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Operations on Imaginary Numbers

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

Imaginary Numbers

•Most imaginary numbers result from findings roots of negative numbers given an EVEN index
only. A purely imaginary number is re presented by the letter i and i is equal to . 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.

•When we encounter the square root of a negative number, it is customary to take the negative
sign out of the radical and convert it to the letter i as follows:

•Furthermore,

Complex Numbers

Complex Numbers are of the form a + bi, where a is a real number and bi a purely
imaginary number with coefficient b. All real numbers can be written in complex form.

For example, 3 + 0i, -2.34 + 0i, etc.

On the other hand, 3 + 2i or -2.34 - 5.1i are complex number containing an
imaginary part and are therefore called imaginary numbers.

Problem 1:

Simplify , if possible, and write in terms of i .

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

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

Problem 2:

Write in terms of i.

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.

However, we can simplify by writing

NOTE: It is customary to write the i in front of the radical!

Some times , we want to change the radical expression to a decimal approximation
(remember it is a non- terminating decimal ) in which case we write

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

Problem 3:

Simplify , if possible, and write in terms of i.

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 -64. Therefore, the solution
to only exists in our imagination.

However, we can simplify by writing

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

Adding and Subtracting Complex Numbers

•Add or subtract the real parts.
•Add or subtract the coefficients of the imaginary parts.

Problem 4:

Add (3 + 6i) + (9 - 2i).
NOTE: When you carry out an arithmetic operation on complex numbers, you
must enclose them in parentheses!

We can rewrite this as follows:

Problem 5:

Subtract (2 + 7i) - (8 - i).
In this case, we MUST observe the minus sign in front of the parentheses.
We first must write 2 + 7i - 8 + i.
The we combine "like" terms to get -6 + 8i .

Please note that i has a coefficient of 1 which is usually not written, but must
be used in addition and subtraction.

Multiplying Complex Numbers

Multiplying complex numbers uses procedures similar to multiplying polynomials!

Problem 6:

Multiply 7(3i).
Here we multiply the coefficients to get 21i.

Problem 7:

Multiply 7i(3i).
Here we multiply the coefficients and the imaginary numbers to get
Since we know that , we can state

Problem 8:

Multiply (2 + 7i)(8 - 3i).
Use the FOIL process to multiply (2 + 7i)(8 - 3i).

hen

Since we know that , we can write
16 - 6i + 56i - 21(-1) = 16 - 6i + 56i + 21
and finally we can combine like terms to get
37 + 50i

Problem 9:

Factor the Sum of Squares
Now we know that the Difference of Squares is factored into (x - 2)(x + 2).
The Sum of Squares, on the other hand is factored into (x - 2i)(x + 2i).
Check:

Use FOIL to multiply (x - 2i)(x + 2i).

then

Since we know that , we can write

and multiplying and combining like terms will result in

Rationalizing a Denominator containing a Complex Number

•Multiply the denominator by its conjugate ***.
•To preserve the value of the fraction, multiply the numerator by the same number.
•Simplify all and write the number in the form a + bi.

*** The conjugate of a complex number a + bi is the complex number a - bi.

NOTE: In Steps 1 and 2 above , we have actually multiplied the fraction by an
equivalent of the number 1!

Problem 10:

Rationalize the denominator of and write in standard form a + bi.
First, we will multiply both the numerator and the denominator by 3 + i, which is the
conjugate of the denominator.

Next, we will use the FOIL method to multiply the complex numbers in the numerator.
Observe that the denominator contains a Difference of Squares!

Since we know that , we can write

and finally, we find that we can express in standard form as

Problem 11:

Rationalize the denominator of and write in standard form a + bi.

First, we will multiply both the numerator and the denominator by 3 + i, which is the
conjugate of the denominator.

Next, we will use the FOIL method to multiply the complex numbers in the numerator.
Observe that the denominator contains a Difference of Squares!

Since we know that , we can write

and finally, we find that we can express in standard form as .

Problem 12:

Rationalize the denominator of and write in standard form a + bi.

First, we will multiply both the numerator and the denominator by 3 + i, which is the
conjugate of the denominator.

Next, we will use the FOIL method to multiply the complex numbers in the numerator.
Observe that the denominator contains a Difference of Squares!

Since we know that , we can write

and

Finally, we find that we can express in standard form as 1 + i.

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