Back to Exercise: Represent complex numbers on the complex plane

Exercises: Represent Complex Numbers on the Complex Plane

Show your work. For polar form, use exact values where possible (e.g., $\cos 60° = \frac{1}{2}$).

Grade 9·20 problems·~35 min·Common Core Math - HS Number and Quantity·standard·hsn-cn-b-4

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Warm-Up: Review What You Know

In the complex plane, the imaginary axis is vertical. What do the tick marks on the imaginary axis represent?

Values $i, 2i, 3i, \ldots$ (complex numbers)

Real numbers $1, 2, 3, \ldots$ (the coefficients of $i$ )

Only integers

The modulus of each complex number on that axis

The complex number $-3i$ is plotted at which coordinates on the complex plane?

$(3, 0)$

$(0, -3)$

$(-3, 0)$

$(0, 3)$

Two different-looking expressions can represent the same complex number, just as $\frac{1}{2}$ and $0.5$ represent the same real number. Which pair of expressions represents the same complex number?

$1 + i$ and $2(\cos 45° + i \sin 45°)$

$\sqrt{2}(\cos 45° + i \sin 45°)$ and $1 + i$

$3 + 4i$ and $5(\cos 90° + i \sin 90°)$

$1 + i$ and $\sqrt{2}(\cos 30° + i \sin 30°)$

Fluency Practice

Which complex number is plotted at the point $(-2, 3)$ on the complex plane?

$2 + 3i$

$-2 + 3i$

$3 - 2i$

$-2 - 3i$

Find the modulus $r$ of $1 + \sqrt{3}\,i$ . Enter the exact value.

What is the argument $\theta$ (in degrees) of $1 + \sqrt{3}\,i$ ? Use the fact that $r = 2$ from the previous problem.

$30°$

$45°$

$60°$

$120°$

Convert $4(\cos 150° + i\sin 150°)$ to rectangular form $a + bi$ .

$4 + 4\sqrt{3}\,i$

$-2\sqrt{3} + 2i$

$2\sqrt{3} - 2i$

$-2 + 2i\sqrt{3}$

Convert $2(\cos 90° + i\sin 90°)$ to rectangular form. What is the imaginary part? Enter a number.

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