W16 - Examples

Complex product, quotient, power using Euler

Complex product, quotient, power using Euler

Define:

$$z=2e^{i\frac{\pi }{2}}w=5e^{i\frac{\pi }{3}}$$

Product $zw$:

$$\begin{array}{c}zw\gg \gg \left(2e^{i\frac{\pi }{2}}\right)\cdot \left(5e^{i\frac{\pi }{3}}\right)\\ \\ \gg \gg (2\cdot 5)\left(e^{i\frac{\pi }{2}}\right)\left(e^{i\frac{\pi }{3}}\right)\gg \gg 10e^{i\frac{\pi }{2}+i\frac{\pi }{3}}\gg \gg 10e^{i\frac{5\pi }{6}}\end{array}$$

Quotient $z/w$:

$$\begin{array}{c}z/w\gg \gg \left(2e^{i\frac{\pi }{2}}\right)/\left(5e^{i\frac{\pi }{3}}\right)\\ \\ \gg \gg \frac{2e^{i\frac{\pi }{2}}}{5e^{i\frac{\pi }{3}}}\gg \gg \frac{2}{5}{e}^{i\frac{\pi }{2}}{e}^{-i\frac{\pi }{3}}\gg \gg \frac{2}{5}{e}^{i\frac{\pi }{6}}\end{array}$$

Power $z^8$:

$$\begin{array}{c}{z}^8\gg \gg {\left(2e^{i\frac{\pi }{2}}\right)}^8\\ \\ \gg \gg 2^8{\left(e^{i\frac{\pi }{2}}\right)}^8\gg \gg 512e^{i\cdot 4\pi }\end{array}$$

Notice:

$$e^{i\cdot 4\pi }\gg \gg {\left(e^{2\pi i}\right)}^2\gg \gg 1^2\gg \gg 1$$

Simplify:

$$512e^{i\cdot 4\pi }\gg \gg 512$$ Link to original

Complex power from Cartesian

Complex power from Cartesian

Compute $(3+3i{)}^4$.

Solution

First convert to exponential form:

$$\begin{array}{c}3+3i\gg \gg 3\sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\right)\\ \\ \gg \gg 3\sqrt{2}{e}^{i\frac{\pi }{4}}\end{array}$$

Compute the power:

$$\begin{array}{c}(3+3i{)}^4\gg \gg {\left(3\sqrt{2}{e}^{i\frac{\pi }{4}}\right)}^4\\ \\ \gg \gg 324e^{i\pi }\gg \gg -324\end{array}$$Link to original

Finding all 4th roots of 16

Finding all 4th roots of 16

Compute all the $4^{\text{th}}$ roots of $16$.

Solution

Write $16=16e^{0i}$.

Evaluate roots formula:

$${\left(16e^{0i}\right)}^{\frac{1}{4}}\gg \gg w_k=16^{\frac{1}{4}}{e}^{i\left(\frac{0}{4}+k\frac{2\pi }{4}\right)}$$

Simplify:

$$\gg \gg 2e^{i\cdot k\frac{\pi }{2}}\gg \gg 2,2i,-2,-2i$$ Link to original

Finding 2nd roots of 2i

Finding 2nd roots of 2i

Find both $2^{\text{nd}}$ roots of $2i$.

Solution

Write $2i=2e^{i\frac{\pi }{2}}$.

Evaluate roots formula:

$$\begin{array}{c}{\left(2e^{i\frac{\pi }{2}}\right)}^{\frac{1}{2}}\gg \gg w_k=\sqrt{2}{e}^{i\left(\frac{\pi /2}{2}+k\frac{2\pi }{2}\right)}\\ \\ \gg \gg \sqrt{2}{e}^{i\left(\frac{\pi }{4}+k\pi \right)}\end{array}$$

Compute the options: $k=0,1$:

$$\gg \gg \sqrt{2}{e}^{i\frac{\pi }{4}},\sqrt{2}{e}^{i\frac{5\pi }{4}}$$

Convert to rectangular:

$$\begin{array}{c}\gg \gg \sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i\right),\sqrt{2}\left(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i\right)\\ \\ \gg \gg 1+i,1-i\end{array}$$ Link to original

Some roots of unity

Some roots of unity

Find the $1^{\text{st}}$ and $2^{\text{nd}}$ and $3^{\text{rd}}$ and $4^{\text{th}}$ and $5^{\text{th}}$ and $6^{\text{th}}$ roots of the number $1$.

Solution

(1) $1^{\text{st}}$

Write $1=e^{0i}$. Evaluate roots formula. There is no possible $k$:

$${\left(e^{0i}\right)}^{\frac{1}{1}}\gg \gg e^{0i}\gg \gg 1$$

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(2) $2^{\text{nd}}$

Write $1=e^{0i}$. Evaluate roots formula in terms of $k$:

$${\left(e^{0i}\right)}^{\frac{1}{2}}\gg \gg w_k=e^{i\left(\frac{0}{2}+k\frac{2\pi }{2}\right)}k=0,1$$

Compute the two options, $k=0,1$:

$$\gg \gg 1,e^{\pi i}\gg \gg 1,-1$$

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(3) $3^{\text{rd}}$

Evaluate roots formula in terms of $k$:

$${\left(e^{0i}\right)}^{\frac{1}{3}}\gg \gg w_k=e^{i\left(\frac{0}{3}+k\frac{2\pi }{3}\right)}$$

Compute the options: $k=0,1,2$:

$$\gg \gg 1,e^{i\frac{2\pi }{3}},e^{i\frac{4\pi }{3}}\gg \gg 1,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}-\frac{\sqrt{3}}{2}i$$

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(4) $4^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{4}}\gg \gg w_k=e^{i\left(\frac{0}{4}+k\frac{2\pi }{4}\right)}$$

Compute the options: $k=0,1,2,3$:

$$1,e^{i\frac{\pi }{2}},e^{i\pi },e^{i\frac{3\pi }{2}}\gg \gg 1,i,-1,-i$$

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(5) $5^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{5}}\gg \gg w_k=e^{i\left(\frac{0}{5}+k\frac{2\pi }{5}\right)}$$

Compute the options: $k=0,1,2,3,4$:

$$1,e^{i\frac{2\pi }{5}},e^{i\frac{4\pi }{5}},e^{i\frac{6\pi }{5}},e^{i\frac{8\pi }{5}}$$

Don’t simplify, it’s not feasible.

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(6) $6^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{6}}\gg \gg w_k=e^{i\left(\frac{0}{6}+k\frac{2\pi }{6}\right)}$$

Compute the options: $k=0,1,2,3,4,5$:

$$1,e^{i\frac{2\pi }{6}},e^{i\frac{4\pi }{6}},e^{i\frac{6\pi }{6}},e^{i\frac{8\pi }{6}},e^{i\frac{10\pi }{6}}$$

Simplify:

$$\gg \gg 1,\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-1,-\frac{1}{2}-\frac{\sqrt{3}}{2}i,\frac{1}{2}-\frac{\sqrt{3}}{2}i$$Link to original