## Instructions on using the complex number calculator

- This complex number calculator dynamically calculates the operations that are needed to be done on given complex numbers. The calculated values are rounded off to
**4 decimal**places for accuracy. - For
**single complex number**, the given**inputs**are**real part**of a complex number,**imaginary part**of the complex number, and**power**of the complex number. - For
**single complex number**, the given**outputs**are**modulus**of the entered complex number,**angle**of the entered complex number, and**real**and**imaginary**parts after taking the desired power of the entered complex number. - For
**single complex number**, this calculator can calculate the**Modulus/Magnitude/Absolute**value of the entered complex number. - For
**single complex number**, this calculator can calculate the**Angle/Argument**value of the entered complex number in**radians**. - For
**single complex number**, this calculator can calculate the**Power**of the entered complex number. This power can also yield**squareroot**and**cuberoot**of the entered complex number if the**entered power value**if**0.5**and**0.3334**, respectively. - For
**two complex numbers**, this calculator can calculate the**sum**of the entered complex numbers. - For
**two complex numbers**, this calculator can calculate the**multiplication**of the entered complex numbers. - For
**two complex numbers**, this calculator can calculate the**division**of the entered complex numbers. - After getting the result from sum, multiplication, and disvision, the real and imaginary values of the resultant can be plugged-in to single complex number calculations to do
**further operations**.

## Formulations

The magnitude/modulus/absolute value, and the angle of the given complex number are found by considering the following figure.

The formulas for the multiplication and division of two complex numbers are also mentioned in Fig. 1.

## Example

We have two complex numbers z_{1} =2-i, and z_{2} =-5+i2.

a) Find the magnitude and angle of both complex numbers.

b) Calculate their square, square root, and cube root individually.

c) Find their sum, multiplication, and division (z_{1}/z_{2}).

### Solution

a)

\[|z_1|=\sqrt{2^2+1^2}=2.2361\]
\[|\theta_1|=\tan^{-1}\frac{-1}{2}=-0.4636\]
\[|z_2|=\sqrt{5^2+2^2}=5.3852\]
\[|\theta_2|=\tan^{-1}\frac{2}{-5}=2.7611\]

b)

\[z_1^2=|z|^2(\cos 2\theta+i\sin 2\theta)=|2.2361|^2(\cos 2\theta+i\sin 2\theta)=3.0005-i3.9998\]
\[z_1^{\frac{1}{2}}=|z|^{\frac{1}{2}}(\cos {\frac{1}{2}}\theta+i\sin {\frac{1}{2}}\theta)=|2.2361|^{\frac{1}{2}}(\cos {\frac{1}{2}}\theta+i\sin {\frac{1}{2}}\theta)=1.4554-i0.3435\]
\[z_1^{\frac{1}{3}}=|z|^{\frac{1}{3}}(\cos {\frac{1}{3}}\theta+i\sin {\frac{1}{3}}\theta)=|2.2361|^{\frac{1}{3}}(\cos {\frac{1}{3}}\theta+i\sin {\frac{1}{3}}\theta)=1.2921-i0.2013\]
\[z_2^2=|z|^2(\cos 2\theta+i\sin 2\theta)=|5.3852|^2(\cos 2\theta+i\sin 2\theta)=21.0008-i19.9997\]
\[z_2^{\frac{1}{2}}=|z|^{\frac{1}{2}}(\cos {\frac{1}{2}}\theta+i\sin {\frac{1}{2}}\theta)=|5.3852|^{\frac{1}{2}}(\cos {\frac{1}{2}}\theta+i\sin {\frac{1}{2}}\theta)=0.4388+i2.2787\]
\[z_2^{\frac{1}{3}}=|z|^{\frac{1}{3}}(\cos {\frac{1}{3}}\theta+i\sin {\frac{1}{3}}\theta)=|5.3852|^{\frac{1}{3}}(\cos {\frac{1}{3}}\theta+i\sin {\frac{1}{3}}\theta)=1.0612+i1.3953\]

c)

\[z_1+z_2=2-i-5+i2=-3+i\]
\[z_1*z_2=(2-i)\times(-5+i2)=2\times(-5)-(-1)\times(2)-i[(-1)\times(-5)+(2)\times(2)]=-8+i9\]
\[\frac{z_1}{z_2}=\frac{2\times(-5)+(-1)\times(2)}{(-5)^2+(2)^2}+i\frac{(-1)\times(-5)-(2)\times(2)}{(-5)^2+(2)^2}=-0.4138+i0.0345\]

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