HSN.CN.B.5: Represent operations geometrically — Practice

1

On the complex plane, addition is vector addition. If $z = 2 + 3i$ and $w = 1 + i$ , compute $z + w$ by adding components. $z + w = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $+ $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $i$ .

real part:

imaginary part:

2

Compute $z - w$ where $z = 2 + 3i$ and $w = 1 + i$ . $z - w = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $+ $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $i$ .

real part:

imaginary part:

3

The conjugate of $z = 3 - 5i$ is $\bar{z} = $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $+ $ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ $i$ . (Enter the imaginary part with its sign.)

real part:

imaginary part:

4

Start with the complex number $1 + 0i$ (at angle $0°$ and modulus $1$ ). Multiply by $i$ once. Where does the result land on the complex plane?

A.

$-1 + 0i$ — on the negative real axis ($180°$)

B.

$0 + i$ — on the positive imaginary axis ($90°$)

C.

$0 - i$ — on the negative imaginary axis ($270°$)

D.

$1 + i$ — in the first quadrant ($45°$)

5

The complex number $z = 1 + i$ has modulus $\sqrt{2}$ and argument $45°$.

Using the multiplication rule $|z \cdot w| = |z| \cdot |w|$ and $\arg(z \cdot w) = \arg(z) + \arg(w)$ , compute $(1 + i)^2$ .

What is the modulus of $(1 + i)^2$ ? Enter the exact value.

Exercises: Represent Operations Geometrically on the Complex Plane

Warm-Up: Review What You Know

Fluency Practice