Multiplying Complex Numbers: A Comprehensive Guide
In mathematics, particularly in the realm of complex analysis, the multiplication of complex numbers involves both arithmetic manipulation and trigonometric functions. This article aims to clearly explain the process of multiplying two complex numbers, providing a thorough understanding of the steps involved and the importance of trigonometric identities. This content is designed to satisfy the Google SEO standards, ensuring rich, informative content and optimal keyword density.
Introduction to Complex Numbers and Their Multiplication
Complex numbers are a fundamental part of algebra, represented in the form (a bi), where (a) and (b) are real numbers, and (i) is the imaginary unit satisfying (i^2 -1). Multiplying complex numbers involves using both the arithmetic multiplication of the real and imaginary parts, as well as the application of trigonometric identities. This process holds true for more complex forms of complex numbers, including those represented in trigonometric form.
Multiplication of Complex Numbers Involving Trigonometric Functions
The problem at hand involves multiplying two complex numbers represented in trigonometric form: [-sqrt{3}i quad text{and} quad frac{1}{8} left(cos frac{5pi}{2} i sin frac{5pi}{2}right)]
Let's break down the process of multiplication step by step:
Simplifying the Trigonometric Angles
The angle (frac{5pi}{2}) can be simplified by recognizing that angles are periodic with a period of (2pi). Therefore, (frac{5pi}{2} 2pi frac{pi}{2}). This simplification allows us to rewrite the expression as follows:
[cos frac{5pi}{2} cos left(2pi frac{pi}{2}right) cos frac{pi}{2} 0] [sin frac{5pi}{2} sin left(2pi frac{pi}{2}right) sin frac{pi}{2} 1]Thus, the expression simplifies to:
[frac{1}{8} left(cos frac{5pi}{2} i sin frac{5pi}{2}right) frac{1}{8} (0 i) frac{1}{8}i]Multiplying the Simplified Expressions
Now, we can multiply (-sqrt{3}i) by (frac{1}{8}i) as follows:
[-sqrt{3}i cdot frac{1}{8}i frac{1}{8} (-sqrt{3}i^2)]Recall that (i^2 -1), so the expression becomes:
[frac{1}{8} (-sqrt{3}(-1)) frac{1}{8} (sqrt{3}) frac{sqrt{3}}{8}]Hence, the result of the multiplication is:
[-frac{1}{8} - frac{sqrt{3}}{8}i]Additional Example: Multiplication Involving Euler’s Formula
Euler’s formula, (e^{itheta} cos theta i sin theta), is a powerful tool in complex number multiplication. Consider the multiplication of the following complex numbers:
[-sqrt{3}i quad text{and} quad frac{1}{8} left(e^{i frac{5pi}{3}}right)]The angle (frac{5pi}{3}) can be simplified using the periodicity of (2pi):
[frac{5pi}{3} 2pi - frac{pi}{3}]Thus, we can rewrite the expression using (cos) and (sin):
[e^{i frac{5pi}{3}} cos frac{5pi}{3} i sin frac{5pi}{3} cos left(2pi - frac{pi}{3}right) i sin left(2pi - frac{pi}{3}right) cos left(-frac{pi}{3}right) i sin left(-frac{pi}{3}right) frac{1}{2} - i frac{sqrt{3}}{2}]The multiplication becomes:
[-sqrt{3}i cdot frac{1}{8} left(frac{1}{2} - i frac{sqrt{3}}{2}right) frac{1}{16} - i frac{sqrt{3}}{16} frac{1}{16} (1 - i sqrt{3})]Therefore, the final result is:
[frac{1}{16} - i frac{sqrt{3}}{16}]Conclusion
Multiplying complex numbers, particularly those involving trigonometric functions, requires a clear understanding of both arithmetic and trigonometric principles. By simplifying angles and applying trigonometric identities, we can effectively multiply complex numbers. Euler’s formula provides a powerful method for simplifying such multiplications, making the process more intuitive and manageable.