Complex Numbers warm up

Solve the following

Complex Numbers

� Any number in form a+bi, where a and b are real numbers and i is imaginary.

What is an imaginary number?

Definition of imaginary numbers:

b2 b2 1 bi

i 1

i2 1

Solving for i

We know that

But what about other exponents?

Warm UP

(3 + 2i)²

(4 + 5i)(2 – 4i)

3i(2 + 5i)

STEPS for Dividing Complex Numbers

Find the conjugate of the denom

The Cycle of i

And so on…

The cycle of i

The Cycle of i

You Try – Remember the cycle-1, -i, 1, i (for exponents 2, 3, 4, 5)

-1

1

-1

Simplify complex numbers

Ex: 4i7i 28i2 28 1 Remember i2 1

28

Ex# 2: 8 5

i 8 i 5 Remember that 1 i

i2 40 12 10

2 10

Try these problems:

1. 5i 3i

2. 3 6 3

3. i13

-15

9i 2

i

When adding or subtracting complex numbers, combine like terms.

Ex: 8 3i 2 5i 8 2 3i 5i

10 2i

Try these on your own

1. (16 4i) (12 3i)2. (3i 7) (34 6i)3. ( 4 7i) (8 9i)4. ( 1 i) ( 3 4i)

ANSWERS:

1. 4 i2. 3i 413. 4 2i4. 2 + 3i

Multiplying complex numbers.

To multiply complex numbers, you use the same procedure as multiplying polynomials.

Examples to do together:

3 12

36 1 36

6i

Lets do another example.

3 2i 5 3i

15 9i 10i 6i2F O I L

15 9i 10i 6 i2 1Next

Answer:

Now try these:

21-i

1. 5 20

2. 4 5i 4 5i 3. (3 2i)2

Next

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Answers:

1. 10i2. 413. 5 12i

Conjugates

In order to simplify a fractional complex number, use a conjugate.

What is a conjugate?

What is a conjugate?

The reverse!

Take the sign in the middle of the terms and change it

Positive becomes negative

Negative becomes positive

Another Example

3 + 2i 3 – 2i

6i -6i

-2 + 4i -2 – 4i

STEPS TO DIVIDE COMPLEX NUMBERS Find the conjugate of the denominator

Multiply the numerator and the denominator by the conjugate

Collect like terms

Simplify (remember, no i² in the answer)

numeratordenominator

Example

3 x + 1

Take the conjugate of the denominatorMultiply the numerator and denominator by conjugate

Simplify

Collect like terms

Another Example

3 2 + i

One more example

2 – 2i1 + 3i

Lets do an example:

Ex: 8i

1 3i

8i1 3i

1 3i1 3i

Rationalize using the conjugate

Next

8i 24i2

1 98i 24

10

4i 125

Reduce the fraction

Lets do another example

Ex: 4 i2i

4 i2i

ii

4i i2

2i2

Next

4i i2

2i2 4i 1 2

Try these problems.

1. 32 5i

2. 3 - i2 - i

1. 2 5i

9

2. 7 i5