9.2: Approximating Square Roots
Introduction
Helping at Little League
Miguel is enjoying his time with the Wildcats. The players have all been very friendly to him and Miguel loves helping out. He gets to hand out towels and water and sometimes collects and organizes equipment.
The Wildcats do a lot of community service. One thing that they do is help out Little League teams. There is one new ball field being created and the Wildcats are going to help design the infield so that it has the correct dimensions. Miguel has been invited to go along.
When they arrive at the field, Miguel and one of the players, Harris, take a measurement to determine the distance from home plate to first base. It measures 58 feet.
“That isn’t correct,” Harris tells Miguel. “The area of the infield for little league should be 3600 square feet. This measurement is inaccurate. We’ll have to help them fix it.”
Miguel is puzzled. What is inaccurate about the 58 feet? Does it need to be a longer distance or a shorter one?
To figure this problem out, Miguel will need to use his knowledge of squares and square roots. This lesson will expand upon what you learned in the last lesson. Then you can help Miguel figure this problem out at the end of the lesson.
What You Will Learn
By the end of this lesson you will be able to complete the following tasks:
- Approximate square roots to the nearest given decimal place.
- Recognize irrational numbers.
- Recognize the set of real numbers as the union of all rational numbers and all irrational numbers.
- Solve real-world problems involving square roots.
Teaching Time
I. Approximate Square Roots to the Nearest Given Decimal Place
In the last lesson we began learning about square roots. Remember that the square root of a number is the number that you multiply by itself to get a product, in essence it is a number that you square. Squaring and square roots are inverse operations. You can square a number to get a product, then take the square root of the product to get back to the original number.
Example
\begin{align*}6^2 = 36\!\\ \sqrt{36} = 6\end{align*}
Here you can see the inverse relationship between squaring a number and finding the square root of a product.
A number like 36 is a perfect square, meaning that the square root is a whole number. Here are some other perfect squares.
16, 25, 36, 49, 64, 81, 100, 121, 144, 169
Notice that if you find the square root of any of these numbers, you will end up with a whole number as the answer.
What if a square root is not a whole number?
This can also happen. When it does, we can approximate the square root of the number. There are a couple of different ways to do this. Let’s take a look.
First, we can approximate the square root using perfect squares.
To do this, we look for the perfect squares that generate a number close to the square root that we are looking for. Let’s look at an example.
Example
\begin{align*}\sqrt{30}\end{align*}
If we are looking for the square root of 30, we first need to find two perfect squares near 30. One should be less than 30 and one should be more than 30.
\begin{align*}5 \times 5 &= 25\\ 6 \times 6 &= 36\end{align*}
25 is the perfect square less than 30.
36 is the perfect square greater than 30.
Because 30 is between 25 and 36, we can say that the approximate square root of 30 is between 5 and 6. It is probably close to 5.5.
This is how we can approximate a square root using perfect squares.
The second way of approximating a square root is to use a calculator. Most calculators have a radical sign on them.
\begin{align*}\sqrt{\Box}\end{align*}
To find the square root of a number, we enter the radical sign, then the value and press enter.
This will give us a decimal approximation of the square root. Many times you will need to round these answers. Let’s look at an example.
We can use 30 again.
Example
\begin{align*}\sqrt{30}\end{align*}
Press the radical button.
Now enter the value, 30.
Next, press enter.
Here is our answer on the calculator. 5.477225575.
To find a final answer, we can round to the nearest tenth. To do this, we round to the 4; it is in the tenths place. The number after the 4 is 7, so we round up to five.
Our answer is 5.5.
WOW! Notice that this value is the same as the answer that we found when estimating using perfect squares.
The third way of approximating a square root is to use something called “tabular interpolation.” Tabular interpolation is using a table. To use this table, we find the approximate value of the square root according to research that has been completed by a mathematician.
Look at this example.
Example
\begin{align*}\sqrt{17}\end{align*}
To find the square root of 17, let’s use a table. Keep in mind that these values have been rounded to the nearest thousandth.
Looking at the table you can see that the square root of 17 is 4.123.
This is our answer.
You can find tables that include numbers all the way up to 100. By finding these tables, you can use them to locate the square root of numbers 1 to 100.
9C. Lesson Exercises
Choose a method to find an approximate square root of each number. Round your answer to the nearest tenth.
- \begin{align*}\sqrt{11}\end{align*}
- \begin{align*}\sqrt{33}\end{align*}
- \begin{align*}\sqrt{41}\end{align*}
Discuss your method of finding each square root. Share this as well as your answer with a friend. Compare your answers for accuracy.
