Using 0 s and 1 s to Represent Stuff


 Ronald Underwood
 1 years ago
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1 Using 0 s and 1 s to Represent Stuff First, how many things can we represent? In one bit, two things. In two bits, four things. In three bits,? How many things do we want to represent? For numbers, at least a couple billion. How many bits? billion. 1
2 Using 0 s and 1 s to Represent Other Things Not just numbers. Colors. Symbols. How many symbols do we want to represent? What about just the decimal symbols 0, 1, 2,..., 9? How many bits does it take? 2
3 But we have some left over that is, in 4 bits, we can represent more symbols that just the decimal digits. How many things can we represent in the number of bits required to represent all the decimal digits? 3
4 4 bits is a convenient grouping. Each group of 4 bits corresponds to a hexadecimal symbol ( digit ) A F... 4
5 Hexadecimal A (10) 1011 B (11) 1100 C (12) 1101 D (13) 1110 E (14) 1111 F (15) 5
6 Hexadecimal is sometimes a useful notation. It corresponds directly to binary, but it doesn t take up as many spaces to write (2) = 2B (16) Note: = = = 2 B (Of course in the computer, it takes up exactly as much space.) What is A3 (16) in binary? (Take them in groups of 4.) What is (2) in hexadecimal? 6
7 Back to the question, What other types of things might we want to represent? How many different things of that type of things are there? 7
8 Characters letters (upper and lower case), digits, symbols,... Just English? Let s start with that. Over 100? Over 200? How many bits? 8
9 In order to use bytes to represent characters, what do we need? 9
10 We need a translation scheme. A table of correspondences A B a b 10
11 ASCII American Standard Code for Information Interchange Encodes 128 specified characters. Based on the English alphabet. Letters az and AZ. The numbers 09. Some basic punctuation symbols. Some control codes. A blank space. How many bits? 11
12 A grouping of 8 bits is convenient. It will allow us to represent all of the common symbols; ASCII fits this just fine, with one extra bit. Other encoding schemes Unicode. A grouping of 8 bits is called a byte. A grouping of 4 bits, that is, a hexadecimal digit, is sometimes called a nibble. Unicode based on single bytes is called UTF8. More bits are necessary for more symbols. Unicode based on double bytes is called UTF
13 Hexadecimal (hex) is nice because there is a simple correspondence to binary. A byte is two hex digits. Simple ASCII code uses 7 bits, so can be used in a byte with one bit left over. Just make the first bit (on the left) of each byte 0, and we can represent the ASCII code simply in one byte. In ASCII, A and a. Add a 0 at the beginning and bytes represent the same things: A and a. Now, if humans are going to look at this kind of coding, hex is much better (2) = 41 (16) (just write the binary in groups) 13
14 Here are some ASCII codes. Fill in the missing representations: Symbol binary hex decimal A B? 42? a ?? b??? ?? 2??? The computer, of course, only uses binary; 0 s and 1 s. 14
15 Since standard ASCII uses only 7 bits, if we use the 8 bits in a byte, we could extend the ASCII coding. For example, é is represented by 82 in hex. What s the binary? ñ is represented by A4 in hex. What s the binary? 15
16 Storage and Transmission Storage of data takes space. How much? Transmitting data from one device to another takes time. How much? 16
17 Size of a Set of Data We measure the size of a set of data in terms of the number of bits or bytes required to represent the data. This depends on how the data are represented of course. Exactly how we represent any data depends on the range of values that the data could assume. We can often come up with a special, compact representation if we know that the data values will always be restricted to some small set of values. The ASCII standard coding was defined to represent 128 different symbols. Although that would require only 7 bits, ASCII codes are usually put into 8 bits or 1 byte. 17
18 Size of ASCII Files Using one byte for each character, a document with 2,000 characters would require 2,000 bytes if we do not need to distinguish fonts; that is, if only need to distinguish A from A or A. If we need to distinguish fonts, the ASCII encoding scheme is not sufficient, and a scheme to allow us to represent more differences may require more than one byte per character. On the other hand, there may be other ways of encoding the data that are more efficient in terms of numbers of bits needed. 18
19 Size of Computer Files Whatever coding scheme is used, even small computer files make require over a thousand bytes, so we often specify their size in kilobytes, which we abbreviate as KB. This is a common unit of measure for the size of a computer file. A short memo or letter in a common storage scheme for text may use 20 or 30 KB, and a lowresolution image, such as a photograph, in a common storage scheme for photographs may use 200 to 300 KB. 19
20 Size of Computer Files Many files take up several million bytes, so a more common unit of measurement is a megabyte, that is, 1,000,000 bytes, denoted MB. A mediumresolution image, such as a photograph, may use 3 to 5 MB. That is also the approximate size of common songs in a standard format for music. A 500 page science textbook also uses a similar number of bytes. The total size of several files are often measured in gigabytes, that is, 1,000,000,000 bytes, denoted GB, or in terabytes, that is, 1,000,000,000,000 bytes, denoted TB. 20
21 Storage Media Media on which computer files may be stored, such as a computer s hard drive, a portable comunication device such as a phone, a flash drive, or a memory card for a camera, are rated in terms of how many bytes can be stored on the medium. The sizes of useful storage devices must be quite large, so most sizes are measured in GB or TB. 21
22 Storage and Transmission Another issue with the size of files is how long it takes to transfer them from on computer to another. One common kind of transfer is over a network. Networks vary greatly in speed. Common measures of the speed of a link within a network is in bits per second, bps, or bytes per second, Bps. The speed of transmission over a link is sometimes called bandwidth. Bandwidth can refer to the maximum rate of transfer of a given network, or it can refer to the actual transfer rate in a given transmission. 22
23 How Fast Does a Computer Compute? The most common measure of the speed of a computer is how many simple binary operations (addition or multiplication) the computer performs on floatingpoint numbers per second. flops = floatingpoint operations per second We may be interested in the maximum flops a computer can attain, on for a given sequence of computations, how many flops the computer averaged. The average PC nowadays in theory can compute at a rate of between 1 and 7 gigaflops (10 9 flops). The fastest computer in the world can compute at a rate of about 40 petaflops ( gigaflops or flops). 23
24 Other Measures of Speed Another measure of the speed of a computer is instructions per second, usually measured in millions, or mips. Within a given class of processors, the frequency of the crystal or the clock speed is a relative measure of the speed of the processor. The processors in PCs nowadays run at a clock speed of a few million hertz. 24
25 Notation pico p nano n 10 9 micro µ 10 6 milli m 10 3 kilo K 10 3 mega M 10 6 giga G 10 9 tera T peta P exa E Notice that it is common to use uppercase for order prefixes greater than 1, for example K for kilo, M for mega, and so on, and lowercase for order prefixes less than 1, for example c for centi, m for milli, and so on. (Not everyone does this, of course.) Notice also that usually we use lowercase for bits, b, and uppercase for bytes, B. 25
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