数字电路ch1 number systems

The value of a decimal number is the sum of the digits after each digit has been multiplied by its weight.
EXAMPLE
1 102 4 101 3 100 143.75 = 7 10 5 10
Weight The position of each digit in a decimal number indicates the magnitude of the quantity represented and can be assigned a weight .The weights for whole numbers are positive powers of ten that increase from right to left, beginning with 10 0= 1. . . . 105 104 103 102 101 100 For fractional numbers, the weights are negative powers of ten that decrease from left to right beginning with 10-1.
Chapter 1:
Number Systems
DIGITAL SYSTEM APPLICATION
OBJECTIVES
After completing this chapter, you should be able to: Count in binary, octal, hexadecimal, and binary-coded decimal (BCD). Convert from decimal to binary and binary to decimal. Convert from binary to octal and octal to binary. Convert from binary to hexadecimal and hexadecimal to binary.
1 2
1.1 BINARY NUMBER SYSTEM
The binary number system is simply another way to represent quantities. The binary system is less complicated than the decimal system because it has only two digits. It may seem more difficult at first because it is unfamiliar to you. The decimal system with its ten digits is a base-ten system; the binary system with its two digits is a base-two system. The two binary digits (bits) are 1 and 0. The position of a 1 or 0 in a binary number indicates its weight, or value within the number, just as the position of a decimal digit determines the value of that digit. The weights in a binary number are based on powers of two.
There is a 2 in 3. Place a 1 in the 2's column and subtract 2 from 3 to see what remains.
Place a 1 in the 1 's column and subtract 1 from 1 to see what remains.
1.2 BINARY TO DECIMAL CONVERSION
The decimal value of any binary number can be found by adding the weights of all bits that are 1 and discarding the weights of all bits that are 0.
101102=2210
or
10110B=22D
EXAMPLE
1.Convert the binary whole number 1101101 to decimal. Solution Determine the weight of each bit that is a 1, and then find the weights to get the decimal number.
In the decimal number system each of the ten digits, 0 through 9, represents a certain quantity. As you know, the ten symbols (digits) do not limit you to expressing only ten different quantities because you use the various digits in appropriate positions within a number to indicate the magnitude of the quantity.
Counting In Binary
Bit The word bit is a contraction of the words binary digit. Each place in a binary number is called a bit. The binary number 10110 is a 5bit binary number. The first place on the right is called the least significant bit (LSB) and the left-most place is called the most significant bit (MSB).
32
16
8
4
2
1
There are no 32's in 23, but there is a 16. Place a 1 in the 16's column, and subtract 16 from 23 to see how much is left to convhts in 7, but there is a 4. Place a 0 in the 8's column and a 1 in the 4's place and subtract 4 from 7 to see what remains.
2310=101112
EXAMPLE
11011012=10910
2. Convert the fractional binary number 0.1011 to decimal. Solution Determine the weight of each bit that is a 1, and then sum the weights to get the decimal fraction.
Using three bits we can count in binary to 111 or 7. Including 000, we have eight different combinations. In general, with N bits we can count up to 2N- 1 for a total of 2N different numbers.
Convert from decimal to BCD and BCD to decimal.
Add and subtract binary numbers.
KEY TERMS
anode binary Binary-Coded Decimal (BCD) bit carry-out cathode hexadecimal light-emitting diode (LED)
carry-in
octal
1.0 DECIMAL NUMBERS
You are familiar with the decimal number system because you use decimal numbers every day. Although decimal numbers are commonplace, their weighted structure is often not understood. In this section, the structure of decimal numbers is reviewed. This review will help you more easily understand the structure of the binary number system, which is important in computers and digital electronics.
EXAMPLE
1. How high can you count using a 4-bit number?
Solution With N =4, we can count up to 24- 1 = 15. 2. How many different numbers can be represented with six bits? Solution With N=6, there are 2N combinations, 26 = 64.