1. Representing Information in Binary (Continued)
SIMS-201
Representing Information in Binary (Continued)

2. Overview Chapter 3 continued: Binary Coded Decimal (BCD)
Representing alphanumeric characters in binary form
Bits, Bytes, and Beyond
Representing real numbers in binary form
Octal numbering system
Hexadecimal numbering system
Conversion between different numbering systems

3. Binary Coded Decimal (BCD)
Representing one decimal number at a time
How can we represent the ten decimal numbers (0-9) in binary code?
Numeral 1 2 3 4 5 6 7 8 9
BCD Representation
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Binary
Coded
Decimal
We can represent any integer by a string of binary digits.
For example, 749 can be represented in binary as:

4. How Many Bits Are Necessary to Represent Something?
1 bit can represent two (21) symbols
either a 0 or a 1
2 bits can represent four (22) symbols
00 or 01 or 10 or 11
3 bits can represent eight (23) symbols
000 or 001 or 011 or 111 or 100 or 110 or 101 or 010
4 bits can represent sixteen (24) symbols
5 bits can represent 32 (25) symbols
6 bits can represent 64 (26) symbols
7 bits can represent 128 (27) symbols
8 bits (a byte) can represent 256 (28) symbols
n bits can represent (2n) symbols!
So…how many bits are necessary for all of us in class to have a unique binary ID? Are two bits enough? Three? Four? Five? Six? Seven?

5. To think about..
Can 64 bits represent twice as many symbols as 32 bits?
32 bit = 232 = 4,294,967,296 symbols
64 bit = 264 = 1.8 x 1019 symbols
128 bit = 2128 = 3.4 x 1038 symbols
Can 8 bits represent twice as many symbols as 4 bits?
8 bit = 28 = 256 symbols
4 bit = 24 = 16 symbols
Remember that we’re dealing with exponents!
8 bit is twice as big as __________?
7 bit!
7 bits can represent 27 possible symbols or 2x2x2x2x2x2x2 = 128
8 bits can represent 28 possible symbols or 2x2x2x2x2x2x2x2 = 256

6. 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.

7. 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 from the textbook)
A complete ASCII chart may be found in appendix A in your book
How do you spell Lecture in ASCII?

8. Extended ASCII Chart
This ASCII chart illustrates Decimal and Hex (Hexadecimal) 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

9. 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

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

11. 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
languages with accents (French, German)
nonLatin alphabets (e.g. Hebrew, Russian)
languages withough alphabets (e.g. Chinese, Japanese)

12. 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) – temperature can take any value between say 60 and 70 degrees F. There is an infinite number of such possible temperatures.
In order to perfectly represent that, a digital thermometer will need an infinite number of bits (which we can neither generate, store, nor transmit an infinite stream)
So: if we are building a digital thermometer, we must make some choices to determine certain parameters:
Precision (number of bits we will use) vs. the cost
Accuracy (how true is our measurement against a given standard)

13. 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 = 180º F
If our thermometer needs to measure in 0.01º increments in order to be considered accurate, then we need to be able to represent 18,000 steps. (There are 180 degrees between -40º to 140º F. Since each degree can have 100 different values (with the 0.01º increments), then 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)

14. Thermometer Coding (One solution)
15 bit code

15. Thermometer Coding (Another solution)

16. Convenient Forms for Binary Codes
Bits, Bytes, and Beyond

17. 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

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

19. 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)

20. 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

21. 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

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

23. Start here
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

24. 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

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

26. 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

27. 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

28. 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 next

29. 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. We recall that, in order to to convert from binary to decimal, we summed together the weights of the positions of the binary number that contained a “1”. Similarly, we sum the weights of the positions in the octal or hex numbers (depending on the base being used).
Binary is base 2, octal is base 8 and hex is base 16
Two examples of octal to decimal conversion:
778 converted to decimal form: 7x81 + 7x80 = 6310
converted to decimal form: 2x83 + 6x82 + 3x81 + 5x80 =
Two examples of hex to decimal conversion:
A516 is converted to decimal form by: 10x x160 = 16510
F8C16 is converted to decimal form by: 15x x x160 =
In order to convert from octal to hex (or vice versa), we first convert octal to binary, and then convert binary to hex.