Finding Coordinates In Complex Plane If X + Iy = (1 + I√3)^100
In the fascinating world of mathematics, complex numbers hold a special allure. They extend the familiar realm of real numbers by incorporating an imaginary component, denoted by the symbol 'i', where i² = -1. This seemingly simple addition opens up a vast landscape of mathematical possibilities, allowing us to solve equations, model physical phenomena, and explore intricate geometric relationships.
This article delves into a specific problem involving complex numbers, where we aim to determine the coordinates of a point P(x, y) represented by the complex number x + iy, given that it is equal to (1 + i√3)^100. This problem not only showcases the power of complex number manipulation but also highlights the elegance of De Moivre's Theorem, a fundamental concept in complex number theory.
Problem Statement
Our journey begins with the following equation:
x + iy = (1 + i√3)^100
where:
- x and y are real numbers representing the coordinates of point P.
- i is the imaginary unit, defined as √-1.
- (1 + i√3) is a complex number in the form a + bi, where a = 1 and b = √3.
- 100 is the exponent, indicating that the complex number (1 + i√3) is raised to the power of 100.
Our objective is to find the values of x and y, which will reveal the precise location of point P in the complex plane. To achieve this, we will embark on a step-by-step exploration, employing key concepts and techniques from complex number theory.
Transforming to Polar Form
The first step in our quest involves transforming the complex number (1 + i√3) from its rectangular form (a + bi) to its polar form (r(cos θ + i sin θ)). This transformation simplifies the exponentiation process and allows us to leverage the power of De Moivre's Theorem.
Finding the Modulus (r)
The modulus, often denoted as 'r', represents the distance of the complex number from the origin in the complex plane. It is calculated using the Pythagorean theorem:
r = √(a² + b²)
In our case, a = 1 and b = √3, so:
r = √(1² + (√3)²) = √(1 + 3) = √4 = 2
Thus, the modulus of the complex number (1 + i√3) is 2.
Determining the Argument (θ)
The argument, denoted as 'θ', represents the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. It can be calculated using the following trigonometric relationships:
cos θ = a/r sin θ = b/r
For our complex number (1 + i√3), we have:
cos θ = 1/2 sin θ = √3/2
From these trigonometric values, we can identify that θ = π/3 radians (or 60 degrees). This means that the complex number (1 + i√3) lies in the first quadrant of the complex plane, making an angle of π/3 with the positive real axis.
Polar Form Representation
Now that we have determined the modulus (r = 2) and the argument (θ = π/3), we can express the complex number (1 + i√3) in its polar form:
1 + i√3 = 2(cos(π/3) + i sin(π/3))
This transformation is crucial as it sets the stage for applying De Moivre's Theorem, which will simplify the process of raising the complex number to the power of 100.
Unleashing De Moivre's Theorem
De Moivre's Theorem is a cornerstone of complex number theory, providing a powerful tool for raising complex numbers in polar form to any integer power. The 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))
This theorem elegantly connects the exponentiation of a complex number to the scaling of its modulus and the multiplication of its argument by the exponent.
Applying the Theorem
In our problem, we need to find (1 + i√3)^100. We have already expressed (1 + i√3) in polar form as 2(cos(π/3) + i sin(π/3)). Now, we can apply De Moivre's Theorem with r = 2, θ = π/3, and n = 100:
(1 + i√3)^100 = [2(cos(π/3) + i sin(π/3))]^100
Using De Moivre's Theorem, we get:
(1 + i√3)^100 = 2^100(cos(100π/3) + i sin(100π/3))
This expression significantly simplifies the problem. We have reduced the task of raising a complex number to the power of 100 to evaluating the cosine and sine of the angle 100π/3.
Simplifying the Angle
The angle 100π/3 is quite large, making it difficult to directly determine its cosine and sine values. To simplify this, we can find an equivalent angle within the range of 0 to 2π (a single revolution) by subtracting multiples of 2π from 100π/3 until we obtain an angle within this range.
