Q1P - Mathematical Methods in Physical Sciences

Question

For each of the following numbers, first, visualize where it is in the complete plane. With a little practice, you can quickly find $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$ 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 numbers 1+i .

Step-by-Step Solution

Answer

The right values for the complex numbers are

$$\begin{gathered} x=1,y=1 \\ r=\sqrt{2} \\ \theta =\frac{\mathrm{\pi }}{4} \\ 1+i=\sqrt{2}\left(\cos \frac{\mathrm{\pi }}{4}+i \sin \frac{\mathrm{\pi }}{4}\right) \\ =\sqrt{2}{e}^{i\frac{\mathrm{\pi }}{4}} \end{gathered}$$

Step 1: Given information and the complex numbers:

The given complex number is 1+i.

The general form of complex numbers is x+iy  .

Step 2: Complex numbers and the complex 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 1+i .

$$\begin{gathered} x=1,y=1 \\ r=\sqrt{1^2+1^2} \\ =\sqrt{2} \\ \theta ={\cos }^{-1}\frac{1}{\sqrt{2}}=\frac{\mathrm{\pi }}{4} \\ 1+i=\sqrt{2}\left(\cos \frac{\mathrm{\pi }}{4}+i \sin \frac{\mathrm{\pi }}{4}\right) \\ =\sqrt{2}{e}^{i\frac{x}{4}} \end{gathered}$$

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