Chapter 8, Problem 5RCC

To determine

the polar form of a complex number z , the modulus of z , the argument of z and graph the complex number z .

Answer to Problem 5RCC

A complex number $z=a+bi$ has the polar from $z=r\left(\cos \theta +i\sin \theta \right)$ . Modulus of a complex number $z=a+bi$ is $\left|z\right|=\sqrt{a^2+b^2}$ . Argument of complex number $z=a+bi$ is $\theta =ta{n}^{-1}\left(\frac{b}{a}\right)$ . Graph was drawn of complex number z .

Explanation of Solution

Calculation:

Explain the steps or procedure to graph a complex number z , write the polar form of a complex number z , the modulus of z and the argument of z .

For real numbers or sets of real numbers, the number line in use has just one dimension.

Complex number, however, have two components a real part and an imaginary part.

This suggests that two axes are needed to graph a complex number, one for the real part and the other for imaginary part. Call these axes the real axis and the imaginary axis, respectively.

The plane determined by these two axes is called the complex plane.

To graph the complex number $\text{'}a+bi\text{'}$ , plot the ordered pair of numbers $\left(a,b\right)$ in the complex plane.

  

A complex number $z=a+bi$ has the polar from

  $z=r\left(\cos \theta +i\sin \theta \right)$ .

Where,

  $\begin{array}{l}r=\left|z\right|\\ =\sqrt{a^2+b^2},\end{array}$

and

  $tan\theta =\frac{b}{a},$

The r is the modulus of z and is an argument of z .

Modulus of a complex number $z=a+bi$ is $\left|z\right|=\sqrt{a^2+b^2}$ .

Argument of complex number $z=a+bi$ is $\theta =ta{n}^{-1}\left(\frac{b}{a}\right)$ .

The argument of z is not unique, but any two arguments of z differ by a multiple of $2\pi$ .

When determining the argument, we must consider the quadrant in which z lies.

Conclusion:

Therefore, a complex number $z=a+bi$ has the polar from $z=r\left(\cos \theta +i\sin \theta \right)$ . Modulus of a complex number $z=a+bi$ is $\left|z\right|=\sqrt{a^2+b^2}$ . Argument of complex number $z=a+bi$ is $\theta =ta{n}^{-1}\left(\frac{b}{a}\right)$ . Graph was drawn of complex number z .