Complex Number Solutions Solving Equations And Polar Form Conversion
Complex numbers, which extend the real number system by including the imaginary unit j, are indispensable in various fields including engineering, physics, and computer science. Mastering operations with complex numbers is crucial for problem-solving in these domains.
In this article, we will explore several problems involving complex numbers. Our first task involves determining the values of a and b given the equation (a + 2j) + (4 - 3j) = 5 + bj. This problem requires us to understand how to add complex numbers and equate their real and imaginary parts. Complex number addition involves combining the real parts and the imaginary parts separately. To solve for unknowns, we then equate the real and imaginary parts on both sides of the equation. Let's delve into a step-by-step solution.
To begin, let's add the complex numbers on the left side of the equation: (a + 2j) + (4 - 3j). We combine the real parts (a and 4) and the imaginary parts (2j and -3j) separately. This gives us (a + 4) + (2j - 3j), which simplifies to (a + 4) - j.
Now, we have the equation (a + 4) - j = 5 + bj. For this equation to hold true, the real parts on both sides must be equal, and the imaginary parts must also be equal. This gives us two separate equations:
- Real parts: a + 4 = 5
- Imaginary parts: -1 = b
Solving the first equation for a, we subtract 4 from both sides: a = 5 - 4, which simplifies to a = 1. The second equation directly gives us the value of b: b = -1.
Therefore, the values of a and b that satisfy the given equation are a = 1 and b = -1. This straightforward algebraic approach highlights the fundamental principle of equating real and imaginary parts in complex number equations. Understanding this principle is crucial for handling more complex problems involving complex numbers.
Next, we address the problem of expressing the complex number (2 - 3j) / (4 + 5j) in the polar form re^(jθ). This form is particularly useful in various applications, including electrical engineering and signal processing. Converting a complex number to polar form involves finding its magnitude (r) and argument (θ). The magnitude represents the distance of the complex number from the origin in the complex plane, while the argument represents the angle it makes with the positive real axis. The formula re^(jθ) elegantly combines these two pieces of information into a single expression, where r is the magnitude, θ is the argument, and e is the base of the natural logarithm.
To express (2 - 3j) / (4 + 5j) in polar form, we first need to simplify the expression by eliminating the complex number in the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bj is a - bj. In our case, the conjugate of 4 + 5j is 4 - 5j.
Multiplying the numerator and the denominator by the conjugate, we get:
(2 - 3j) / (4 + 5j) = [(2 - 3j) * (4 - 5j)] / [(4 + 5j) * (4 - 5j)]
Expanding the numerator, we have:
(2 - 3j) * (4 - 5j) = 2 * 4 + 2 * (-5j) - 3j * 4 - 3j * (-5j) = 8 - 10j - 12j + 15j^2
Since j^2 = -1, this simplifies to:
8 - 10j - 12j - 15 = -7 - 22j
Expanding the denominator, we use the fact that (a + bj) * (a - bj) = a^2 + b^2:
(4 + 5j) * (4 - 5j) = 4^2 + 5^2 = 16 + 25 = 41
Now we have the simplified complex number:
(-7 - 22j) / 41 = -7/41 - (22/41)j
To convert this to polar form, we need to find the magnitude r and the argument θ.
The magnitude r is given by:
r = √[(-7/41)^2 + (-22/41)^2] = √(49/1681 + 484/1681) = √(533/1681) ≈ 0.563
The argument θ is given by:
θ = arctan[(-22/41) / (-7/41)] = arctan(22/7)
Since the complex number lies in the third quadrant (both real and imaginary parts are negative), we need to add π to the arctangent result:
θ = arctan(22/7) + π ≈ 1.265 + π ≈ 4.407 radians
Therefore, the polar form of (2 - 3j) / (4 + 5j) is approximately 0. 563e^(4.407j).
This process underscores the importance of complex conjugates in simplifying complex number expressions and the method for converting complex numbers from rectangular to polar form. This conversion is fundamental in various applications where the magnitude and phase of a complex number are of primary interest.
Finally, let's solve the equation Z - 3 + 5j = 0 for the complex variable Z. This is a straightforward algebraic problem that demonstrates how to isolate a complex variable in an equation. Solving such equations is a basic skill in complex number algebra and is crucial for more advanced topics. The key here is to treat the complex number Z as a single entity and perform algebraic manipulations to isolate it, much like solving for a variable in a real number equation.
To solve for Z in the equation Z - 3 + 5j = 0, we simply need to isolate Z on one side of the equation. We can do this by adding 3 - 5j to both sides of the equation:
Z - 3 + 5j + (3 - 5j) = 0 + (3 - 5j)
This simplifies to:
Z = 3 - 5j
Therefore, the solution to the equation Z - 3 + 5j = 0 is Z = 3 - 5j. This solution is a complex number with a real part of 3 and an imaginary part of -5. This simple example illustrates the basic principles of solving linear equations involving complex numbers. The ability to manipulate and solve these equations is essential for handling more complex problems in mathematics, physics, and engineering.
In this article, we tackled several problems involving complex numbers, from solving equations to converting complex numbers to polar form. These examples highlight the fundamental operations and concepts in complex number algebra. By understanding how to add complex numbers, equate their real and imaginary parts, use conjugates to simplify expressions, and convert between rectangular and polar forms, one can effectively solve a wide range of problems involving complex numbers. The ability to work with complex numbers is an invaluable skill in various fields, and a solid understanding of these concepts is crucial for success in these areas. The solutions provided here serve as a foundation for further exploration and application of complex numbers in more advanced contexts.
