Digital-electronics 简明教程

Digital Electronics - Number Systems

digital number system 是一个位置数字系统，具有一些称为数字的符号。它提供了一组完整的数字、运算符和规则来执行操作。

A digital number system is a positional number system that has some symbols called digits. It provides a complete set of digits, operators, and rules to perform operations.

在数字系统中，所用数字的数量决定了数字系统的基数。例如，二进制数字系统有两个数字（0 和 1），因此二进制数字系统的基数为 2。

In a digital number system, the number of digits used determines the base of the number system. For example, the binary number system has two digits (0 and 1), hence, the base of the binary number system is 2.

数字系统构成了现代计算技术和数字电子的基础。它们用于使用数字系统来表示、处理和操纵信息。

Digital number systems form the foundation of the modern computing technologies and digital electronics. They are used to represent, process, and manipulate the information using a digital system.

在本章中，我们将讨论不同类型数字系统的基本概念。

In this chapter, we will discuss the fundamental concepts of different types of digital number systems.

Types of Digital Number Systems

在数字电子学中，主要使用以下四种类型的数字系统 -

In digital electronics, the following four types of digital number systems are mainly used −

  1. Binary Number System

  2. Decimal Number System

  3. Octal Number System

  4. Hexadecimal Number System

让我们详细讨论这些数字系统中的每一个。

Let’s discuss each of these number systems in detail.

Binary Number System

二进制数字系统是所有数字系统实现和工作背后的基本构建块。

Binary number system is the fundamental building block behind the implementation and working of all digital systems.

二进制数字系统有两种符号或数字，即 0 和 1。因此，这两个数字用于表示信息并执行所有数字运算。每一位二进制数字称为位。

Binary number system has two symbols or digits, i.e., 0 and 1. Hence, these two digits are used to represent information and perform all the digital operations. Each binary digit is called a bit.

由于二进制数字系统中使用了两个数字，因此其基数为 2。因此，二进制数的值计算为 2 的幂的和。

Since there are two digits are used in the binary number system, hence its base is 2. Therefore, the value of a binary number is calculated as the sum of powers of 2.

二进制数字用于数字系统中表示其开和关状态。其中，0 用于表示数字系统的关状态，1 用于表示系统的开状态。

Binary digits are used in digital system to represent their ON and OFF states. Where, 0 is used to represent the OFF state of the digital system and 1 is used to represent the ON state of the system.

总体而言，二进制数字系统构成了计算、数字通信和数字信息存储的基础。

Overall, the binary number system forms the foundation of computation, digital communication, and digital information storage.

Example

考虑 binary number 1101.011 。此数的整数部分为 1101，分数部分为 0.011。整数部分的数字 1、0、1 和 1 的权重分别为 20、21、22、23。类似地，分数部分的数字 0、1 和 1 的权重分别为 2-1、2-2、2-3。

Consider the binary number 1101.011. The integer part of this number is 1101 and the fractional part of this number is 0.011. The digits 1, 0, 1 and 1 of the integer part have weights of 20, 21, 22, 23 respectively. Similarly, the digits 0, 1 and 1 of fractional part have weights of 2-1, 2-2, 2-3 respectively.

Mathematically ，我们可以写成，

Mathematically, we can write it as,

\mathrm{1101.011 \: = \: (1 \: \times \: 2^{3}) \: + \:(1 \: \times \: 2^{2}) \: + \: (0 \: \times \: 2^{1}) \: + \: (1 \: \times \: 2^{0}) \: + \: (0 \: \times \: 2^{−1}) \: + \: (1 \: \times \: 2^{−2}) \: + \: (1 \: \times \: 2^{−3})}

简化右侧项后，我们将得到一个十进制数，它相当于左侧的二进制数。

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of binary number on left-hand side.

Decimal Number System

十进制数字系统本质上不是一个数字系统。但它被广泛用于以人类可读格式表示数字信息。

Decimal number system is not inherently a digital number system. But it is widely used to represent the digital information in a human readable format.

