Powers of i and Complex Numbers

Imaginary and Complex Numbers

What are imaginary numbers?

The Imaginary Numbers

Multiple Choice

Your Turn!

$$\sqrt{-9}$$  

• $$\pm3$$  

• $$\pm3i$$  

• $$\pm9i$$  

• $$\pm9$$  
  

Multiple Choice

$$x^2+81=0$$  

• $$\pm\sqrt{9i}$$  

• $$\pm9$$  

• $$\pm9i$$  

• $$\pm3i$$  

How do you think you would solve this?

Multiple Choice

Your Turn!

$$i^{22}$$  

• $$\sqrt{-1}$$  

• $$-1$$  

• $$-i$$  

• 1

Multiple Choice

$$\sqrt{-8}\cdot\sqrt{24}$$  

• $$i\sqrt{24}$$  

• $$8\sqrt{3}$$  

• $$8i\sqrt{3}$$  

• $$60i$$  

Match

Match the following non-real roots with their complex number form: $$a\cdot i$$ where $$a$$ is a real number.

• $$\sqrt[]{-25}$$

• $$\sqrt[]{-36}$$

• $$\sqrt[]{-100}$$

• $$\sqrt[]{-45}$$

• $$\sqrt[]{-\frac{144}{625}}$$

• $$5i$$

• $$6i$$

• $$10i$$

• $$3\sqrt[]{5}\cdot i$$

• $$\frac{12}{25}i$$

Multiple Choice

$$4i\cdot7i$$  

• $$28i$$  

• -28

• 28

• $$14i$$  

Multiple Choice

$$\left(2i\right)^3\cdot\left(5i\right)$$  

• 40

• $$40i$$  

• 10

• $$10i$$  

Complex Number

A complex number has the form a +bi, where a is the real component and b is the imaginary component.

Ex.

6-3i

0+5i

8+0i

-3+7i

Labeling

Label the

REAL part __

and the

IMAGINARY part __

of the complex number.

Drag labels to their correct position on the image

REAL part

IMAGINARY part

Labeling

Label the

REAL part __

and the

IMAGINARY part __

of the complex number.

Drag labels to their correct position on the image

REAL part

IMAGINARY part

Match

Match the complex number with its STANDARD FORM:

$$a+bi$$

• $$-29+\sqrt[]{-49}$$

• $$\frac{2}{5}-\frac{\sqrt[]{-81}}{7}$$

• $$-131+\sqrt[]{-441}$$

• $$242-\sqrt[]{-289}$$

• $$114-\sqrt[]{-243}$$

• $$-29+7i$$

• $$\frac{2}{5}-\frac{9}{7}i$$

• $$-131+21i$$

• $$242-17i$$

• $$114-9\sqrt[]{3}\cdot i$$

Operations with Complex Numbers

Operations with complex numbers are similar to expressions with variables. You can combine real components with each other and imaginary components with each other.

When multiplying imaginary components, you add exponents, similar to multiplying variables.

Adding/Subtracting Complex Numbers

• Combine real terms

• Combine imaginary components

• Distribute negative first (when subtracting)

Adding Complex Numbers

(2+3i)+(8-4i)

=10+-i (Add 2+8 and 3i+-4i)

=10-i

Subtracting Complex Numbers

• (4-5i)-(1-8i)

• =(4-5i)+(-1+8i) (Distribute negative)

• =3+3i (Combine like terms)

Multiple Choice

Add the two complex numbers:

$$\left(-7+16i\right)+\left(-18+13i\right)$$

• $$-23-5i$$

• $$-25-29i$$

• $$-25+29i$$

• $$-11+29i$$

Multiple Choice

Subtract the two complex numbers:

$$\left(14+9i\right)-\left(23+11i\right)$$

• $$-9-20i$$

• $$37+20i$$

• $$-9+20i$$

• $$-9-2i$$

Multiple Choice

Add the two complex numbers:

$$\left(5+7i\right)+\left(8+3i\right)$$

• $$12+11i$$

• $$13+10i$$

• $$-2+5i$$

• $$13-10i$$

Multiple Choice

Subtract the two complex numbers:

$$\left(-26-18i\right)-\left(42-39i\right)$$

• $$-68-21i$$

• $$-68-57i$$

• $$16+21i$$

• $$-68+21i$$

Multiplying Complex Numbers

• Remember to FOIL

• i²=-1

• Simplify as much as possible.

Multiplying Complex Numbers

(8+2i)(7-3i)

=56-24i+14i-6i²

=56-10i-6(-1)

=56-10i+6

=62-10i

Multiple Choice

(5-i)(5-i)

• 15-8i
• 36
• 24+10i
• 24-10i

Multiple Choice

(-3-i)(6-i)

• -21-7i
• -19-3i
• -15-5i
• -17-9i

​Complex Conjugates

Fill in the Blanks

What is the complex conjugate of $10-5i$  ?

Fill in the Blanks

What is the complex conjugate of $21+i$  ?

Fill in the Blanks

What is the complex conjugate of $-3i$  ?

​Dividing Complex Numbers

Multiple Choice

Simplify.

$$\left(3-2i\right)\div\left(-2-2i\right)$$  

• $$\frac{7+4i}{5}$$  

• $$\frac{-3+2i}{5}$$  

• $$\frac{-1+5i}{4}$$  

• $$\frac{3i+2}{2}$$  

Multiple Choice

Simplify.

$$\left(-1+2i\right)\div\left(-3+4i\right)$$  

• $$\frac{-12-16i}{25}$$  

• $$\frac{11-2i}{25}$$  

• $$\frac{-6i+8}{25}$$  

• $$\frac{2-14i}{25}$$  

Multiple Choice

Simplify.

$$\left(-1-3i\right)\div\left(-i\right)$$  

• $$-i+3$$  

• $$-1-3i$$  

• $$\frac{-4-3i}{2}$$  

• $$-9i$$