Q45P - Mathematical Methods in Physical Sciences

Question

Solve for all possible values of the real numbers  and in the following equations ${\left(x+iy\right)}^{\mathbf{2}}\mathbf{=}{\left(x-iy\right)}^{\mathbf{2}}$.

Step-by-Step Solution

Answer

When x is 0 , y is a real number, and when y is 0 ,x  is any real number. 

Step 1: Given information

It is given that ${\left(x+iy\right)}^2={\left(x-iy\right)}^2$

Step 2: Definition of a complex number

Every complex number can be described as: 

 

z=a+bi

 

Where a and b are both real numbers, z is the complex number, and i is known as iota, which makes z a complex number.

Step 3: Begin by stating the given information

Evaluate the left- and right-hand sides of the equation:

 $$\begin{gathered} {\left(x+iy\right)}^2=x^2+2xyi-y^2 \\ {\left(x-iy\right)}^2=x^2-2xyi-y^2 \end{gathered}$$

  

The equation can be written as:

 $$\begin{gathered} x^2+2xyi-y^2=x^2-2xyi-y^2 \\ 2xyi+2xyi=0 \end{gathered}$$

Step 5: Equate like terms

Equate like terms from both sides:

 

$$\begin{gathered} 4xyi=0 \\ 4xy=0 \end{gathered}$$

 

Thus, when x is 0, y is a real number, and when y is 0, x is any real number.