Q2P - Mathematical Methods in Physical Sciences

Question

For each of the following numbers, first visualize where it is in the complex plane. With a little practice, you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }\mathbf{ }$in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also, plot the complex conjugate of the number i-1.

Step-by-Step Solution

Answer

The right value for the complex numbers are

$$\begin{gathered} x=-1,y=1 \\ r=\sqrt{2} \\ \theta =\frac{3\mathrm{\pi }}{2} \\ 1-i=\sqrt{2}\left(\cos \frac{3\mathrm{\pi }}{2}+i \sin \frac{3\mathrm{\pi }}{2}\right) \\ =\sqrt{2}{e}^{i\frac{3\mathrm{\pi }}{2}} \end{gathered}$$

Step 1: Given information and the general form of complex numbers:

The given complex number is i-1  .

The general form of complex numbers is x+iy.

Step 2: Complex numbers and the given conjugate:

The general form of complex numbers is x+iy .

$$\begin{gathered} x+iy=r\left(\cos \theta +i \sin \theta \right) \\ =r{e}^{i\theta } \\ r=\sqrt{x^2+y^2} \\ \theta ={\cos }^{-1}\frac{x}{r} \end{gathered}$$ 

 

 In the given number i-1 .

$x=-1,y=1$

$$\begin{gathered} r=\sqrt{{\left(-1\right)}^2+1^2} \\ =\sqrt{2} \\ \theta ={\cos }^{-1}\frac{-1}{\sqrt{2}} \\ =\frac{3\mathrm{\pi }}{2} \\ i-1=\sqrt{2}\left(\cos \frac{3\mathrm{\pi }}{2}+ i \sin \frac{3\mathrm{\pi }}{2}\right) \\ =\sqrt{2}{e}^{i\frac{3x}{2}} \end{gathered}$$

Plot the given complex number and the given conjugate of the number on the graph.