Linear Algebra Examples

\sqrt{2} + \sqrt{2} i

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

Step 1

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

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

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

Step 2

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

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

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

z = a + b i

$z = a + b i$

Step 3

Substitute the actual values of

a = \sqrt{2}

$a = \sqrt{2}$ and

b = \sqrt{2}

$b = \sqrt{2}$ .

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

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

Step 4

Find

| z |

$| z |$ .

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Step 4.1

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 4.1.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

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

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

Step 4.1.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.1.4

Cancel the common factor of

2

$2$ .

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Step 4.1.4.1

Cancel the common factor.

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

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

Step 4.1.4.2

Rewrite the expression.

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

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

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

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

Step 4.1.5

Evaluate the exponent.

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

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

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

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

Step 4.2

Rewrite

{\sqrt{2}}^{2}

${\sqrt{2}}^{2}$ as

2

$2$ .

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Step 4.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{2}

$\sqrt{2}$ as

2^{\frac{1}{2}}

$2^{\frac{1}{2}}$ .

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

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

Step 4.2.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

| z | = \sqrt{2 + 2^{\frac{2}{2}}}

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

Step 4.2.4

Cancel the common factor of

2

$2$ .

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Step 4.2.4.1

Cancel the common factor.

| z | = \sqrt{2 + 2^{\frac{2}{2}}}

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

Step 4.2.4.2

Rewrite the expression.

| z | = \sqrt{2 + 2}

$| z | = \sqrt{2 + 2}$

| z | = \sqrt{2 + 2}

$| z | = \sqrt{2 + 2}$

Step 4.2.5

Evaluate the exponent.

| z | = \sqrt{2 + 2}

$| z | = \sqrt{2 + 2}$

| z | = \sqrt{2 + 2}

$| z | = \sqrt{2 + 2}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Add

2

$2$ and

2

$2$ .

| z | = \sqrt{4}

$| z | = \sqrt{4}$

Step 4.3.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

| z | = \sqrt{2^{2}}

$| z | = \sqrt{2^{2}}$

| z | = \sqrt{2^{2}}

$| z | = \sqrt{2^{2}}$

Step 4.4

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

| z | = 2

$| z | = 2$

| z | = 2

$| z | = 2$

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{\sqrt{2}}{\sqrt{2}})

$θ = \arctan (\frac{\sqrt{2}}{\sqrt{2}})$

Step 6

Since inverse tangent of

\frac{\sqrt{2}}{\sqrt{2}}

$\frac{\sqrt{2}}{\sqrt{2}}$ produces an angle in the first quadrant, the value of the angle is

\frac{π}{4}

$\frac{π}{4}$ .

θ = \frac{π}{4}

$θ = \frac{π}{4}$

Step 7

Substitute the values of

θ = \frac{π}{4}

$θ = \frac{π}{4}$ and

| z | = 2

$| z | = 2$ .

2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))

$2 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$