Trung tâm đào tạo thiết kế vi mạch Semicon


  • ĐĂNG KÝ TÀI KHOẢN ĐỂ TRUY CẬP NHIỀU TÀI LIỆU HƠN!
  • Đăng ký
    *
    *
    *
    *
    *
    Fields marked with an asterisk (*) are required.
semi4_volunteering.jpg

Digital Number System part I

Email In PDF.

Numbering System

Many number systems are in use in digital technology. The most common are the decimal, binary, octal, and hexadecimal systems.

 The decimal system is clearly the most familiar to us because it is a tool that we use every day. Examining some of its characteristics will help us to better understand the other systems. In the next few pages we shall introduce four numerical representation systems that are used in the digital system. There are other systems, which we will look at briefly.

  • Decimal
  • Binary
  • Octal
  • Hexadecimal

Decimal System

The decimal system is composed of 10 numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Using these symbols as digits of a number, we can express any quantity. The decimal system is also called the base-10 system because it has 10 digits.

 

103

102

101

100

 

10-1

10-2

10-3

=1000

=100

=10

=1

.

=0.1

=0.01

=0.001

Most Significant Digit

     

Decimal point

   

Least Significant Digit

Even though the decimal system has only 10 symbols, any number of any magnitude can be expressed by using our system of positional weighting.

Decimal Examples

  • 3.1410
  • 5210
  • 102410
  • 6400010

Binary System
In the binary system, there are only two symbols or possible digit values, 0 and 1. This base-2 system can be used to represent any quantity that can be represented in decimal or other base system.

 

23

22

21

20

 

2-1

2-2

2-3

=8

=4

=2

=1

.

=0.5

=0.25

=0.125

Most Significant Digit

     

Binary point

   

Least Significant Digit

Binary Counting

The Binary counting sequence is shown in the table:

23

22

21

20

Decimal

0

0

0

0

0

0

0

0

1

1

0

0

1

0

2

0

0

1

1

3

0

1

0

0

4

0

1

0

1

5

0

1

1

0

6

0

1

1

1

7

1

0

0

0

8

1

0

0

1

9

1

0

1

0

10

1

0

1

1

11

1

1

0

0

12

1

1

0

1

13

1

1

1

0

14

1

1

1

1

15

Representing Binary Quantities

presented in binary form. Binary quantities can be represented by any device that has only two operating states or possible conditions. E.g.. a switch is only open or closed. We arbitrarily (as we define them) let an open switch represent binary 0 and a closed switch represent binary 1. Thus we can represent any binary number by using series of switches.

Typical Voltage Assignment

Binary 1: Any voltage between 2V to 5V
Binary 0:
 Any voltage between 0V to 0.8V

Not used: Voltage between 0.8V to 2V in 5 Volt CMOS and TTL Logic, this may cause error in a digital circuit. Today's digital circuits works at 1.8 volts, so this statement may not hold true for all logic circuits.

We can see another significant difference between digital and analog systems. In digital systems, the exact voltage value is not important; eg, a voltage of 3.6V means the same as a voltage of 4.3V. In analog systems, the exact voltage value is important.

The binary number system is the most important one in digital systems, but several others are also important. The decimal system is important because it is universally used to represent quantities outside a digital system. This means that there will be situations where decimal values have to be converted to binary values before they are entered into the digital system.

In additional to binary and decimal, two other number systems find wide-spread applications in digital systems. The octal (base-8) and hexadecimal (base-16) number systems are both used for the same purpose- to provide an efficient means for representing large binary system.

Octal System

The octal number system has a base of eight, meaning that it has eight possible digits: 0,1,2,3,4,5,6,7.

83

82

81

80

 

8-1

8-2

8-3

=512

=64

=8

=1

.

=1/8

=1/64

=1/512

Most Significant Digit

     

Octal point

   

Least Significant Digit

Octal to Decimal Conversion

  • 2378 = 2 x (82) + 3 x (81) + 7 x (80) = 15910
  • 24.68 = 2 x (81) + 4 x (80) + 6 x (8-1) = 20.7510
  • 11.18 = 1 x (81) + 1 x (80) + 1 x (8-1) = 9.12510
  • 12.38 = 1 x (81) + 2 x (80) + 3 x (8-1) = 10.37510

Hexadecimal System

The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It uses the digits 0 through 9 plus the letters A, B, C, D, E, and F as the 16 digit symbols.

163

162

161

160

 

16-1

16-2

16-3

=4096

=256

=16

=1

.

=1/16

=1/256

=1/4096

Most Significant Digit

     

Hexa Decimal point

   

Least Significant Digit

Hexadecimal to Decimal Conversion

  • 24.616 = 2 x (161) + 4 x (160) + 6 x (16-1) = 36.37510
  • 11.116 = 1 x (161) + 1 x (160) + 1 x (16-1) = 17.062510
  • 12.316 = 1 x (161) + 2 x (160) + 3 x (16-1) = 18.187510

  Bạn Có Đam Mê Với Vi Mạch hay Nhúng      -     Bạn Muốn Trau Dồi Thêm Kĩ Năng

Mong Muốn Có Thêm Cơ Hội Trong Công Việc

Và Trở Thành Một Người Có Giá Trị Hơn

Bạn Chưa Biết Phương Thức Nào Nhanh Chóng Để Đạt Được Chúng

Hãy Để Chúng Tôi Hỗ Trợ Cho Bạn. SEMICON  

Lần cập nhật cuối ( Thứ ba, 29 Tháng 3 2022 00:16 )