Precalculus Examples

$| - 10 + 24 i |$

Step 1

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

$\sqrt{{(- 10)}^{2} + 24^{2}}$

Step 2

Raise $- 10$ to the power of $2$ .

$\sqrt{100 + 24^{2}}$

Step 3

Raise $24$ to the power of $2$ .

$\sqrt{100 + 576}$

Step 4

Add $100$ and $576$ .

$\sqrt{676}$

Step 5

Rewrite $676$ as $26^{2}$ .

$\sqrt{26^{2}}$

Step 6

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

$26$

Step 7

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 8

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 9

Substitute the actual values of $a = 26$ and $b = 0$ .

$| z | = \sqrt{0^{2} + 26^{2}}$

Step 10

Find $| z |$ .

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Step 10.1

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

$| z | = \sqrt{0 + 26^{2}}$

Step 10.2

Raise $26$ to the power of $2$ .

$| z | = \sqrt{0 + 676}$

Step 10.3

Add $0$ and $676$ .

$| z | = \sqrt{676}$

Step 10.4

Rewrite $676$ as $26^{2}$ .

$| z | = \sqrt{26^{2}}$

Step 10.5

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

$| z | = 26$

Step 11

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

$θ = \arctan (\frac{0}{26})$

Step 12

Since inverse tangent of $\frac{0}{26}$ produces an angle in the first quadrant, the value of the angle is $0$ .

$θ = 0$

Step 13

Substitute the values of $θ = 0$ and $| z | = 26$ .

$26 (\cos (0) + i \sin (0))$