II. Recognize Irrational Numbers
When you were finding square roots using a calculator, you probably noticed that each time you found the square root of a number that was not a perfect square. There were many decimal places following the decimal point.
This means that the square root of that number was not a whole number. It was a whole number and some fraction of another.
The key thing to notice about each of the numbers that we worked with in the last section is that they had many decimal places. When we find the square root of a number that is not a perfect square it brings another type of number into our work.
Let’s look at an example.
Example
\begin{align*}\sqrt{16} = 4\end{align*}
Sixteen is a perfect square. So the square root is a whole number.
Example
\begin{align*}\sqrt{50}\end{align*}
We use a calculator and here is our answer: 7.071067812...
Yes. There are a lot of digits after the decimal point, and you can see that the dots after the last digit tell us that the numbers go on and on and on. We can round to the nearest tenth to see that 7.1 is a likely approximation for this square root.
This square root is a decimal that does not end. Numbers like this one are called irrational numbers. Anytime that you have a decimal that does not repeat and does not end; it is a member of the group of numbers called irrational numbers.
Often the little dots at the end of the decimal will let you know that you are working with an irrational number.
III. Recognize the Set of Real Numbers as the Union of all Rational Numbers and all Irrational Numbers
Not all numbers are irrational numbers, some are rational numbers. You have learned about rational numbers in another section of this book. Let’s review what defines a rational number for just a minute.
Rational Numbers are numbers that can be written in the form of \begin{align*}\frac{a}{b}\end{align*}. They can be whole numbers, fractions, negative numbers, decimals and repeating decimals.
Here are some examples of rational numbers.
\begin{align*}.56 \qquad \frac{1}{2} \qquad .6\bar{6} \qquad 33\frac{1}{3} \qquad 9 \qquad -23\end{align*}
Irrational numbers complete the group of numbers that are not rational numbers. This means that any decimal that does not end is irrational. The square root of any number that is not a perfect square is irrational.
There is one famous irrational number. Do you know what it is?
The most famous irrational number is a number called pi or \begin{align*}\pi\end{align*}. We use the number 3.14 to represent pi, but it is really an irrational number without an end point. It just goes on and on and on. Pi is a special ratio that you will learn about when we work with circles very soon.
Rational and Irrational Numbers make up the set of Real Numbers. Numbers are considered real numbers whether they are rational or irrational. So, you can look at a number, know that it is a real number and then classify it further as rational or irrational.
9D. Lesson Exercises
Identify each of the following numbers are rational or irrational.
- 4.567....
- \begin{align*}\sqrt{25}\end{align*}
- \begin{align*}\sqrt{13}\end{align*}
Take a few minutes to check your work with a peer.
IV. Solve Real-World Problems Involving Square Roots
We can use square roots to solve problems too. Anytime we are working with squares or with patterns of squares, square roots can be very useful.
Look at this diagram.
Example
Kelly wants to use this tile to finish the floor in her room. The room is a square room and the area of the floor is 324 square feet. She wants to put 8” tiles along the edge of the floor. What is the length of one side of the room? How many tiles can she fit along one side?
We have to look at what information we have been given.
We know that the area of the room is a square and that it is 324 square feet.
We know that she is using 8” tiles along the border of the room and that she wants to figure out how many she will need.
First, we need to figure out the length of one side of the room.
This room is a square, so we can find the square root of the area of the room and that will tell us the side length.
\begin{align*}\sqrt{324} = 18 \ feet\end{align*}
Each side of the room is 18 feet.
Now we can figure out the tiles.
The tiles are measured in inches. The length of the room is in feet, so we have to first convert feet back to inches. There are 12 inches in 1 foot, so we multiply \begin{align*}18 \times 12\end{align*}.
\begin{align*}18 \times 12 = 216”\end{align*}
Now each tile is 8” so we divide 216 by 8
\begin{align*}216 \div 8 = 27 \ tiles\end{align*}
She will need 27 tiles for each side of the room.
Anytime that you have a problem with square in it; you can use square roots to help you to solve the problem. Now let’s go back to the problem in the introduction and use what we have learned to solve it.
Real–Life Example Completed
Helping at Little League
Here is the original problem once again. Reread it and underline the important information.
Miguel is enjoying his time with the Wildcats. The players have all been very friendly to him and Miguel loves helping out. He gets to hand out towels and water and sometimes collects and organizes equipment.
The Wildcats do a lot of community service. One thing that they do is help out Little League teams. There is one new ball field being created and the Wildcats are going to help design the infield so that it has the correct dimensions. Miguel has been invited to go along.
