1. Information Representation
Computer Architecture Section 001
Information Representation
Slides from (George Mason University)
Week #11

2. Overview Chapter 3: Bits vs. Bytes
Representing real numbers in binary form
Representing negative numbers in binary form
Octal numbering system
Hexadecimal numbering system
Conversion between different numbering systems
Representing alphanumeric characters in binary form

3. Bits vs. Bytes 1 Byte = 8 Bits
Transmission Rates Measured in Bits Per Second
1 Byte = 8 Bits
“Bits” are often used in terms of a data rate, or speed of information flow:
56 Kilobit per second modem (56 Kbps)
A T-1 is Megabits per second (1.544 Mbps or 1544 Kbps)
“Bytes” are often used in terms of storage or capacity--computer memories are organized in terms of 8 bits
256 Megabyte (MB) RAM
40 Gigabyte (GB) Hard disk
Computer Storage and Memory Measured in Bytes

4. Practical Use Everyday stuff measured in bits: 32-bit sound card
64-bit video accelerator card
128-bit encryption in your browser

5. Note! The Multipliers for Bits and Bytes are Slightly Different.
“Kilo” or “Mega” have slightly different values when used with bits per second or with bytes.
When Referring to Bytes (as in computer memory)
Kilobyte (KB) = 1,024 bytes
Megabyte (MB) 220 = 1,048,576 bytes
Gigabyte (GB) = 1,073,741,824 bytes
Terabyte (TB) = 1,099,511,627,776 bytes
When Referring to Bits Per Second (as in transmission rates)
Kilobit per second (Kbps) = 1000 bps (thousand)
Megabit per second (Mbps) = 1,000,000 bps (million)
Gigabit per second (Gbps) = 1,000,000,000 bps (billion)
Terabit per second (Tbps) = 1,000,000,000,000 bps (trillion)

6. More Multipliers for Measuring Bytes
Kilobyte (K) = 1,024 bytes
Megabyte (M) 220 = 1,048,576 bytes
Gigabyte (G) = 1,073,741,824 bytes
Terabytes (T) 240 = 1,099,511,627,776 bytes
Petabytes (P) 250 = 1,125,899,906,842,624 bytes
Exabytes (E) = 1,152,921,504,606,846,976 bytes
Zettabytes (Z) 270 = 1,180,591,620,717,411,303,424 bytes
Yottabytes (Y) 280 = 1,208,925,819,614,629,174,706,176 bytes

7. Representing Real Numbers in Binary Form
Previously, we learned how to represent integers in binary form
Real numbers can be represented in binary form as well
We will illustrate this with a thermometer example:
A Mercury thermometer reflects temperature that can continuously vary over its range of measurement (an analog device)
A digital thermometer would require an infinite number of bits to accomplish the same thing
So: if we are building a digital thermometer, we must make some choices and determine some parameters:
Precision (number of bits we will use) vs. the cost
Accuracy (how true is our measurement against a given standard)

8. Suppose we want to measure the temperature range: -40º F to 140º F
What is the range we wish to measure?
How many bits are we willing to use?
Suppose we want to measure the temperature range: -40º F to 140º F
Total measurement range = 180 º F
If our thermometer measures in 0.01º increments, we need to represent 18,000 steps (There are 180 º degrees between -40 º to 140 ºF. Since each degree can have 100 different values, the number of possible values that the thermometer can measure is: 180x100=18,000)
We can accomplish this with a 15 bit code (A 15 bit code can represent 215=32,768 different values-since we have 18,000 different values we have to represent, a code with 14 bits will not suffice, as a 14-bit code can only represent 214=16,384 values)

9. Thermometer Coding (One solution)
15 bit code

10. Thermometer Coding (Another solution)

11. Representing Negative Numbers in Binary Form
Negative numbers may also be represented in binary
This may be accomplished a number of ways:
The leftmost bit i.e.: MSB may be used to represent the sign. i.e.: 0 if positive, 1 if negative. ex: 1110 is a negative number because MSB is “1”. In decimal, this will correspond to -6. Positive 6 (+6) may then be represented by: 0110
There is a problem with this scheme in computer logic, because addition of these numbers in binary will result in:
0110
+1110
The result is not 0, it is: 410 with a 1 overflow
Positive number
Negative number
overflow

12. The result is 0, with a 1 overflow
This would cause errors in computing
To overcome this problem, we can represent the negative number by using the 2’s complement notation:
Take the complement (flip the bit values) of the positive number with the MSB as a sign bit (a 0 makes the representation positive)
The binary complement of: 0110 is: 1001
Add a binary 1
= 1010 (-6)
The addition of the original number (+6) and it’s 2’s complemented value (-6) should give zero:
0110
+1010
Step1
Step2
Positive number
Negative number
overflow
The result is 0, with a 1 overflow

13. Example: Express the decimal number -5 in 5-bit binary notation.
First determine the 5-bit binary representation of 5:
00101
Then complement (change 1s to 0s and 0s to 1s) each bit:
11010
Finally, add 1:
11011
-5 = in 2’s complement notation
(To reverse the process take the complement and add 1!)
Note that the 2’s complement of a 2’s complement of a number equals the original number.

