modulus of complex number

简明释义

复数模数

英英释义

例句

1.The modulus of complex number 复数的模 helps us understand the stability of a system in control theory.

在控制理论中，modulus of complex number 复数的模帮助我们理解系统的稳定性。

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

要找到原点到复平面中某点的距离，我们计算该点的modulus of complex number 复数的模。

3.To solve this equation, we need to calculate the modulus of complex number 复数的模 first.

为了解这个方程，我们需要先计算modulus of complex number 复数的模。

4.The formula for the modulus of complex number 复数的模 is derived from the Pythagorean theorem.

计算modulus of complex number 复数的模的公式源于勾股定理。

5.In electrical engineering, the modulus of complex number 复数的模 is used to represent the amplitude of an alternating current.

在电气工程中，modulus of complex number 复数的模用于表示交流电的幅度。

作文

In the realm of mathematics, particularly in complex number theory, one of the fundamental concepts is the modulus of complex number. This term refers to the absolute value or magnitude of a complex number, which is represented in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. To understand this concept more thoroughly, it is essential to delve into the definition and applications of the modulus of complex number. The modulus of complex number is denoted as |z|, where z is the complex number. Mathematically, it can be calculated using the formula: |z| = √(a² + b²). This formula arises from the Pythagorean theorem, as the complex number can be visualized as a point (a, b) in a two-dimensional plane known as the complex plane. The distance from the origin (0, 0) to the point (a, b) represents the modulus of complex number, giving us a clear geometric interpretation. Understanding the modulus of complex number is crucial not only for theoretical mathematics but also for practical applications in engineering and physics. For instance, in electrical engineering, complex numbers are used to represent alternating current circuits. The modulus of complex number indicates the amplitude of the current or voltage, which is vital for analyzing circuit behavior. Moreover, the modulus of complex number plays a significant role in complex analysis, which is a branch of mathematics that studies functions of complex variables. In this context, the modulus helps in understanding the behavior of complex functions, especially when dealing with limits, continuity, and differentiability. The properties of the modulus of complex number also extend to inequalities, such as the triangle inequality, which states that for any two complex numbers z1 and z2, |z1 + z2| ≤ |z1| + |z2|. This property highlights the relationship between complex numbers and their geometrical interpretations. Furthermore, the modulus of complex number is essential in polar representation of complex numbers. A complex number can be expressed in polar form as r(cos θ + i sin θ), where r is the modulus of complex number and θ is the argument (or angle) of the complex number. This representation simplifies many operations, such as multiplication and division of complex numbers, making it easier to handle them in various mathematical contexts. In conclusion, the modulus of complex number is a fundamental concept in mathematics that bridges the gap between algebra and geometry. Its applications span across various fields, including engineering, physics, and advanced mathematics. By grasping the significance of the modulus of complex number, students and professionals alike can enhance their understanding of complex numbers and their myriad applications in the real world. Whether it is through theoretical exploration or practical application, the modulus of complex number remains an indispensable tool in the mathematician's toolkit. As we continue to explore the fascinating world of complex numbers, it becomes evident that the modulus of complex number is not merely a mathematical abstraction but a crucial element that underpins many scientific principles and technological advancements.

在数学的领域中，特别是在复数理论中，一个基本概念是复数的模。这个术语指的是复数的绝对值或大小，复数的形式为 a + bi，其中 'a' 是实部，'b' 是虚部。要更深入地理解这个概念，有必要探讨复数的模的定义和应用。 复数的模用 |z| 表示，其中 z 是复数。数学上，它可以使用公式计算：|z| = √(a² + b²)。这个公式源于毕达哥拉斯定理，因为复数可以被视为在一个称为复平面的二维平面中的点 (a, b)。从原点 (0, 0) 到点 (a, b) 的距离表示复数的模，这为我们提供了一个清晰的几何解释。 理解复数的模不仅对理论数学至关重要，而且对工程和物理等实际应用也同样重要。例如，在电气工程中，复数用于表示交流电路。复数的模表示电流或电压的幅度，这对于分析电路行为至关重要。 此外，复数的模在复分析中也发挥着重要作用，这是研究复变量函数的数学分支。在这种情况下，模帮助理解复函数的行为，尤其在处理极限、连续性和可微性时。复数的模的性质还扩展到不等式，例如三角不等式，指出对于任何两个复数 z1 和 z2，有 |z1 + z2| ≤ |z1| + |z2|。这个性质突显了复数及其几何解释之间的关系。 此外，复数的模在复数的极坐标表示中至关重要。复数可以表示为极坐标形式 r(cos θ + i sin θ)，其中 r 是复数的模，θ 是复数的幅角（或角度）。这种表示法简化了许多运算，例如复数的乘法和除法，使得在各种数学上下文中处理它们变得更加容易。 总之，复数的模是数学中的一个基本概念，它架起了代数与几何之间的桥梁。它的应用跨越多个领域，包括工程、物理和高等数学。通过掌握复数的模的重要性，学生和专业人士都可以增强对复数及其在现实世界中众多应用的理解。无论是通过理论探索还是实际应用，复数的模始终是数学家工具箱中不可或缺的工具。 随着我们继续探索复数的迷人世界，显而易见的是，复数的模不仅仅是一个数学抽象，而是支撑许多科学原理和技术进步的关键元素。