十进制数字系统是一个基数为 10 的数字系统，有 10 个唯一数字，即 0、1、2、3、4、5、6、7、8 和 9。这是人类使用的一种自然方式来表示信息的标准数字系统。但是，数字系统无法直接处理以十进制形式表示的信息，因此它被转换为二进制形式然后处理。

Decimal number system is a base 10 number system having 10 unique digits i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the standard number system used by human beings to represent information in a natural way. However, a digital system cannot directly process the information represented in decimal form, so it is converted into binary form and then processed.

十进制数字系统的基数为 10。因此，十进制数的值由 10 的幂的和计算得出。

The base of the decimal number system is 10. So, the value of a decimal number is calculated by the sum of powers of 10.

Example

考虑 decimal number 1358.246 。此数的整数部分为 1358，分数部分为 0.246。数字 8、5、3 和 1 的权重分别为 (10)0、(10)1、(10)2 和 (10)3。类似地，数字 2、4 和 6 的权重分别为 (10)-1、(10)-2 和 (10)-3。

Consider the decimal number 1358.246. The integer part of this number is 1358 and the fractional part of this number is 0.246. The digits 8, 5, 3 and 1 have weights of (10)0, (10)1, (10)2 and (10)3 respectively. Similarly, the digits 2, 4 and 6 have weights of (10)-1, (10)-2 and (10)-3 respectively.

Mathematically ，我们可以写成，

Mathematically, we can write it as,

\mathrm{1358.246 \: = \: (1 \: \times \: 10^{3}) \: + \:(3 \: \times \: 10^{2}) \: + \: (5 \: \times \: 10^{1}) \: + \: (8 \: \times \: 10^{0}) \: + \: (2 \: \times \: 10^{−1}) \: + \: (4 \: \times \: 10^{−2}) \: + \: (6 \: \times \: 10^{−3})}

简化右侧项后，我们将得到十进制数，它在左侧。

After simplifying the right-hand side terms, we will get the decimal number, which is on the left-hand side.

Octal Number System

八进制数字系统是另一种数字系统类型，用于数字电子领域来表示信息。它是一个基数为 8 的数字系统，有八个唯一数字，即 0、1、2、3、4、5、6 和 7。

The octal number system is another type of digital number system used in the field of digital electronics to represent information. It is a base 8 number system having eight unique digits i.e., 0, 1, 2, 3, 4, 5, 6, and 7.

需要注意的是，八进制数字系统等效于 3 位二进制数字系统，因为 23 = 8。因此，此数字系统可用于计算和数字电子应用。

It is important note that the octal number system is equivalent to 3-bit binary number system as 23 = 8. Hence, this number system can be used in computing and digital electronic applications.

八进制数的值由 8 的幂的和得出，因为 8 是八进制数字系统的基数。

The value of an octal number is obtained by the sum of powers of 8, as 8 is the base of the octal number system.

八进制数字系统在数字电子领域用于表示紧凑形式的二进制信息、Linux 或 Unix 系统中的权限、IPv6 地址、二进制机器代码指令、错误检测算法中，等等。

Octal number system is used in the field of digital electronics to represent binary information in compact form, permissions in Linux or Unix systems, IPv6 address, binary machine code instructions, in error detection algorithms, etc.

Example

考虑 octal number 1457.236 。此数的整数部分为 1457，分数部分为 0.236。数字 7、5、4 和 1 的权重分别为 (8)0、(8)1、(8)2 和 (8)3。类似地，数字 2、3 和 6 的权重分别为 (8)-1、(8)-2、(8)-3。

Consider the octal number 1457.236. Integer part of this number is 1457 and fractional part of this number is 0.236. The digits 7, 5, 4 and 1 have weights of (8)0, (8)1, (8)2 and (8)3 respectively. Similarly, the digits 2, 3 and 6 have weights of (8)-1, (8)-2, (8)-3 respectively.

Mathematically ，我们可以写成，

Mathematically, we can write it as,

1457.236 = (1 × 8^3) + (4 × 8^2) + (5 × 8^1) + (7 × 8^0) + (2 × 8^-1) + (3 × 8^-2) + (6 × 8^-3)

\mathrm{1457.236 \: = \: (1 \: \times \: 8^{3}) \: + \:(4 \: \times \: 8^{2}) \: + \: (5 \: \times \: 8^{1}) \: + \: (7 \: \times \: 8^{0}) \: + \: (2 \: \times \: 8^{−1}) \: + \: (3 \: \times \: 8^{−2}) \: + \: (6 \: \times \: 8^{−3})}

化简右侧项后，我们将得到一个小数，它等价于左侧的八进制数。

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of octal number on the left-hand side.

