5.9: Imaginary + Complex Numbers

-Defining i

-Simplifying negative radicands

-Powers of i

-Solving equations

-Complex numbers

-Operations with complex #s

Imaginary units

• Consider the equation 2x2 + 2 = 0

• You end up with x2 = -1

• There is no real number that, when squared, equals -1

• We define the imaginary unit, i, to be the square root of -1… and i2 = -1

• We can then simplify negative radicands (if the index is even) in terms of i

Simplify .

Answer:

Simplify .

Answer:

Simplify.

a.

b.

Answer:

Answer:

Combining terms with negative radicands

• Recall that by definition, i equals the square root of negative 1 and i2 = -1

• When combining two or more terms with negative radicands, always rewrite each radical in terms of i first!!!

Answer: = 6

Simplify .

Answer:

Simplify .

Answer: –15

Answer:

Simplify.

a.

b.

Higher powers of i

• i raised to ANY power equals either 1, -1, i or –i

• For this reason, your answer should NEVER contain i raised to a power

• To simplify, rewrite as i2 raised to a power, or as i * (i2 raised to a power)

• Ex: i14 = (i2)7 = (-1)7 = -1

• Ex. i29 = i * i28 = i*(i2)14 = i* (-1)14 = i*1 = i

Simplify

Multiplying powers

Power of a Power

Answer:

Answer: i

Simplify .

Solving equations with squared term

• Isolate the squared term/expression first

• Then take the square root of each side!

• REMEMBER when you take the root yourself, stick the ± in front

• Then simplify the radical, using i if necessary

Solve

Answer:

Original equation

Subtract 20 from each side.

Divide each side by 5.

Take the square root of each side.

Solve

Answer:

Complex Numbers

• A complex number is a number that can be written in the form a + bi , where a and b are real numbers

• That is, a complex number contains two parts, a real part (a) and an imaginary part (bi)

• Examples: 4 + 5i, 7 – 2i • Also: 4 (can be written as 4 + 0i)• Also: -3i (can be written as 0 – 3i)

Equality of complex numbers

• Two complex numbers a + bi and c + di are equal iff a = c and b = d

• If confused, set the coefficients of the I term equal to each other and solve for the variable

• Then you can set the “real” parts equal and solve

Find the values of x and y that make the equationtrue.

Set the real parts equal to each other and the imaginary parts equal to each other.

Real parts

Divide each side by 2.

Imaginary parts

Answer:

Find the values of x and y that make the equationtrue.

Answer:

Operations with complex #s

• Adding/subtracting – just add/subtract the “real” components and the imaginary components

• Multiplying – distribute or use FOIL.. Just remember that i2 = -1

• Rationalizing (may need to use the COMPLEX CONJUGATE)

Simplify .

Answer:

Commutative and AssociativeProperties

Simplify .

Commutative and Associative Properties

Answer:

Simplify.

a.

b.

Answer:

Answer:

Application

• Complex #s are used with electricity.. Except they use j instead of i (the letter i is used elsewhere)

• E = I * Z, where E is the voltage, I is the current, and Z is the impedance

• Not that important to know.. Just an example of multiplying complex #s

Answer: The voltage is volts.

Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formulaFind the voltage in a circuit with current 1 + 4 j ampsand impedance 3 – 6 j ohms.

Electricity formula

FOIL

Multiply.

Add.

Electricity In an AC circuit, the voltage E, current I, and impedance Z are related by the formula E = I • Z. Find the voltage in a circuit with current 1 – 3 j ampsand impedance 3 + 2 j ohms.

Answer: 9 – 7 j

andare conjugates.

Multiply.

Answer: Standard form

Simplify .

Simplify .

Multiply.

Answer: Standard form

Multiply by

Simplify.

a.

b.

Answer:

Answer: