sqrt {25} since 17-16=1 and 25-17=8. Since 4 and \sqrt{17} are very close, we will estimate \sqrt{17}\approx 4.1. Doubling our estimate, we get:

\displaystyle 4.1\times 2\displaystyle =\displaystyle 8.2

This means 2\sqrt{17} will be around 8.2.

Estimate the value of \sqrt {50} by comparing to the closest square numbers that are bigger and smaller than \sqrt {50}.

\sqrt{49}\lt \sqrt{50} \lt \sqrt{64}

7\lt \sqrt{50} \lt 8

The value of \sqrt {50} is closer to \sqrt {49} than \sqrt {64} since 50-49=1 and 64-50=14. This means \sqrt{50} will be close to 7.

Plotting the values on the number line, we have:

78910
Reflect and check

From left to right, the points on the number line are B. \sqrt{50}, C. 2\sqrt{17}, and A. \pi^2.

Example 2

Approximate \sqrt[3]{95} to the nearest tenth without using a calculator.

Worked Solution
Create a strategy

We can use cubes to approximate the value. First, we will find the two integers \sqrt[3]{95} lies between, then cube the tenths that are between those integers.

Apply the idea

First we need to find two perfect cubes close to 95.

\displaystyle 4^3\displaystyle =\displaystyle 6464 \lt 95
\displaystyle 5^3\displaystyle =\displaystyle 125125 \gt 95
\displaystyle 64 \lt 95\displaystyle <\displaystyle 125Compare the values
\displaystyle \sqrt[3]{64} \lt \sqrt[3]{95}\displaystyle <\displaystyle \sqrt[3]{125}Cube root the numbers
\displaystyle 4 \lt \sqrt[3]{95}\displaystyle <\displaystyle 5Evaluate the cube roots of the perfect cubes

Next, we need to determine whether \sqrt[3]{95} is closer to 4 or 5.

\displaystyle 125-95\displaystyle =\displaystyle 30The difference between the two largest values
\displaystyle 95-64\displaystyle =\displaystyle 31The difference between the two smallest values

This shows us \sqrt[3]{95} is closer to 5. Since \sqrt[3]{95} is closer to 5, we can start cubing 4.5 and the tenths above it until we determine which tenths \sqrt[3]{95} lies between.

\displaystyle 4.5^3\displaystyle =\displaystyle 91.125
\displaystyle 4.6^3\displaystyle =\displaystyle 97.336
\displaystyle 91.125 \lt 95\displaystyle <\displaystyle 97.336Compare the values
\displaystyle \sqrt[3]{91.125} \lt \sqrt[3]{95}\displaystyle <\displaystyle \sqrt[3]{97.336}Cube root the numbers
\displaystyle 4.5\lt\sqrt[3]{95}\displaystyle <\displaystyle 4.6Evaluate the cube roots of the perfect cubes

Finally, we need to determine whether \sqrt[3]{95} is closer to 4.5 or 4.6.

\displaystyle 97.336-95\displaystyle =\displaystyle 2.336The difference between the two largest values
\displaystyle 95-91.125\displaystyle =\displaystyle 3.875The difference between the two smallest values

This shows us \sqrt[3]{95} is closer to 4.6.

Therefore, \sqrt[3]{95}\approx 4.6.

Reflect and check

If we were asked to estimate the value of \sqrt[3]{95} to two decimal places, we would need to repeat this process one more time.

Since \sqrt[3]{95} is closer to 4.6, we can start by cubing 4.55, 4.56, 4.57, etc. until we find two cubes that 95 lies between. After that, we will look at the differences in the numbers to determine which one 95 is closer to. The cube root of that number will be the estimate of \sqrt[3]{95} to two decimal places.

Idea summary

We can estimate the values of irrational numbers and represent them on the number line.

To estimate square roots or cube roots, we can follow these steps:

  • Determine the closest squares or cubes that are bigger and smaller than the number underneath the root symbol.

  • Evaluate the square root or cube root to find what two integers the given number lies between.

To determine which squares the given square root or cube root lies closet to, we can find the differences between the squares and the number underneath the root symbol.

Outcomes

8.NS.A.2

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g. π^2).

8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

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