# 2.5: Square Roots and Real Numbers

**At Grade**Created by: CK-12

## Learning Objectives

- Find square roots.
- Approximate square roots.
- Identify irrational numbers.
- Classify real numbers.
- Graph and order real numbers.

## Find Square Roots

The **square root** of a number is a number which, when multiplied by itself, gives the original number. In other words, if , we say that is the square root of .

**Note:** Negative numbers and positive numbers both yield positive numbers when squared, so each positive number has both a positive and a negative square root. (For example, 3 and -3 can both be squared to yield 9.) The positive square root of a number is called the **principal square root**.

The square root of a number is written as or sometimes as . The symbol is sometimes called a **radical sign**.

Numbers with whole-number square roots are called **perfect squares**. The first five perfect squares (1, 4, 9, 16, and 25) are shown below.

You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. To find the square root of that number, simply take one of each pair of matching factors and multiply them together.

**Example 1**

Find the principal square root of each of these perfect squares.

a) 121

b) 225

c) 324

d) 576

**Solution**

a) , so .

b) , so .

c) , so .

d) , so .

For more practice matching numbers with their square roots, try the Flash games at http://www.quia.com/jg/65631.html.

When the prime factors don’t pair up neatly, we “factor out” the ones that do pair up and leave the rest under a radical sign. We write the answer as , where is the product of half the paired factors we pulled out and is the product of the leftover factors.

**Example 2**

Find the principal square root of the following numbers.

a) 8

b) 48

c) 75

d) 216

**Solution**

a) . This gives us one pair of 2’s and one leftover 2, so .

b) , so , or .

c) , so .

d) , so , or .

Note that in the last example we collected the paired factors first, **then** we collected the unpaired ones under a single radical symbol. Here are the four rules that govern how we treat square roots.

**Example 3**

Simplify the following square root problems

a)

b)

c)

d)

**Solution**

a)

b)

c)

d)

## Approximate Square Roots

Terms like and (square roots of prime numbers) cannot be written as **rational numbers**. That is to say, they cannot be expressed as the ratio of two integers. We call them **irrational numbers**. In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.

To find approximate values for square roots, we use the or button on a calculator. When the number we plug in is a perfect square, or the square of a rational number, we will get an exact answer. When the number is a non-perfect square, the answer will be irrational and will look like a random string of digits. Since the calculator can only show some of the infinitely many digits that are actually in the answer, it is really showing us an **approximate answer**—not exactly the right answer, but as close as it can get.

**Example 4**

Use a calculator to find the following square roots. Round your answer to three decimal places.

a)

b)

c)

d)

**Solution**

a)

b)

c)

d)

You can also work out square roots by hand using a method similar to long division. (See the web page at http://www.homeschoolmath.net/teaching/square-root-algorithm.php for an explanation of this method.)

## Identify Irrational Numbers

Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, is rational because it equals 7, but can’t be reduced farther than . That factor of is irrational, making the whole expression irrational.

**Example 5**

*Identify which of the following are rational numbers and which are irrational numbers.*

a) 23.7

b) 2.8956

c)

d)

e)

**Solution**

a) 23.7 can be written as , so it is rational.

b) 2.8956 can be written as , so it is rational.

c) We know from the definition of that the decimals do not terminate or repeat, so is irrational.

d) . We can’t reduce it to a form without radicals in it, so it is irrational.

e) This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as .

You can see from this example that any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, 0.439 can be expressed as , or just . Also, any decimal that repeats is rational, and can be expressed as a fraction. For example, can be expressed as , which is equivalent to .

## Classify Real Numbers

We can now see how real numbers fall into one of several categories.

If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its **simplest form** is one, then it is an **integer**. If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)

If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is **irrational**.

**Example 6**

*Classify the following real numbers.*

a) 0

b) -1

c)

d)

e)

**Solution**

a) Integer

b) Integer

c) Irrational (Although it’s written as a fraction, is irrational, so any fraction with in it is also irrational.)

d) Irrational

e) Rational (It simplifies to , or .)

## Lesson Summary

- The
**square root**of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers and , if , then . - A square root can have two possible values: a positive value called the
**principal square root**, and a negative value (the opposite of the positive value). - A
**perfect square**is a number whose square root is an integer. - Some mathematical properties of square roots are:
- Square roots of numbers that are not perfect squares (or ratios of perfect squares) are
**irrational numbers**. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point. - Computing a square root on a calculator will produce an
**approximate solution**since the calculator only shows a finite number of digits after the decimal point.

## Review Questions

- Find the following square roots
**exactly without using a calculator**, giving your answer in the simplest form.- (Hint: The division rules you learned can be applied backwards!)

- Use a calculator to find the following square roots. Round to two decimal places.
- Classify the following numbers as an integer, a rational number or an irrational number.
- Place the following numbers in numerical order, from lowest to highest.
- Use the marked points on the number line and identify each proper fraction.