Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版:英文原文完整保留,中文紧随对应单元;术语采用 v20260916 确认表,自动语义审校结果可追溯。

Basic Combinations

基本组合

Since the raising of a number n to a power p may be defined as multiplying n times itself p times, it follows that

由于将一个数n提升到幂p可以定义为将n乘以自身p次,因此可以得出

The rule for raising a power to a power can also be deduced

幂的幂法则也可以推导出来
Index
索引
 
HyperPhysics*****HyperMath *****Exponents
HyperPhysics ***** HyperMath ***** 指数
R Nave
Go Back
返回





Powers of Ten

十的幂

For very large or very small numbers, it convenient to use the "scientific notation", i.e., to express the numbers in terms of a power of 10. Standard prefixes are used to denote certain powers of 10:

对于非常大或非常小的数,使用“科学记数法”会更方便,即用10的幂来表示这些数。常用的前缀用于表示特定的10的幂:
Index
索引
 
HyperPhysics*****HyperMath *****Exponents
HyperPhysics ***** HyperMath ***** 指数
R Nave
Go Back
返回





Radical Combinations

自由基组合
Index
索引
 
HyperPhysics*****HyperMath *****Exponents
HyperPhysics ***** HyperMath ***** 指数
R Nave
Go Back
返回





Euler Relationship

欧拉关系

The trigonometric functions are related to a complex exponential by the Euler relationship

三角函数与复指数之间存在欧拉关系

From these relationships the trig functions can be expressed in terms of the complex exponential:

从这些关系可以将三角函数用复指数表示为:
Index
索引
 
HyperPhysics*****HyperMath *****Exponents
HyperPhysics ***** HyperMath ***** 指数
R Nave
Go Back
返回





Hyperbolic Functions

双曲函数

The hyperbolic functions can be defined in terms of exponentials.

双曲函数可以用指数函数来定义。

There are closed-form expressions for the inverse hyperbolic functions:

反双曲函数有闭式表达式:

An application of the hyperbolic tangent is in calculating the speed of ocean waves. The hyperbolic cosine and its inverse are encountered in fluid friction in the case of quadratic drag.

双曲正切函数的应用之一是计算海洋波速。在二次阻力情况下,双曲余弦及其反函数会出现在流体摩擦中。
Index
索引
 
HyperPhysics*****HyperMath *****Exponents
HyperPhysics ***** HyperMath ***** 指数
R Nave
Go Back
返回