Real Numbers

We are going to start this lesson with a simple definition that is extremely important to your algebra knowledge.

A number x is a square root of a number y if x² = y.

For example: 6² = 36 therefore 6 is the square root of 36.

You can use the definition above to find the exact square root of some nonnegative numbers. You can also find the approximate square root of other nonnegative numbers.

The radical symbol $\sqrt{x}$ indicates that a nonnegative square root exists. It is also called the principal square root. The expression under the radical symbol is the call the radicand.

Radical symbol---> $\sqrt{x}$ <--- Radicand

The combination of a radical symbol and a radicand form a radical.

Watch the video below to see these ideas in more detail, then go on to the examples.

YouTube Link

Example 1: Find the simplified form of each expression.

a. $\sqrt{64}$

The solution here is 8 because 82 = 64

b. $\sqrt{\frac{9}{25}}$

The solution here is $\frac{3}{5}$ because ${(\frac{3}{5})}^{2}$ = $\frac{9}{25}$

c. $\sqrt{121}$

The solution here is 11 because 112 = 121

Practice problem 1: Find the simplified form of each expression.

Click each prompt below to reveal the answer.

$\sqrt{81}$

9

$\sqrt{\frac{1}{49}}$

$\frac{1}{7}$

$\sqrt{\frac{36}{25}}$

$\frac{6}{5}$

The square of an integer is called a perfect square. For example 64 is a perfect square for 8 because 8² = 64. When a radicand is not a perfect square, you will need to estimate the square root of the radicand.

Example 2: Estimate the square roots.

a. $\sqrt{130}$

You know that

112 = 121

122 = 144

Therefore, the answer here has to fall between 11 and 12. Since 130 is closer to 121 than it is to 144, the answer has to be closer to 11. So, would you say it was about 11.4? That would be a good estimate. Or you could use a calculator to find the best answer. When you use a calculator, you will see that the answer is actually 11.40175425...Therefore, your estimate of 11.4 was fairly accurate.

If you do not have a calculator handy, you can just estimate to the best of your ability. No one expects estimates to be exacts answers, so if you are reasonably close, you can count it as correct.

b. $\sqrt{220}$

You know that

142 = 196

152 = 225

The answer falls between 14 and 15. 220 is a lot closer to 225 compared to 196. So the answer is close to 15. So, without a calculator, you can estimate it at about 14.9.

If you use a calculator, you will get 14.83239697...Therefore, the estimate of 14.9 is accurate enough even though 14.8 would have been slightly better.

Practice problem 2: Estimate the square roots.

Click each prompt below to reveal the answer.

$\sqrt{93}$

9.64365...

$\sqrt{381}$

19.51922...

Let's go on now with some more definitions. Hopefully you've seen most of these before.

You can classify numbers using sets. A set is a well-defined collection of similar objects. Each of those objects is called an element of the set. A subset of a set consists of certain objects from the given set. You can list the elements or objects of a set within braces or, as some like to call, them "curly brackets" - as shown below.

{ }

A useful set you may have heard of is the set of all rational numbers.
A Rational Number is any number that you can write as a fraction in the form of $\frac{m}{n}$ where m and n are integers and n $\neq$ 0. A rational number in decimal form either terminates such as 7.45 or is a repeating decimal such as 7.45656565656... which you can write as $7.4 \bar{56}$ .

Some more useful sets:

Natural Numbers: {1, 2, 3, 4,...}

Whole Numbers: {0, 1, 2, 3, 4,...}

Integers: {...-2, -1, 0, 1, 2,...}

An Irrational Number cannot be represented as a quotient or a fraction of two integers. In decimal form, irrational numbers do not terminate or repeat.

Here are a couple of examples: 0.6134259828400124... and $π$ = 3.14159265...

Now getting back to square roots, there are some square roots that are rational numbers and there are some that are irrational numbers.

If the whole number is not a perfect square, its square root is irrational.

Rational: $\sqrt{9} = 3$ or $\sqrt{144} = 12$

Irrational: $\sqrt{5} = 2.23606797749979...$ or $\sqrt{14} = 3.741657386773941...$

All rational and irrational numbers belong to the same set of numbers called Real Numbers.

Example 3: Classify the given number in as many subsets of real numbers as possible.

a. 19

natural numbers, whole numbers, integers and rational numbers

b. -4.3532

rational numbers

c. $\sqrt{42}$

irrational numbers

Practice problem 3: Classify the number in as many subsets of real numbers as possible.

Click each prompt below to reveal the answer.

67

natural numbers, whole numbers, integers and rational numbers

$\frac{4}{9}$

rational numbers

$\sqrt{121}$

natural numbers, whole numbers, integers and rational numbers

One last definition...

An Inequality is a mathematical sentence that compares the values of two expressions using an inequality symbol such as

$< l e s s t h a n > g r e a t e r t h a n$

$\leq l e s s t h a n o r e q u a l t o \geq g r e a t e r t h a n o r e q u a l t o$

Example 4: What inequality symbol would you use to compare $\sqrt{15} a n d \frac{153}{50}$ ?

You need to estimate both of these number to make the determination.

$\sqrt{15} = 3.87...$ (since 15 is just a little less than the perfect square 16)

$\frac{153}{50} = 3.06$ (since 153 is just a little over 3x50, the value should be just over 3)

since 3.87 is bigger than 3.06 we would use the greater than symbol >

$\sqrt{15} > \frac{153}{50}$

Practice problem 4:

Click the question prompt below to reveal the answer.

What inequality symbol would you use to compare $\frac{243}{20}$ and $\sqrt{148}$ ?

$\frac{243}{20} < \sqrt{148}$

Example 5: Place the following numbers in order from least to greatest.

$7, \sqrt{45}, \frac{43}{6}, 6.8, \sqrt{51}$

We need to change all of these numbers to the same form. You should choose decimals.

7 = 7

$\sqrt{45} = 6.71$

$\frac{43}{6} = 7.17$

6.8 = 6.8

$\sqrt{51} = 7.14$

When we arrange the decimal values from least to greatest, we see the correct order would be: $\sqrt{45}, 6.8, 7, \sqrt{51}, \frac{43}{6}$

Practice problem 5: Place the following numbers in order from least to greatest.

Click the question prompt below to reveal the answer.

$\sqrt{30}, 5.7, \frac{65}{12}, \sqrt{33}, 6$

$\frac{65}{12}, \sqrt{30}, 5.7, \sqrt{33}, 6$