%2F(1-i)-%5Em%3D1.jpeg "(1+i)/(1-i) ^m=1")
Solving the Equation: (1+i)/(1-i)^m = 1
This article explores the equation (1+i)/(1-i)^m = 1, where i is the imaginary unit (√-1) and m is an integer. We will analyze the equation, find solutions for m, and delve into the underlying concepts.
Understanding Complex Numbers and Exponents
Before tackling the equation, let's review some essential concepts:
- Complex Numbers: Complex numbers are expressed as a + bi, where a and b are real numbers, and i is the imaginary unit. The number a is called the real part, and b is called the imaginary part.
- Exponents: When raising a complex number to a power, we multiply the number by itself the specified number of times. For example, (1 + i)² = (1 + i)(1 + i) = 1 + 2i + i² = 2i.
Simplifying the Equation
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Simplify (1+i)/(1-i): We can simplify this fraction by multiplying both the numerator and denominator by the conjugate of the denominator (1+i): (1+i)/(1-i) * (1+i)/(1+i) = (1 + 2i + i²)/(1 - i²) = (2i)/(2) = i
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Substitute the simplified expression: Our equation now becomes: i^m = 1.
Solving for m
The imaginary unit i has the following cyclical powers:
- i¹ = i
- i² = -1
- i³ = -i
- i⁴ = 1
Notice that i⁴ = 1, and the powers of i repeat in a cycle of 4.
To solve i^m = 1, we need to find the values of m that result in a power of i equal to 1.
Therefore, the solutions for m are all multiples of 4: m = 4k, where k is any integer.
Conclusion
The equation (1+i)/(1-i)^m = 1 has infinitely many solutions for m, which are all multiples of 4. This solution arises from understanding the cyclic nature of powers of the imaginary unit i.