Find All Complex Number Solutions z=3-4i

$z = 3 - 4 i$

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

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

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

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

Substitute the actual values of $a = 3$ and $b = - 4$ .

$| z | = \sqrt{{(- 4)}^{2} + 3^{2}}$

Find $| z |$ .

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Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{16 + 3^{2}}$

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

$| z | = \sqrt{16 + 9}$

Add $16$ and $9$ .

$| z | = \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

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

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

$| z | = 5$

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

$θ = \arctan (\frac{- 4}{3})$

Since inverse tangent of $\frac{- 4}{3}$ produces an angle in the fourth quadrant, the value of the angle is $- 0.92729521$ .

$θ = - 0.92729521$

Substitute the values of $θ = - 0.92729521$ and $| z | = 5$ .

$5 (\cos (- 0.92729521) + i \sin (- 0.92729521))$

Replace the right side of the equation with the trigonometric form.

$z = 5 (\cos (- 0.92729521) + i \sin (- 0.92729521))$

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Name

Name	three hundred twenty-three million nine hundred eighty-nine thousand nine hundred thirty-nine

Interesting facts

323989939 has 4 divisors, whose sum is 327550360
The reverse of 323989939 is 939989323
Previous prime number is 91

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 55
Digital Root 1

Name

Name	one billion eight hundred fifty-two million eight hundred thirteen thousand eight hundred twenty-seven

Interesting facts

1852813827 has 8 divisors, whose sum is 2074398480
The reverse of 1852813827 is 7283182581
Previous prime number is 131

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 45
Digital Root 9

Name

Name	seventy million ninety-three thousand nine hundred fifty-four

Interesting facts

70093954 has 8 divisors, whose sum is 120161088
The reverse of 70093954 is 45939007
Previous prime number is 7

Basic properties

Is Prime? no
Number parity even
Number length 8
Sum of Digits 37
Digital Root 1

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