%5E2018.jpeg "(1-i)^2018")
Simplifying (1-i)^2018
This article explores how to simplify the complex number (1-i)^2018. We'll utilize the properties of complex numbers and De Moivre's Theorem to achieve this.
Understanding Complex Numbers
A complex number is a number 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).
Polar Form
Complex numbers can also be represented in polar form. This form uses the magnitude (or modulus) and the angle (or argument) of the complex number.
The magnitude, denoted as r, is the distance from the origin to the point representing the complex number in the complex plane. It is calculated using the formula: r = √(a² + b²)
The angle, denoted as θ, is the angle between the positive real axis and the line connecting the origin to the point representing the complex number. It is calculated using the formula: θ = arctan(b/a)
The polar form of a complex number is given by: r(cos θ + i sin θ)
De Moivre's Theorem
De Moivre's Theorem states that for any complex number in polar form, r(cos θ + i sin θ), and any integer n: (r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ))
Simplifying (1-i)^2018
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Convert to Polar Form:
- Find the magnitude: r = √(1² + (-1)²) = √2
- Find the angle: θ = arctan(-1/1) = -45° (or 315° in the range [0, 360°))
- Therefore, (1-i) = √2(cos(-45°) + i sin(-45°))
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Apply De Moivre's Theorem:
- (1-i)^2018 = (√2(cos(-45°) + i sin(-45°)))^2018
- (1-i)^2018 = (√2)^2018 (cos(-45° * 2018) + i sin(-45° * 2018))
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Simplify:
- (1-i)^2018 = 2^1009 (cos(-90900°) + i sin(-90900°))
- Note: -90900° is equivalent to 180° (since -90900° / 360° = -252.5, which leaves a remainder of 1 when divided by 4).
- Therefore, (1-i)^2018 = 2^1009 (cos(180°) + i sin(180°))
- Finally, (1-i)^2018 = 2^1009 (-1 + 0i) = -2^1009
Conclusion
By utilizing the polar form and De Moivre's Theorem, we have successfully simplified (1-i)^2018 to -2^1009. This highlights the power of these tools in simplifying complex number expressions, especially when dealing with large exponents.