modulus of complex number

简明释义

复数模

英英释义

例句

1.When solving complex equations, it's essential to consider the modulus of complex number 复数的模 to understand their geometric interpretation.

在解复杂方程时，考虑modulus of complex number 复数的模对于理解它们的几何解释是至关重要的。

2.The modulus of complex number 复数的模 can be computed using the formula √(a² + b²) for a complex number a + bi.

对于复数 a + bi，modulus of complex number 复数的模可以通过公式 √(a² + b²) 计算。

3.In electrical engineering, the modulus of complex number 复数的模 represents the amplitude of an AC signal.

在电气工程中，modulus of complex number 复数的模表示交流信号的幅度。

4.In polar coordinates, the modulus of complex number 复数的模 corresponds to the radius from the origin.

在极坐标中，modulus of complex number 复数的模对应于从原点的半径。

5.To find the distance from the origin to a point in the complex plane, we need to calculate the modulus of complex number 复数的模.

要找出复平面中某点到原点的距离，我们需要计算该点的modulus of complex number 复数的模。

作文

In the realm of mathematics, particularly in complex analysis, one of the fundamental concepts is the modulus of a complex number. A complex number is typically expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The modulus of a complex number, denoted as |z|, is a measure of the distance of the point represented by that complex number from the origin in the complex plane. This distance can be calculated using the formula |z| = √(a² + b²). Understanding this concept is crucial for various applications in engineering, physics, and computer science.The importance of the modulus of a complex number extends beyond mere theoretical interest. In electrical engineering, for example, complex numbers are used to analyze alternating current (AC) circuits. The modulus of a complex number helps engineers understand the amplitude of voltage or current in these systems. When dealing with phasors, which are complex representations of sinusoidal functions, the modulus of a complex number indicates the peak value of the waveform.Moreover, in signal processing, the modulus of a complex number plays a significant role in transforming signals from time domain to frequency domain using techniques like the Fourier Transform. The modulus of a complex number provides information about the magnitude of different frequency components in a signal, which is essential for filtering, compression, and other signal manipulation tasks.In mathematics, the modulus of a complex number also has geometric interpretations. When plotted on the complex plane, the modulus of a complex number corresponds to the radius of the circle formed by the complex number with respect to the origin. This geometric perspective aids in visualizing operations involving complex numbers, such as multiplication and division, which can be interpreted as transformations of these circles.Furthermore, the concept of the modulus of a complex number is not limited to basic calculations. It leads to more advanced topics such as the Argand diagram, polar coordinates, and even complex functions. For instance, converting a complex number from its rectangular form (a + bi) to its polar form involves calculating its modulus of a complex number and its argument (the angle formed with the positive real axis). This conversion is particularly useful when performing multiplication or division of complex numbers, as it simplifies the process significantly.In conclusion, the modulus of a complex number is a pivotal concept in mathematics and its applications. Its ability to represent the magnitude of complex numbers provides essential insights into various fields, including engineering, physics, and computer science. By understanding the modulus of a complex number, we gain a deeper appreciation for the complexities of both theoretical and applied mathematics. As we continue to explore the vast world of complex numbers, the modulus of a complex number will remain a key element in our mathematical toolkit, facilitating our understanding and application of this intriguing area of study.

在数学领域，尤其是复分析中，一个基本概念是复数的模。复数通常以形式a + bi表示，其中'a'代表实部，'b'代表虚部。复数的模，记作|z|，是表示该复数所代表的点与复平面原点之间距离的度量。这个距离可以通过公式|z| = √(a² + b²)来计算。理解这一概念对于工程、物理和计算机科学等各种应用至关重要。复数的模的重要性不仅仅限于理论兴趣。在电气工程中，例如，复数用于分析交流（AC）电路。复数的模帮助工程师理解这些系统中电压或电流的幅度。在处理相量时，相量是正弦函数的复数表示，复数的模表示波形的峰值。此外，在信号处理中，复数的模在使用傅里叶变换等技术将信号从时域转换到频域时发挥着重要作用。复数的模提供了信号中不同频率分量的幅度信息，这对于滤波、压缩和其他信号操作任务至关重要。在数学中，复数的模也有几何解释。当在复平面上绘制时，复数的模对应于该复数与原点之间形成的圆的半径。这种几何视角有助于可视化涉及复数的运算，如乘法和除法，这可以被解释为这些圆的变换。此外，复数的模的概念并不限于基本计算。它引出了更高级的话题，如阿根图、极坐标甚至复函数。例如，将复数从矩形形式（a + bi）转换为极坐标形式涉及计算其复数的模和其幅角（与正实轴形成的角度）。这种转换在执行复数的乘法或除法时特别有用，因为它显著简化了过程。总之，复数的模是数学及其应用中的关键概念。它表示复数的幅度，提供了对工程、物理和计算机科学等多个领域的重要见解。通过理解复数的模，我们对理论和应用数学的复杂性有了更深入的认识。随着我们继续探索复数的广阔世界，复数的模将始终是我们数学工具箱中的一个关键元素，促进我们对这一引人入胜的研究领域的理解和应用。