Product Of Complex Numbers Z And W Explained

Introduction

In this article, we delve into the fascinating world of complex numbers and explore how to find the product of two complex numbers given in polar form. The specific problem we will address involves finding the product of two complex numbers, z and w, where $z=38\left(\cos \left(\frac{\pi}{8}\right)+i \sin \left(\frac{\pi}{8}\right)\right)$ and $w=2\left(\cos \left(\frac{\pi}{16}\right)+i \sin \left(\frac{\pi}{16}\right)\right)$. This problem not only provides a practical application of complex number multiplication but also reinforces the understanding of their polar representation. Understanding how to multiply complex numbers in polar form is crucial for various applications in mathematics, physics, and engineering. Before diving into the solution, let's briefly recap some foundational concepts about complex numbers and their polar representation.

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). The real part of the complex number is a, and the imaginary part is b. Complex numbers can be represented graphically on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This representation allows us to visualize complex numbers and their operations geometrically. Complex numbers have a rich algebraic structure, supporting operations like addition, subtraction, multiplication, and division, each with unique properties and interpretations. For example, adding complex numbers involves adding their real and imaginary parts separately, while multiplication involves applying the distributive property and simplifying using i² = -1.

The polar form of a complex number offers an alternative way to represent complex numbers, using magnitude and argument instead of real and imaginary parts. A complex number z = a + bi can be written in polar form as z = r(cos θ + i sin θ), where r is the magnitude (or modulus) of z, and θ is the argument (or angle) of z. The magnitude r is the distance from the origin to the point representing z in the complex plane, and it is calculated as r = √(a² + b²). The argument θ is the angle formed between the positive real axis and the line segment connecting the origin to the point representing z and can be found using trigonometric functions. Specifically, cos θ = a/r and sin θ = b/r. The polar form is particularly useful when performing multiplication and division of complex numbers, as it simplifies these operations significantly. When multiplying two complex numbers in polar form, we multiply their magnitudes and add their arguments. This property is a cornerstone of complex number arithmetic and is instrumental in solving various mathematical problems.

In the subsequent sections, we will apply these concepts to find the product of z and w, demonstrating the elegance and efficiency of using the polar form for complex number multiplication. By understanding and utilizing the properties of complex numbers and their polar representation, we can tackle more complex problems and gain a deeper appreciation for the mathematical structures that govern these numbers.

Solution

To find the product zw, we will use the property that the product of two complex numbers in polar form is obtained by multiplying their magnitudes and adding their arguments. Given the complex numbers $z=38\left(\cos \left(\frac{\pi}{8}\right)+i \sin \left(\frac{\pi}{8}\right)\right)$ and $w=2\left(\cos \left(\frac{\pi}{16}\right)+i \sin \left(\frac{\pi}{16}\right)\right)$, we can identify their magnitudes and arguments.

The magnitude of z is 38, and its argument is $\frac{\pi}{8}$. The magnitude of w is 2, and its argument is $\frac{\pi}{16}$. To find the product zw, we multiply the magnitudes and add the arguments:

$$zw = 38 \cdot 2 \left(\cos \left(\frac{\pi}{8} + \frac{\pi}{16}\right) + i \sin \left(\frac{\pi}{8} + \frac{\pi}{16}\right)\right)$$

First, let's multiply the magnitudes:

$$38 \cdot 2 = 76$$

Next, let's add the arguments:

$$\frac{\pi}{8} + \frac{\pi}{16} = \frac{2\pi}{16} + \frac{\pi}{16} = \frac{3\pi}{16}$$

Now, we can substitute these results back into the expression for zw:

$$zw = 76\left(\cos \left(\frac{3\pi}{16}\right) + i \sin \left(\frac{3\pi}{16}\right)\right)$$

Thus, the product of z and w is $76\left(\cos \left(\frac{3\pi}{16}\right) + i \sin \left(\frac{3\pi}{16}\right)\right)$. This result demonstrates how multiplying complex numbers in polar form simplifies the process, as it avoids the need to expand and combine terms using the distributive property. The magnitude of the product is the product of the magnitudes, and the argument of the product is the sum of the arguments. This property is a cornerstone of complex number arithmetic and is essential for solving various problems involving complex numbers.

In summary, we found the product of the complex numbers z and w by multiplying their magnitudes and adding their arguments. This approach is highly efficient and underscores the utility of the polar representation of complex numbers. The final result, $76\left(\cos \left(\frac{3\pi}{16}\right) + i \sin \left(\frac{3\pi}{16}\right)\right)$, clearly illustrates the outcome of this multiplication. Understanding these principles is crucial for more advanced topics in complex analysis and related fields.

Alternative Approaches

While the polar form provides the most direct method for multiplying complex numbers in this format, it's worth discussing alternative approaches and the reasons why they might be less efficient in this particular case. One such approach involves converting the complex numbers from polar form to rectangular form (i.e., a + bi), performing the multiplication, and then converting the result back to polar form if needed. This method, though valid, is generally more cumbersome and prone to errors, especially when dealing with angles that do not have simple trigonometric values.

