Section 2



Section 2.4: Imaginary Unit and Complex Numbers

Imaginary Unit:

Some polynomial equations have complex (non-real) solutions, when a negative number is under the radical symbol. For example,

there is no real solution to [pic] or [pic].

Mathematicians (for fun, maybe?!) created a new system of numbers using the imaginary unit, i, defined as [pic]. With this new system of numbers, radicals of negative numbers can now be simplified!

Therefore;

[pic]

[pic]

[pic] where ‘i’ is NOT under the radical sign!

[pic] where ‘i’ is NOT under the radical sign!

[pic] [pic] [pic] [pic]

[pic]

[pic]

[pic]

Definition of Complex Numbers:

A complex number z is a number of the form z = a + b i

where a and b are real numbers and i is the imaginary unit defined by [pic]

a is called the real part of z and b is the imaginary part of z.

Operations on Complex Numbers:

Addition and Subtraction is similar to grouping like terms where real parts are combined with real parts and imaginary parts are combined with imaginary parts.

Example:

Express in the form of a complex number a + b i.

• (2 + 3i) + (-4 + 5i) =

(2 - 4) + (3i + 5i) = - 2 + 8i (-2 is the real part of this complex number, 8i is the

real imaginary imaginary part)

EXAMPLES: 1] (3i) + (-5 + 6i) = -5 +9i 2] (2) + (-2 + 9i) = 0 + 9i = 9i

3] (2 - 5i) - (-4 - 5i) = 6 +0i = 6

Multiplication of Complex Numbers

The multiplication of two complex numbers is performed using properties similar to those of the real numbers (FOIL) and distributive property. Remember that

Simplify each expression and express in the form of a complex number a + bi.

EXAMPLE 1.: 5(2 + 7i) = 10 + 35i

EXAMPLE 2.: (6 − i)(5i) = 30i – 5i2 = 5 + 30i

EXAMPLE 3: (2 − i)(3 + i) = 6 +2i – 3i - i² = 6 – i – (-1) = 7 – i

EXAMPLE 4: (5 + 3i)²= 25 +30i + 9i² = 25 +30i +9(– 1) = 16 + 30i

Division of Complex Numbers

Earlier, in Geometry and Adv Algebra we learned how to rationalize the denominator of an expression like the expression below. We multiplied numerator and denominator by the “conjugate” of the denominator, 3 + √2:

[pic]

EXPRESS [pic] The conjugate of 4 − 2j is 4 + 2j.

TRY TO SIMPLIFY: [pic] Answer: [pic]

PRACTICE QUIZ : (Simplify each radical using imaginary numbers.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. i³ 7. [pic] 8. [pic]

(Simplify and write final answer in standard form (a +bi).

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

(Write the following quotients in standard form.

1. [pic] 2. [pic] 3. [pic]

ANSWER KEY :

(Simplify each radical using imaginary numbers.

1.[pic] = [pic] 2. [pic]= [pic] 3. [pic][pic] 4. [pic][pic]

5. [pic] [pic] =[pic] 6. i³ [pic] 7. [pic] [pic] 8. [pic][pic]

(Simplify and write final answer in standard form (a +bi).

2. [pic][pic] 2. [pic][pic] 3. [pic][pic] 4. [pic][pic] 5. [pic] [pic] 6. [pic][pic]

7. [pic] [pic] 8. [pic] [pic] 9. [pic][pic]

(Write the following quotients in standard form.

2. [pic] [pic] 2. [pic][pic] 3. [pic][pic]

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[pic]

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