When they arrive at the field, Miguel and one of the players Harris take a measurement to determine the distance from home plate to first base. It measures 58 feet.
“That isn’t correct,” Harris tells Miguel. “The area of the infield for little league should be 3600 square feet. This measurement is inaccurate. We’ll have to help them fix it.”
Miguel is puzzled. What is inaccurate about the 58 feet? Does it need to be a longer distance or a shorter one?
Harris told Miguel that the area of the infield needs to be 3600 square feet for Little League. To figure out what is inaccurate about the current measure of 58 feet, Miguel will need to find the square root of 3600 square feet.
Finding the square root will give you the measurement of the distance between the bases. This is the distance that is squared to find the overall area of the infield.
\begin{align*}\sqrt{3600}\end{align*}
To find this square root, don’t worry about 3600, drop the zeros and find the square root of 36.
\begin{align*}6 \times 6 = 36\end{align*}
Now add back one zero.
\begin{align*}\sqrt{3600} = 60 \ feet\end{align*}
The distance between each of the bases should be 60 feet. With a current distance of 58 feet, the distance is short two feet between each base.
Miguel is ready to help Harris correct the problem thanks to squares and square roots!
Vocabulary
Here are the vocabulary words found in this lesson.
- Square Root
- A number which equals a given product when multiplied by itself.
- Perfect Square
- a number whose square root is a whole number.
- Tabular Interpolation
- using a table to find approximate square roots.
- Irrational Numbers
- the set of numbers whose decimal digits do not end, they continue indefinitely. A square root that is not from a perfect square is irrational.
- Rational Numbers
- the set of numbers that can be written in \begin{align*}a/b\end{align*} form.
- Real Numbers
- the set of numbers that includes all numbers whether they are rational or irrational.
Technology Integration
Khan Academy Approximating Square Roots
James Sousa, Example of Estimating Square Roots with a Calculator
James Sousa, Another Example of Estimating Square Roots with a Calculator
Other Videos:
- http://www.onlinemathlearning.com/rational-irrational-numbers.html – This webpage contains two great videos. One is on rational numbers and determining rational numbers and one is on irrational numbers.
Time to Practice
Directions: Find each an approximate answer for each square root. You may round your decimal answer to the nearest tenth.
1. \begin{align*}{\overline{ ) {8 }}}\end{align*}
2. \begin{align*}{\overline{ ) {11 }}}\end{align*}
3. \begin{align*}{\overline{ ) {24 }}}\end{align*}
4. \begin{align*}{\overline{ ) {31 }}}\end{align*}
5. \begin{align*}{\overline{ ) {37 }}} \end{align*}
6. \begin{align*}{\overline{ ) {43 }}}\end{align*}
7. \begin{align*}{\overline{ ) {59 }}}\end{align*}
8. \begin{align*}{\overline{ ) {67 }}}\end{align*}
9. \begin{align*}{\overline{ ) {73 }}}\end{align*}
10. \begin{align*}{\overline{ ) {80 }}}\end{align*}
11. \begin{align*}{\overline{ ) {95 }}}\end{align*}
12. \begin{align*}{\overline{ ) {97 }}}\end{align*}
Directions: Identify each of the following numbers as rational or irrational.
13. .345....
14. 2
15. -9
16. -122
17. 3.456....
18. \begin{align*}\sqrt{25}\end{align*}
19. \begin{align*}\sqrt{16}\end{align*}
20. \begin{align*}\sqrt{12}\end{align*}
21. \begin{align*}\sqrt{38}\end{align*}
22. -4.56
23. \begin{align*}\pi\end{align*}
24. \begin{align*}- \frac{4}{5}\end{align*}
25. 9.8712....
Directions: Estimate each square root using perfect squares. Which answer is true for each square root?
26. \begin{align*}\sqrt{8}\end{align*}
a) \begin{align*}2 < x < 3\end{align*}
b) \begin{align*}7 < x < 9\end{align*}
c) \begin{align*}49 < x < 81\end{align*}
27. \begin{align*}\sqrt{93}\end{align*}
a) \begin{align*}8 < x < 9\end{align*}
b) \begin{align*}9 < x < 10\end{align*}
c) \begin{align*}90 < x < 100\end{align*}
28. \begin{align*}\sqrt{140}\end{align*}
a) \begin{align*}13 < x < 14\end{align*}
b) \begin{align*}12 < x < 13\end{align*}
c) \begin{align*}11 < x < 12\end{align*}