it-101 section 001 lecture #4 introduction to information technology
TRANSCRIPT
IT-101Section 001
Lecture #4
Introduction to Information Technology
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
Bits vs. Bytes
“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 1.544 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
Practical Use
Everyday stuff measured in bits: 32-bit sound card 64-bit video accelerator card 128-bit encryption in your
browser
Note! The Multipliers for Bits and Bytes are Slightly Different.
When Referring to Bytes (as in computer memory)
Kilobyte (KB) 210 = 1,024 bytes
Megabyte (MB) 220 = 1,048,576 bytes
Gigabyte (GB) 230 = 1,073,741,824 bytes
Terabyte (TB) 240 = 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)
““Kilo” or “Mega” have slightly different values when used with bits Kilo” or “Mega” have slightly different values when used with bits per second or with bytes.per second or with bytes.
More Multipliers for Measuring Bytes
Kilobyte (K) 210 = 1,024 bytes
Megabyte (M) 220 = 1,048,576 bytes
Gigabyte (G) 230 = 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) 260 = 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
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)
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)
Thermometer Coding (One solution)
140.00º F 100011001010000 139.99º F 100011001001111 139.98º F 100011001001110 139.97º F
100011001001101
0.00º F
000111110100000
-39.98º F 000000000000010 -39.99º F 000000000000001 -40.00º F 000000000000000
15 bit code
Thermometer Coding (Another solution)
0.04º F 0000000000001000.03º F 0000000000000110.02º F 0000000000000100.01º F 0000000000000010.00º F 000000000000000-0.01º F 111111111111111-0.02º F 111111111111110-0.03º F 111111111111101
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
1 0100 The result is not 0, it is: 410 with a 1 overflow
Negative numberPositive number
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
1001 + 1 = 1010 (-6)The addition of the original number (+6) and it’s 2’s
complemented value (-6) should give zero: 0110
+1010 1 0000
Negative number
Positive number
– The result is 0, with a 1 overflow
overflow
Step1
Step2
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 = 11011 in 2’s complement notation-5 = 11011 in 2’s complement notation(To reverse the process take the complement and add (To reverse the process take the complement and add 1!)1!)
Note that the 2’s complement of a 2’s complement of a Note that the 2’s complement of a 2’s complement of a number equals the original number.number equals the original number.
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 111111.
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
Octal numeral
Bit pattern
0 000
1 001
2 010
3 011
4 100
5 101
6 110
7 111
For example, the number 1010010101112 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: 51278 in octal form
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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
Decimal Octal Hex Binary
0 0 0 0000
1 1 1 0001
2 2 2 0010
3 3 3 0011
4 4 4 0100
5 5 5 0101
6 6 6 0110
7 7 7 0111
8 10 8 1000
9 11 9 1001
10 12 A 1010
11 13 B 1011
12 14 C 1100
13 15 D 1101
14 16 E 1110
15 17 F 1111
For example, the number 1010110101112 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
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Conversion between different numbering systems
Conversions are possible between different numbering systems:1-Binary to Decimal and vice versa2-Binary to Octal and vice versa3-Binary to Hex and vice versa4-Octal to Decimal and vice versa5-Hex to Decimal and vice versa6-Octal to Hex and vice versa
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
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 26358 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).
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 as @, $, %, etc. Commands that are used by computers to
represent carriage returns, line feeds, etc
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: 73 78 70 84 32 49 48 49 (decimal) or
–1001001 1001110 1000110 1010100 0100000 0110001 0110000 0110001 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?
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 0100 0100 in binary
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: 1001 1101
ASCII conversion example
Let us convert You & I, to decimal, hex and binary using the ASCII code table :
Y: 8910 5916 10110012
o: 11110 6F16 11011112
u: 11710 7516 11101012
Space: 3210 2016 01000002
&: 3810 2616 01001102
Space: 3210 2016 01000002
I: 7310 4916 10010012
,: 4410 2C16 01011002
You & I, in Hex: 59 6F 75 20 26 20 49 2C
You & I, in decimal: 89 111 117 32 38 32 73 44
You & I, in binary: 1011001 1101111 1110101 0100000
0100110 0100000 1001001 0101100
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
www.unicode.org