Hexadecimal Number System

十六进制数系统是一个 16 进制数系统。它有 16 个数字，即 0 到 9 和 A 到 F。其中，A 表示 10，B 表示 11，C 表示 12，D 表示 13，E 表示 14，F 表示 15。十六进制数系统与 4 位二进制数系统等价，因为 2^4 = 16。因此，十六进制数的值可以通过 16 的幂次和计算。

The hexadecimal number system is a base 16 number system. It has 16 digits, 0 to 9 and A to F. Where, A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. The hexadecimal number system is equivalent to a 4-bit binary number system as 24 = 16. Thus, the value of a hexadecimal number can be calculated by the sum of powers of 16.

在数字电子领域，十六进制数系统用于内存地址表示、数字颜色表示、低级计算机编程、编码、汇编语言编程、微控制器、键盘等。十六进制数系统在数字表示和人类可读性之间创造了一种平衡。

In the field of digital electronics, the hexadecimal number system is used in memory address representation, digital colors representation, low level computer programming, encoding, assembly language programming, microcontrollers, keyboards, etc. Hexadecimal number system creates a balance between digital representation and human readability.

Example

考虑 hexadecimal number 1A05.2C4 。此数字的整数部分为 1A05，小数部分为 0.2C4。数字 5、0、A 和 1 的权重分别为 (16)0、(16)1、(16)^2 和 (16)^3。同样，数字 2、C 和 4 的权重分别为 (16)-1、(16)-2 和 (16)^-3。

Consider the hexadecimal number 1A05.2C4. The integer part of this number is 1A05 and the fractional part of this number is 0.2C4. The digits 5, 0, A and 1 have weights of (16)0, (16)1, (16)2 and (16)3 respectively. Similarly, the digits 2, C and 4 have weights of (16)-1 , (16)-2 and (16)-3 respectively.

Mathematically ，我们可以写成，

Mathematically, we can write it as,

1A05.2C4 = (1 × 16^3) + (10 × 16^2) + (0 × 16^1) + (5 × 16^0) + (2 × 16^-1) + (12 × 16^-2) + (4 × 16^-3)

\mathrm{1A05.2C4 \: = \: (1 \: \times \: 16^{3}) \: + \:(10 \: \times \: 16^{2}) \: + \: (0 \: \times \: 16^{1}) \: + \: (5 \: \times \: 16^{0}) \: + \: (2 \: \times \: 16^{−1}) \: + \: (12 \: \times \: 16^{−2}) \: + \: (4 \: \times \: 16^{−3})}

化简右侧项后，我们将得到一个小数，它等价于左侧的十六进制数。

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of the hexadecimal number on the left-hand side.

Advantages of Digital Number Systems

以下是数字数系统的一些主要优点：

The following are some key advantages of digital number systems −

  1. Digital number systems provide a simple and consistent way of representing and understanding information.

  2. Digital number systems allow to develop efficient methods for storage and transmission of digital information.

  3. Digital number systems provide methods of representing different types of information like text, numbers, images, etc.

  4. Digital number systems allow to convert information from one form to full fill the needs of applications.

  5. Digital number systems create compatibility between hardware and software.

Applications of Digital Number Systems

数字数系统用于各种数字电子领域，如计算、互联网、通信、信号处理等。以下是数字数系统应用的一些示例：

Digital number systems are used in various digital electronic fields such as computing, internet, communication, signal processing, and more. Here are a few examples of applications of digital number systems −

  1. Information Representation

  2. Digital Communication

  3. Storage and Transmission of Digital Data and Information

  4. Algorithm Development

  5. System Programming, etc.

Conclusion

在本章中，我们讨论了数字数系统的一些基本概念。了解数字数系统对于设计、实现和故障排除数字系统至关重要。数字数系统提供了在数字系统中表示和操作信息的不同方法。

In this chapter, we discussed the basic concepts of digital number systems. The understanding of digital number systems is essential for designing, implementing, and troubleshooting the digital systems. Digital number systems provide different methods of representing and manipulating information in digital systems.