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Back to Exercise: Represent operations geometrically

Exercises: Represent Operations Geometrically on the Complex Plane

Show your work. Use exact values where possible. For polar form, express angles in degrees.

Grade 9·22 problems·~40 min·Common Core Math - HS Number and Quantity·standard·hsn-cn-b-5
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A

Warm-Up: Review What You Know

1

Which of the following describes the modulus ∣z∣|z| of a complex number z=a+biz = a + bi?

A.

The imaginary part bb

B.

The distance from zz to the origin: a2+b2\sqrt{a^2 + b^2}

C.

The real part aa

D.

The argument θ\theta

2

The complex number z=1+iz = 1 + i has modulus 2\sqrt{2} and argument $45°$. What is its polar form?

A.

2(cos⁡45°+isin⁡45°)2(\cos 45° + i\sin 45°)

B.

2(cos⁡45°+isin⁡45°)\sqrt{2}(\cos 45° + i\sin 45°)

C.

2(cos⁡90°+isin⁡90°)\sqrt{2}(\cos 90° + i\sin 90°)

D.

1(cos⁡45°+isin⁡45°)1(\cos 45° + i\sin 45°)

3

What is the argument (in degrees) of the complex number −1+3 i-1 + \sqrt{3}\,i?

A.

$60°$

B.

$120°$

C.

−60°-60°

D.

$240°$

B

Fluency Practice

1

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

2

Compute z−wz - w where z=2+3iz = 2 + 3i and w=1+iw = 1 + i. $z - w = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $+ $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   ii.

3

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

4

Start with the complex number 1+0i1 + 0i (at angle $0°$ and modulus 11). Multiply by ii once. Where does the result land on the complex plane?

A.

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

B.

0+i0 + i — on the positive imaginary axis ($90°$)

C.

0−i0 - i — on the negative imaginary axis ($270°$)

D.

1+i1 + i — in the first quadrant ($45°$)

5

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

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

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

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