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umber Systems
Inside today’s computers, data is represented as 1’s and 0’s. These 1’s and 0’s might be stored magnetically on a disk, or as a state in a transistor. To perform useful operations on these 1’s and 0’s we have to organize them together into patterns that make up codes. This lecture is about ways to represent data in the codes and to illustrate different number systems that are used today. This is how data is fundamentally stored in the computer.
Each 1 and 0 is called a bit. 4 bits is called a nibble. 8 bits is called a byte. 16/32/64 bits is called a word. The number depends upon the number of bits that can simultaneously be operated upon by the processor.
Since we only have 1’s and 0’s available, the number system we will use inside the computer is the binary number system. We are used to the decimal number system. In decimal, we have ten symbols (0-9) that we can use. When all of the symbols are used up, we need to combine together two symbols to represent the next number (10). The most significant digit becomes our first digit, and the least significant digit cycles back to
- The least significant digit increments until the symbols are used up, and which point the most significant digit is incremented to the next larger digit. This is just the familiar process of counting!
The same process applies to the binary number system, but we only have two symbols available (0 and 1). When we run out of symbols, we need to combine them and increment the digits just as we would in a decimal, base 10, number system: Let’s start counting to illustrate:
Decimal Binary 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001
Binary to Decimal Conversion
The binary number system is positional where each binary digit has a weight based upon its position relative to the least significant bit (LSB). To convert a binary number to decimal, sum together the weights. Each weight is calculated by multiplying the bit by 2 i, where i is the index relative to the LSB.
1 1 0 1 1 24 23 22 21 20 Weight = 12^0 + 12^1 + 02^2 + 12^3 + 12^4 = 11 + 12 + 04 + 18 + 116 = 1+2+8+16 = 27 (base 10, decimal)
What is 11111 in base 10? What is 10110101 in base 10?
Decimal to Binary Conversion
To convert a decimal number to its equivalent binary number representation, the easiest method is to use repeated division by 2. Each time, save the remainder. The first remainder is the least significant bit. The last remainder is the most significant bit.
What is 27 in binary?
27 / 2 = 13 remainder 1 LSB 13 / 2 = 6 remainder 1 6 / 2 = 3 remainder 0 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1 MSB
Our final number is 11011.
If using a calculator, watch to see if the quotient has a fraction. If it is 0.5 then the remainder was 1, and repeat the process with the whole part minus the fraction.
What is 83 in binary? What is 255 in binary?
Hexadecimal umbering
The hexadecimal numbering system is the most common system seen today in representing raw computer data. This is because it is very convenient to represent 8 bits, or one byte of data.
254 / 16 = 15 remainder 14 = E 15 / 16 = 0 remainder 15 = F So we get FE for 254.
Again, note that if a calculator is being used, you may multiple the fraction remainder by 16 to produce the remainder.
What is 423 in hex?
Hex to Binary Conversion, Binary to Hex
Going from hex to binary is similar to the process of converting from octal to binary. One must simply look up or compute the binary pattern of 4 bits for each hex code, and concatenate the codes together.
To convert FE to binary:
F = 1111 E = 1110 So FE in binary is 1111 1110
The same process applies in reverse by grouping together 4 bits at a time and then look up the hex digit for each group.
Binary 11000100101 broken up into groups of 4: 0110 0010 0101 (note the 0 added as padding on the MSB to get up to 4 bits) 6 2 5 = 625 Hex
Alphanumeric Codes
So far we have only been discussing the representation of numbers. We would also like to represent alphanumeric characters. This includes letters, numbers, punctuation marks, and other special characters like % or / that appear on a typical keyboard.
The most common code in the United States is ASCII, the American Standard Code for Information Interchange. ASCII is comprised of a 7 bit code. This means that we have 27 or 128 possible code groups. This is plenty for English as well as control functions such as the carriage return or linefeed functions.
Another code is EBCDIC, extended binary coded decimal interchange code. EBCDIC was used on some mainframe systems but is now out of favor.
Page 128 of your book has a complete listing of ASCII codes. A small sample is shown below.
Character ASCII Hex ASCII binary Integer Equivalent A 41 01000001 65 B 42 01000010 66 C 43 01000011 67 Z 5A 01011010 90
Note that the sequence of bits “01000001” could represent many things: it could be the number 65, or it might be the letter ‘A’. Or it could be something else entirely different! Ultimately it is up to the software in the computer to know what a sequence of bits represents and treat it accordingly.
Since most computers store data in bytes, or 8 bits, there is 1 bit that may go unused (or used for parity, described below).
Could we represent something like Chinese in ASCII, which has many more than 128 characters? The answer is we can’t, because there are way more than 128 characters in Chinese.
One way to represent languages such as Chinese is to use a different code. UNICODE is an international standard used to represent multiple languages. To address the shortcomings of ASCII, UNICODE uses 16 bits per character. This allows 2^16 or 65536 characters. When this is insufficient, there are extensions to support over a million characters. Most of the work in UNICODE has gone in efforts to define the codes for the particular languages. If you would like more information, visit http://www.unicode.org.
Some example Unicode characters for Tibetan are shown below.
Note that this parity method only works for single bit errors. If there are multiple errors we may be out of luck. Unfortunately, this is often the case due to the bursty nature of most errors. For these cases, we need stronger methods of error detection and correction such as a checksum, cyclic redundancy check or LRC. (which we won’t describe here).
Using even parity, what would be the parity bit for 111010110100?
Representing images, sound
Our pattern of bits can represent more than just numbers. They can represent anything we want them to as long as the meaning can be mapped to a sequence of bits.
Images are represented by a bitmap, or a sequence of bits that represent different colors. (Todo: more detail on bitmap representation in class)
Sound is represented as a sample of intensities from a sound wave.
