W15 Regular

Due date: Sunday 4/26, 11:59pm

Complex algebra

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Complex arithmetic

Write each of these expressions in the form $a+bi$.

(a) ${(2i)}^3$ $$ (b) $\sqrt{-4}\sqrt{-16}$

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Solution

Solutions---0290-03

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Complex solutions of quadratic equations

Find all solutions and write them in the form $z=a+bi$.

(a) $16x^2+9=0$ $$ (b) $x^2+\frac{1}{3}x+\frac{1}{9}=0$

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Solution

Solutions---0290-04

Complex exponential

03 $★$

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Polar and exponential form

Write down Euler’s Formula.

Now write each of the following complex numbers (i) in polar form, and (ii) in exponential form.

(a) $2-2\sqrt{3}i$ $$ (b) $6i$

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Solution

Solutions---0300-03

04 $★$

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Complex products and quotients using polar

For each pair of complex numbers $z$ and $w$, compute:

$$zw,\frac{z}{w},\frac{1}{z}$$

(a) $z=1+\sqrt{3}i,w=\sqrt{3}+i$

(b) $z=2\sqrt{3}-2i,w=6i$

(Use polar forms with $\theta \in [\mathrm{0,2}\pi )$.)

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Solution

Solutions---0300-04

05 $★$

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Complex powers using polar

Using De Moivre’s Theorem, write each number in the form $a+bi$.

(a) ${(1+i)}^{16}$ $$ (b) ${(\sqrt{3}-i)}^5$

(First convert to polar/exponential, then compute the power, then convert back.)

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Solution

Solutions---0300-05

Complex roots

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Complex roots using polar

Find each of the indicated roots.

(a) The four $4^{\text{th}}$ roots of $1$.

(b) The three cube ($3^{\text{rd}}$) roots of $\sqrt{2}+\sqrt{2}i$.

Try to write your answer in $a+bi$ form if that is not hard, otherwise leave it in polar form.

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Solution

Solutions---0310-02