(1+i/1-i)^4

Jasonbradley

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Jun 16, 2024

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Simplifying (1 + i / 1 - i)^4

This article will guide you through the simplification of the complex number expression (1 + i / 1 - i)^4.

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i.e., i² = -1).

Simplifying the Expression

1. Rationalize the denominator: The first step is to get rid of the complex number in the denominator. We do this by multiplying both the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of 1 - i is 1 + i.

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

2. Expand and simplify: Now we expand both the numerator and denominator:

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

3. Raise to the power of 4: Finally, we raise the simplified expression to the power of 4:

   i⁴ = (i²)² = (-1)² = 1

Conclusion

Therefore, the simplified form of (1 + i / 1 - i)^4 is 1.

While the initial expression looks complex, by applying the rules of complex numbers and simplifying step by step, we arrive at a surprisingly simple result. This demonstrates the power of simplification in mathematics and the beauty of working with complex numbers.