# AN INTRODUCTION TO IRRATIONAL AND
IMAGINARY NUMBERS

**Please note that the Texas Instrument TI 30X II B or TI
30X II S calculators will be used for all**

calculator examples.

**Review: Rational Numbers **

Any type of number that can be written as the quotient of
two Integers. This includes all

terminating and repeating decimals, fractions , and the Integers.

**Irrational Numbers**

Any type of number that cannot be written as the quotient
of two Integers . They are non-terminating

decimal numbers.

Most irrational numbers result from findings roots of
certain other numbers. However,

there are also some irrational numbers that occur naturally, such as the number
π

(approximately 3.14) and the number **e** (approximately 2.72).

** Real Numbers **

The Real Numbers include all of the Rational and
Irrational Numbers.

**Imaginary Numbers**

Most imaginary numbers result from findings roots of
negative numbers given an **EVEN**

index only. The very basic imaginary number is given the letter i and i is
equal to

There is no real number that can be squared to get a
result of -1. Therefore,

the solution to only exists in our
imagination.

Please note that given an odd index, roots of negative
numbers result in rational or

irrational numbers.

**Finding Irrational Numbers**

Example 1 - Review of Unit 3:

Find the square root of 144.

According to the calculator
, where 12 is a terminating decimal,
specifically

an integer, which is a rational number.

Remember that 12(12) does equal 144 !!!

**Calculator Tip: You can find the square root in three
(3) different ways .**

**1. Easiest - Press the 2nd key and then the x^2 key.**

**Warning: Left parenthesis will open when you activate
, specifically**

you will see ( . After you type in the
radicand, you MUST type in the

right parenthesis, namely ")", before you press ENTER.

**2. You might also remember from Unit 3 and Unit 4, that**

**. Therefore, you can
also use the following input**

**Warning: You must enclose 1/2 in parentheses to ensure
that the**

radicand gets raised to the appropriate power ! See Order of Operation in

Unit 4.

**3. You can also activate
to find the square root. Simply press the 2nd
key first**

and then the carat key. Of course, you must remember that the index of a square

root is 2.

**Warning: You first have to type the index, then press
the appropriate key**

sequence, and then type in the radicand.

Example 2 - Review of Unit 7:

Find the cube root of -27.

According to the calculator
, where -3 is a terminating decimal,
specifically

an integer, which is a rational number.

Remember that -3(-3)(-3) does equal -27 !!!

**Calculator Tip: You might also remember from Unit 3 and
Unit 4, that**

**. Therefore, you can
use the following input (easiest):**

**Warning: You must enclose both -27 and 1/3 in
parentheses. See Order**

of Operation in Unit 4.

**You can also activate
to find roots of any index. You have to
press the 2nd key**

first and then the carat ^ key.

**Warning: You first have to type the index, then press
the appropriate key**

sequence, and then type in the radicand.

**Example 3:**

Find the cube root of 144.

Use the following easiest calculator input:

According to the calculator
, where 5.241482788 is a non-terminating

decimal, which is an irrational number.

We CANNOT find a number written as the quotient of two
integers that, when

cubed, results in 144.

**Please note that the calculator eventually rounds to a
certain number of decimal**

places. That does not mean that the decimal terminated.

**Example 4:**

Find the cube root of -7.

Use the following easiest calculator input:

According to the calculator,where
-1.912931183 is a non-terminating

decimal, which is an irrational number. Note that the index is odd, therefore,

the root is NOT imaginary!

We CANNOT find a number written as the quotient of two
integers that, when

cubed, results in -7.

**Example 5:**

Given the number 81, find its square root, cube root, and
4th root.

square root: ... a
rational number (Unit 3)

because 9(9) = 81

cube root: ... an
irrational number

We CANNOT find a number written as the quotient of

two integers that, when cubed, results in 81.

4th root: ... a
rational number (Unit 3)

because 3(3)(3)(3) = 81

**Finding Imaginary Numbers**

**Example 1:**

Find the square root of -81.

Use the following easiest calculator input:

The calculator tells us **Domain Error!**

is an imaginary number
because the INDEX IS EVEN and the radicand is

negative.

There is no real number that can be squared to get a
result of -81. Therefore,

the solution to only exists in our
imagination.

**When we encounter the square root of a negative number,
it is customary to take**

the negative sign out of the radicand and convert it to the letter "i" as
follows:

**. There is an
assumed multiplication sign between the number i and**

the radical expression .

Since the number 81 is a perfect square , we can further
write .

It is customary to write the factor i AFTER a number once
the radical sign is eliminated.

**Example 2:**

Find the square root of -3.

Use the following easiest calculator input:

The calculator tells us **Domain Error!**

is an imaginary number
because the INDEX IS EVEN and the radicand is negative.

There is no real number that can be squared to get a result of -3. Therefore,

the solution to only exists in our
imagination.

Again we finish by writing
. Sometimes, we want to change the radical

expression to a decimal approximation (remember it is a non-terminating decimal)
in which case we write

Remember that it is customary to write the factor i AFTER
a number once the radical sign

is eliminated .

**Example 3:**

Given the number -64, find its square root and cube root.

square root: ... an
imaginary number (the index is even)

cube root: ... a
rational number (the index is odd)

because -4(-4)(-4) = -64