(1+i/1-i)^3-(1-i/1+i)^3=x+iy

Jasonbradley

3 min read

·

Jun 16, 2024

28

Share

Solving the Complex Equation: (1+i/1-i)^3-(1-i/1+i)^3=x+iy

This article explores the solution to the complex equation (1+i/1-i)^3-(1-i/1+i)^3=x+iy, where x and y are real numbers. We will simplify the complex expressions and ultimately determine the values of x and y.

Simplifying the Complex Expressions

Let's start by simplifying the individual complex fractions within the equation.

Step 1: Rationalizing the denominator.

• For (1+i)/(1-i), multiply both numerator and denominator by the conjugate of the denominator (1+i):

  (1+i)/(1-i) * (1+i)/(1+i) = (1 + 2i + i^2)/(1 - i^2) = (2i)/(2) = i

• For (1-i)/(1+i), follow the same process:

  (1-i)/(1+i) * (1-i)/(1-i) = (1 - 2i + i^2)/(1 - i^2) = (-2i)/(2) = -i

Step 2: Substituting simplified fractions into the equation.

Now, the equation becomes:

(i)^3 - (-i)^3 = x + iy

Step 3: Evaluating the cubes of complex numbers.

• (i)^3 = i * i * i = -i
• (-i)^3 = -i * -i * -i = -i

Step 4: Simplifying the equation.

Our equation now is:

-i - (-i) = x + iy

Step 5: Solving for x and y.

This simplifies to:

0 = x + iy

Therefore, x = 0 and y = 0.

Conclusion

We have successfully solved the complex equation (1+i/1-i)^3-(1-i/1+i)^3=x+iy and found that the solution is x = 0 and y = 0. This demonstrates that the equation results in a complex number with a real component of 0 and an imaginary component of 0, effectively reducing to 0.