(2i/1+i)^2 Is Equal To

Jasonbradley

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Jun 16, 2024

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Simplifying Complex Numbers: (2i / (1 + i))^2

This article explores the simplification of the complex number expression (2i / (1 + i))^2.

Understanding Complex Numbers

A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, where i² = -1.

Simplifying the Expression

1. Rationalizing the Denominator: We start by simplifying the expression inside the parentheses. To do this, we multiply both the numerator and denominator of 2i / (1 + i) by the conjugate of the denominator, which is (1 - i).

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

   This simplifies to:

   (2i - 2i²) / (1 - i²)
   

   Substituting i² = -1:

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

2. Squaring the Simplified Expression: Now, we have simplified the expression inside the parentheses to (i + 1). We need to square this:

   (i + 1)² = (i + 1)(i + 1)
   

   Expanding the product:

   i² + 2i + 1
   

   Again, substituting i² = -1:

   -1 + 2i + 1 = 2i
   

Conclusion

Therefore, (2i / (1 + i))² = 2i.