# INTRODUCTION TO IRRATIONAL AND IMAGINARY NUMBERS

**PLEASE NOTE THAT YOU CANNOT USE A CALCULATOR ON THE
ACCUPLACER -
ELEMENTARY ALGEBRA TEST! YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS
WITHOUT A CALCULATOR!**

**Natural Numbers**

The numbers used for counting. That is, the numbers

**{1, 2, 3, 4, ...}.**

**Integers**

The numbers

**{... , - 4, -3, -2, -1, 0, 1, 2, 3, 4, ...}.**

**Whole Numbers or Nonnegative Integers**

The numbers

**{0, 1, 2, 3, 4, ...}.**

**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 numbers that are NOT perfect

powers. 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).

NOTE: The results when natural numbers are squared, cubed, or raised to

any power are also referred to as

**perfect powers !**The root of a perfect power

is a natural number.

**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. A purely imaginary number is represented by the letter i and i is equal to

index only

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

or irrational numbers.

NOTE: There is no real number that can be squared to get a result of

**-1.**

Therefore, the solution to only exists in our imagination.

**Finding Rational, Irrational, and Imaginary Numbers**

**Problem 1:**

If possible, find the square root of

**144.**

, where

**12**is a terminating decimal, specifically an integer, which is a

rational number.

Remember that

**12(12)**does equal

**144**!!!

**Problem 2:**

If possible, find the cube root of

**-27**.

, where

**-3**is a terminating decimal, specifically an integer, which is a

rational number.

Remember that

**-3(-3)(-3)**does equal

**-27**!!!

**Problem 3:**

If possible, find the cube root of

**144**rounded to three decimal places.

Here we notice that the number

**144**is not a perfect cube! That is, we CANNOT find a

number written as the quotient of two integers that, when cubed, results in 144!

**NOTE: For a problem like this , the ACCUPLACER test will make a**

calculator available to you!

calculator available to you!

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

decimal, which is an irrational number.

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

places. That does not mean that the decimal terminated.

places. That does not mean that the decimal terminated.

Since we are asked to round the answer to three decimal places, we find to be

approximately equal to

**5.241**.

**Problem 4:**

If possible, find the cube root of** -7** rounded to three decimal places.

Again, **-7** is not a perfect cube.

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**.

Since we are asked to round the answer to three decimal places, we find
to be

approximately equal to **-1.913**.

**Problem 5:**

Given the number **81**, find its **square root, cube root**, and **4th root,** if
possible. Round

to three decimal places, if necessary.

**square root:** ... a rational number because **
9(9) = 81**

**cube root:** ... an irrational number because we
CANNOT find

a number written as the quotient of two integers that, when cubed, results in
81.

Since we are asked to round the answer to three decimal places, we find
to be

approximately equal to **4.327.**

**4th root: ** ... a rational number because **
3(3)(3)(3) = 81**

**Problem 6:**

If possible, find the square root of** -81**.

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.

**Problem 7:**

If possible, find the square root of **-3**.

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.

**Problem 8:**

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

**square root:** ... an imaginary number because
the index is even

**cube root:** ... a rational number because the
index is odd and **-4(-4)(-4) =**

**-64**

**Simplifying Radical Expressions **

**Please note that the word "simplify" takes on many meanings in mathematics .
Often you must figure out its meaning from the mathematical expression you are
asked to "simplify." Here are are asked to "simplify" instead of to finding the
root of
a number.**

Before we begin, we must know that radical expressions can also be written as

exponential expression. Following are the conversions:

Furthermore, is equivalent to

**Problem 9:**

Write as an exponential expression and simplify.

and . As you can see the index 4 becomes the denominator of a

fractional power with a numerator of

**1**.

**Problem 10:**

Write as an the exponential expression and simplify.

and . As you can see the index

**3**becomes the denominator of a

fractional power with a numerator of

**1**.

**Problem 11:**

Write as an exponential expression and simplify.

and
. As you can see the index **2** (it is customary
to not write it) becomes

the denominator of a fractional power with a numerator of **1**.

**Problem 12:**

Write as an exponential expression and
simplify.

As you can see the index **2** (it is
customary to not write it) becomes the

denominator of a fractional power with a numerator of **10**. and then we can
reduce the

exponential fraction.

**Problem 13:**

Write as an exponential expression and
simplify.

As you can see the index **2** becomes the
denominator of a fractional power with

a numerator of **1**.

Using one of the Laws of Exponents we can further simplify to get the following:

**Problem 14:**

Write as an exponential expression and
simplify.

As you can see the index **4** becomes
the denominator of a fractional power

with a numerator of **1**.

Using one of the Laws of Exponents we can
further simplify to get the following:

which can be further simplified to .

**NOTE: It is expected that you have permanently
committed to memory
the following values:**

**Problem 15:**

Write as an exponential expression and
simplify

As you can see the index** 3** becomes
the denominator of a fractional

power with a numerator of **1**.

Using one of the Laws of Exponents we can further simplify to get the following:

which can be further simplified to
.

**Problem 16:**

Write as an exponential expression.

As you can see the index **3** becomes the
denominator of a fractional power with a

numerator of **2**.

**Problem 17:
**

Write as an exponential expression.

As you can see the index

**4**becomes the denominator of a fractional power with a

numerator of

**3**.

**Problem 18:
**Write as an exponential expression.

As you can see the index

**2**becomes the denominator of a fractional power with a

numerator of

**3**.

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