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

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (1+i/1-i)^2 + (1-i/1+i)^2

This article will delve into simplifying the complex expression: (1+i/1-i)^2 + (1-i/1+i)^2.

Understanding the Expression

The expression involves complex numbers and requires a few steps to simplify. Here's a breakdown:

• Complex Numbers: Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit, where i² = -1.
• Fractions with Complex Numbers: We need to manipulate fractions containing complex numbers, which involves multiplying both numerator and denominator by the conjugate of the denominator.
• Squaring Complex Numbers: Squaring a complex number involves multiplying it by itself.

Simplifying the Expression

Let's break down the simplification process step-by-step:

1. Simplify the fractions:

   • (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 + i²) / (1 - i²) 
     

     Since i² = -1, we get:

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

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

     (1 - i) / (1 + i) * (1 - i) / (1 - i) = (1 - 2i + i²) / (1 - i²)
     

     Again, since i² = -1:

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

2. Substitute the simplified fractions back into the original expression:

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

3. Square the complex numbers:

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

Conclusion

Therefore, the simplified form of the expression (1+i/1-i)^2 + (1-i/1+i)^2 is -2.