100π/3 = 33π + π/3
Since 33π is an odd multiple of π, we can rewrite it as:
33π = 16(2π) + π
Therefore:
100π/3 = 16(2π) + π + π/3 = 16(2π) + 4π/3
Since 16(2π) represents 16 full revolutions, it does not affect the cosine and sine values. Thus, we can focus on the angle 4π/3:
cos(100π/3) = cos(4π/3) sin(100π/3) = sin(4π/3)
The angle 4π/3 lies in the third quadrant of the complex plane, where both cosine and sine values are negative. We can determine their values as follows:
cos(4π/3) = -1/2 sin(4π/3) = -√3/2
Final Expression
Now that we have simplified the angle and determined its cosine and sine values, we can substitute them back into our expression:
(1 + i√3)^100 = 2^100(cos(4π/3) + i sin(4π/3))
(1 + i√3)^100 = 2^100(-1/2 - i√3/2)
This expression provides the final form of (1 + i√3)^100 in polar form.
Unveiling the Coordinates of Point P(x, y)
Our ultimate goal is to find the coordinates of point P(x, y), which are represented by the real and imaginary parts of the complex number (1 + i√3)^100. To achieve this, we need to convert the expression we obtained in the previous step from polar form back to rectangular form (x + iy).
Distributing and Simplifying
We have:
(1 + i√3)^100 = 2^100(-1/2 - i√3/2)
Distributing 2^100, we get:
(1 + i√3)^100 = -2^99 - i2^99√3
Now, we can directly identify the real and imaginary parts:
x = -2^99 y = -2^99√3
The Coordinates of Point P
Therefore, the coordinates of point P(x, y) are:
P(x, y) = (-2^99, -2^99√3)
This result pinpoints the exact location of point P in the complex plane. It lies in the third quadrant, far from the origin, with both its x and y coordinates being negative and having extremely large magnitudes due to the presence of 2^99.
Conclusion
In this exploration, we successfully determined the coordinates of point P(x, y) given that x + iy = (1 + i√3)^100. We navigated the realm of complex numbers, transforming between rectangular and polar forms, and masterfully applied De Moivre's Theorem. This journey not only provided us with the solution to a specific problem but also illuminated the elegance and power of complex number theory.
By transforming the complex number to polar form, applying De Moivre's Theorem, and converting back to rectangular form, we were able to efficiently calculate the 100th power of a complex number. This problem serves as a testament to the importance of understanding fundamental concepts and techniques in complex number manipulation.
The coordinates of point P, (-2^99, -2^99√3), reveal its position in the complex plane, showcasing the profound connection between algebra and geometry in the world of complex numbers.
FAQ Section
What are Complex Numbers?
Complex numbers are numbers that extend the real number system by including an imaginary unit, denoted as 'i', where i² = -1. They are expressed in the form a + bi, where a and b are real numbers, and 'a' is the real part and 'b' is the imaginary part.
What is the Polar Form of a Complex Number?
The polar form of a complex number represents it in terms of its distance from the origin (modulus) and the angle it makes with the positive real axis (argument). A complex number a + bi can be written in polar form as r(cos θ + i sin θ), where r = √(a² + b²) and θ is the angle such that cos θ = a/r and sin θ = b/r.
What is De Moivre's Theorem?
De Moivre's Theorem is a fundamental theorem in complex number theory that 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))
This theorem simplifies the process of raising complex numbers in polar form to integer powers.
How do you convert a complex number from rectangular form to polar form?
To convert a complex number from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)), follow these steps:
- Calculate the modulus (r): r = √(a² + b²).
- Determine the argument (θ) using trigonometric relationships: cos θ = a/r and sin θ = b/r. The angle θ can be found using the arctangent function, but care must be taken to ensure the correct quadrant is chosen based on the signs of a and b.
How do you convert a complex number from polar form to rectangular form?
To convert a complex number from polar form (r(cos θ + i sin θ)) to rectangular form (a + bi), simply use the following relationships:
a = r cos θ b = r sin θ
What is the significance of complex numbers in mathematics and other fields?
Complex numbers have significant applications in various fields, including mathematics, physics, engineering, and computer science. They are used to:
- Solve equations that have no real solutions.
- Model alternating current circuits in electrical engineering.
- Represent quantum mechanical systems in physics.
- Perform signal processing in computer science.
- Describe fluid dynamics and aerodynamics.
Complex numbers provide a powerful tool for solving problems and modeling phenomena in a wide range of disciplines.
