Linear Algebra Examples

$| - 1 - 2 i |$

Step 1

Use the formula $| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

$\sqrt{{(- 1)}^{2} + {(- 2)}^{2}}$

Step 2

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

$\sqrt{1 + {(- 2)}^{2}}$

Step 3

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

$\sqrt{1 + 4}$

Step 4

Add $1$ and $4$ .

$\sqrt{5}$

Step 5

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 6

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 7

Substitute the actual values of $a = \sqrt{5}$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(\sqrt{5})}^{2}}$

Step 8

Find $| z |$ .

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Step 8.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(\sqrt{5})}^{2}}$

Step 8.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$| z | = \sqrt{0 + {(5^{\frac{1}{2}})}^{2}}$

Step 8.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$| z | = \sqrt{0 + 5^{\frac{1}{2} \cdot 2}}$

Step 8.2.3

Combine $\frac{1}{2}$ and $2$ .

$| z | = \sqrt{0 + 5^{\frac{2}{2}}}$

Step 8.2.4

Cancel the common factor of $2$ .

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Step 8.2.4.1

Cancel the common factor.

$| z | = \sqrt{0 + 5^{\frac{2}{2}}}$

Step 8.2.4.2

Rewrite the expression.

$| z | = \sqrt{0 + 5}$

Step 8.2.5

Evaluate the exponent.

$| z | = \sqrt{0 + 5}$

Step 8.3

Add $0$ and $5$ .

$| z | = \sqrt{5}$

Step 9

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

$θ = \arctan (\frac{0}{\sqrt{5}})$

Step 10

Since inverse tangent of $\frac{0}{\sqrt{5}}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 11

Substitute the values of $θ = 0$ and $| z | = \sqrt{5}$ .

$\sqrt{5} (\cos (0) + i \sin (0))$