(1+i/1-i)^2

Jasonbradley

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Jun 16, 2024

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

This article will guide you through simplifying the complex expression (1 + i)/(1 - i) ^ 2. We will utilize basic complex number operations and algebraic manipulation.

Understanding Complex Numbers

Before we begin, let's recall some key concepts about complex numbers:

• Complex number: A complex number is a number of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, satisfying i^2 = -1.
• Complex conjugate: The complex conjugate of a complex number a + bi is a - bi. The product of a complex number and its conjugate is always a real number.

Simplifying the Expression

1. Simplifying the denominator: We start by simplifying the denominator. We can eliminate the complex number in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

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

2. Expanding the denominator: We expand the denominator using the difference of squares formula: (a + b)(a - b) = a^2 - b^2

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

3. Simplifying using i^2 = -1:

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

4. Expanding the numerator: We expand the numerator using the distributive property:

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

5. Further simplification:

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

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

Conclusion

By applying the properties of complex numbers and algebraic manipulation, we successfully simplified the complex expression (1 + i)/(1 - i) ^ 2 to 1 - i. This process illustrates the power of using complex conjugates to eliminate complex terms in the denominator.