14. In-class Examples
Determine the 5-bit binary equivalent of -7 in 2’s complement notation and show how to reverse the process.
Determine the decimal value of the 6-bit 2’s complement number given by

15. Octal Numbering system
There are other ways to “count ” besides the decimal and binary systems. One example is the octal numbering system (base 8).
The Octal numbering system uses the first 8 numerals starting from 0
The first 20 numbers in the octal system are:
0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17, 20
21,22,23
Because there are 8 numerals and 8 patterns that can be formed by 3 bits, a single octal number may be used to represent a group of 3 bits

16. Octal numeral
Bit pattern
000 1
001 2
010 3
011 4
100 5
101 6
110 7
111

17. For example, the number may be converted to octal form by grouping the bits into 3, and looking at the table. Starting from the right, count 3 digits to the left. The first 3-bits are: 111, corresponding to 7. The next 3-bits are: 010, corresponding to 2. The next are 001, corresponding to 1 and the last 3-bits are 101, corresponding to 5
Hence, the above binary number may be represented by: in octal form
Start here

18. Hexadecimal Numbering system
The hex system uses 16 numerals, starting with zero
The standard decimal system only provides 10 different symbols
So, the letters A-F are used to fill out a set of 16 different numerals
We can use hex to represent a grouping of 4 bits

19. Decimal
Octal
Hex
Binary
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8 10
1000 9 11
1001 12 A
1010 13 B
1011 14 C
1100 15 D
1101 16 E
1110 17 F
1111

20. Start here
For example, the number may be converted to hexadecimal form by grouping the bits into 4, and looking at the table. Starting from the right, count 4 digits to the left. The first 4-bits are: 0111, corresponding to 716. The next 4-bits are: 1101, corresponding to D16. The last bits are: 1010, corresponding to A16
Hence, the above binary number may be represented by: AD716 in hexadecimal form

21. Conversion between different numbering systems
Conversions are possible between different numbering systems:
1-Binary to Decimal and vice versa
2-Binary to Octal and vice versa
3-Binary to Hex and vice versa
4-Octal to Decimal and vice versa
5-Hex to Decimal and vice versa
6-Octal to Hex and vice versa

22. We learned how to do # 1 in detail in the previous lecture
We learned how to do # 2 and 3 in this lecture
We will learn how to do # 4, 5 and 6 now

23. Remember that binary is base 2, octal is base 8 and hex is base 16
To convert octal and hex to decimal, we apply the same technique as converting binary to decimal. Remember that we summed together the weights of the various positions in the binary number which contained a “1” to convert from binary to decimal. Similarly, we sum the weights of the various positions in the octal or hex numbers depending on the base being used
Remember that binary is base 2, octal is base 8 and hex is base 16
Two examples of octal to decimal coversion:
778 converted to decimal form: 7x81+7x80 =6310
converted to decimal form: 2x83+ 6x82+3x81+5x80=143710
Two examples of hex to decimal coversion:
A516 is converted to decimal form by: 10x161+5x160 =16510
F8C16 is converted to decimal form by: 15x162+8x161+12x160 =398010
To convert from octal to hex (or hex to octal), first convert octal (or hex) to binary, and then convert binary to hex (or octal).

24. Representing Alphanumeric Characters in Binary form (ASCII)
It is important to be able to represent text in binary form as information is entered into a computer via a keyboard
Text may be encoded using ASCII
ASCII can represent:
Numerals
Letters in both upper and lower cases
Special “printing” symbols such $, %, etc.
Commands that are used by computers to represent carriage returns, line feeds, etc

25. ASCII is an acronym for American Standard Code for Information Interchange
Its structure is a 7 bit code (plus a parity bit or an “extended” bit in some implementations
ASCII can represent 128 symbols (27 symbols)
INFT 101 is: (decimal) or
in binary (using Appendix A)
A complete ASCII chart may be found in appendix A in your book
How do you spell Lecture in ASCII?

26. Extended ASCII Chart
This ASCII chart illustrates Decimal and Hex representation of numbers, text and special characters
Hex can be easily converted to binary
Upper case D is 4416
416 is 01002
Upper case D is then in binary

27. Extended ASCII (Cont…)
Another example:
You want to represent the Yen sign (¥)
From the table: 9D
916 = 910 = 10012
D16 = 1310 = 11012
The ¥ sign in binary is:

28. ASCII conversion example
Let us convert You & I, to decimal, hex and binary using the ASCII code table : Y:
o: F u:
Space:
&: I:
,: C

29. You & I, in Hex: You & I, in decimal: You & I, in binary:
59 6F C
You & I, in decimal:
You & I, in binary:

30. Other Text Codes
Extended Binary Coded Decimal Interchange Code (EBCDIC) used by IBM-- 8 bit (28 bits) 256 symbols
Unicode is 16 bit (216) 65,536 symbols
World Wide Web supports many languages
Unicode supports Latin, Russian, Cherokee and other alphabet representations