Linear Algebra Examples

$\frac{2}{3} - \frac{i}{3}$

Step 1

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 2

The modulus of a complex number is the distance from the origin on the complex plane.

$| z | = \sqrt{a^{2} + b^{2}}$ where $z = a + b i$

Step 3

Substitute the actual values of $a = \frac{2}{3}$ and $b = - \frac{1}{3}$ .

$| z | = \sqrt{{(- \frac{1}{3})}^{2} + {(\frac{2}{3})}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.1

Apply the product rule to $- \frac{1}{3}$ .

$| z | = \sqrt{{(- 1)}^{2} {(\frac{1}{3})}^{2} + {(\frac{2}{3})}^{2}}$

Step 4.1.2

Apply the product rule to $\frac{1}{3}$ .

$| z | = \sqrt{{(- 1)}^{2} (\frac{1^{2}}{3^{2}}) + {(\frac{2}{3})}^{2}}$

Step 4.2

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 (\frac{1^{2}}{3^{2}}) + {(\frac{2}{3})}^{2}}$

Step 4.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$| z | = \sqrt{\frac{1^{2}}{3^{2}} + {(\frac{2}{3})}^{2}}$

Step 4.4

One to any power is one.

$| z | = \sqrt{\frac{1}{3^{2}} + {(\frac{2}{3})}^{2}}$

Step 4.5

Raise $3$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{9} + {(\frac{2}{3})}^{2}}$

Step 4.6

Apply the product rule to $\frac{2}{3}$ .

$| z | = \sqrt{\frac{1}{9} + \frac{2^{2}}{3^{2}}}$

Step 4.7

Raise $2$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{9} + \frac{4}{3^{2}}}$

Step 4.8

Raise $3$ to the power of $2$ .

$| z | = \sqrt{\frac{1}{9} + \frac{4}{9}}$

Step 4.9

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 4}{9}}$

Step 4.10

Add $1$ and $4$ .

$| z | = \sqrt{\frac{5}{9}}$

Step 4.11

Rewrite $\sqrt{\frac{5}{9}}$ as $\frac{\sqrt{5}}{\sqrt{9}}$ .

$| z | = \frac{\sqrt{5}}{\sqrt{9}}$

Step 4.12

Simplify the denominator.

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Step 4.12.1

Rewrite $9$ as $3^{2}$ .

$| z | = \frac{\sqrt{5}}{\sqrt{3^{2}}}$

Step 4.12.2

Pull terms out from under the radical, assuming positive real numbers.

$| z | = \frac{\sqrt{5}}{3}$

Step 5

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

$θ = \arctan (\frac{- \frac{1}{3}}{\frac{2}{3}})$

Step 6

Since inverse tangent of $\frac{- \frac{1}{3}}{\frac{2}{3}}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.4636476$ .

$θ = - 0.4636476$

Step 7

Substitute the values of $θ = - 0.4636476$ and $| z | = \frac{\sqrt{5}}{3}$ .

$\frac{\sqrt{5}}{3 (\cos (- 0.4636476) + i \sin (- 0.4636476))}$