To convert z from polar form to rectangular form, we use the identities a = r cos θ and b = r sin θ. For z, r = 38 and θ = π/8. Thus,

$$a_z = 38 \cos \left(\frac{\pi}{8}\right)$$

$$b_z = 38 \sin \left(\frac{\pi}{8}\right)$$

Similarly, for w, r = 2 and θ = π/16, so

$$a_w = 2 \cos \left(\frac{\pi}{16}\right)$$

$$b_w = 2 \sin \left(\frac{\pi}{16}\right)$$

Multiplying the complex numbers in rectangular form involves multiplying (a_z + ib_z) and (a_w + ib_w), which gives:

$$(a_z + ib_z)(a_w + ib_w) = (a_z a_w - b_z b_w) + i(a_z b_w + b_z a_w)$$

Substituting the values we calculated earlier:

$$(38 \cos \left(\frac{\pi}{8}\right) + i 38 \sin \left(\frac{\pi}{8}\right))(2 \cos \left(\frac{\pi}{16}\right) + i 2 \sin \left(\frac{\pi}{16}\right))$$

This expression would require us to compute the values of cos(π/8), sin(π/8), cos(π/16), and sin(π/16), which are not standard angles and would likely involve approximations or the use of trigonometric identities to simplify. The arithmetic would be significantly more involved than simply adding the arguments in polar form.

After performing the multiplication in rectangular form, we would then need to convert the result back to polar form, which requires finding the magnitude and argument of the resulting complex number. This involves calculating the square root of the sum of squares of the real and imaginary parts, and then finding the arctangent of the ratio of the imaginary part to the real part. This process adds further complexity and computational effort.

Another approach might involve using Euler's formula, which states that e^iθ = cos θ + i sin θ. We can rewrite z and w using Euler's formula:

$$z = 38 e^{i\frac{\pi}{8}}$$

$$w = 2 e^{i\frac{\pi}{16}}$$

Then, the product zw is:

$$zw = 38 e^{i\frac{\pi}{8}} \cdot 2 e^{i\frac{\pi}{16}} = 76 e^{i(\frac{\pi}{8} + \frac{\pi}{16})} = 76 e^{i\frac{3\pi}{16}}$$

Using Euler's formula simplifies the multiplication process to some extent, as it directly gives us the magnitude and argument of the product. However, we still need to convert the result back to the standard polar form, which involves recognizing that e^i(3π/16) = cos(3π/16) + i sin(3π/16). While this approach is more streamlined than the rectangular form method, it still involves an extra step compared to the direct multiplication in polar form.

In conclusion, while alternative approaches such as converting to rectangular form or using Euler's formula are valid, they are generally less efficient for this particular problem. The direct multiplication in polar form, by multiplying magnitudes and adding arguments, provides the most straightforward and computationally efficient solution. This highlights the importance of choosing the right method for a given problem, based on the specific form of the complex numbers and the operations involved.

Conclusion

In this comprehensive exploration, we have successfully determined the product of two complex numbers, $z=38\left(\cos \left(\frac{\pi}{8}\right)+i \sin \left(\frac{\pi}{8}\right)\right)$ and $w=2\left(\cos \left(\frac{\pi}{16}\right)+i \sin \left(\frac{\pi}{16}\right)\right)$, by leveraging the properties of complex numbers in their polar form. Our analysis revealed that the product zw is given by $76\left(\cos \left(\frac{3\pi}{16}\right) + i \sin \left(\frac{3\pi}{16}\right)\right)$. This solution not only provides a concrete answer to the problem but also underscores the elegance and efficiency of using the polar representation for complex number multiplication.

We began by introducing the fundamental concepts of complex numbers, including their representation in both rectangular (a + bi) and polar (r(cos θ + i sin θ)) forms. We highlighted the significance of the magnitude r and the argument θ in the polar representation, and how these components offer a geometric interpretation of complex numbers on the complex plane. Understanding these foundational concepts is crucial for grasping the operations involving complex numbers and their applications in various fields.

Our detailed solution demonstrated the key principle that multiplying complex numbers in polar form involves multiplying their magnitudes and adding their arguments. This property significantly simplifies the multiplication process, allowing us to bypass the more cumbersome algebraic manipulations required in the rectangular form. By applying this principle, we efficiently computed the product zw, thereby showcasing the utility of the polar form in complex number arithmetic. This method is not only efficient but also provides a clear and intuitive understanding of the geometric effect of complex number multiplication, where magnitudes are scaled and angles are combined.

Furthermore, we explored alternative approaches to solving this problem, such as converting the complex numbers to rectangular form, performing the multiplication, and then converting back to polar form. While this method is valid, it involves more complex calculations and is generally less efficient, particularly when dealing with angles that do not have simple trigonometric values. We also discussed the use of Euler's formula (e^iθ = cos θ + i sin θ) as another alternative, which simplifies the multiplication to some extent but still requires additional steps to convert the result back to the standard polar form. By comparing these methods, we reinforced the understanding that the most appropriate approach depends on the specific form of the complex numbers and the operations involved. The direct multiplication in polar form remains the most straightforward and computationally efficient method for this type of problem.

In conclusion, this exploration has provided a thorough understanding of how to multiply complex numbers in polar form, highlighting the efficiency and elegance of this method. The result, $76\left(\cos \left(\frac{3\pi}{16}\right) + i \sin \left(\frac{3\pi}{16}\right)\right)$, serves as a testament to the power of complex number properties and their applications in mathematical problem-solving. This understanding is essential for more advanced topics in mathematics, physics, and engineering, where complex numbers play a crucial role. By mastering these fundamental concepts, students and practitioners can confidently tackle more complex problems and appreciate the depth and beauty of complex number theory.