IT-101
Section 001
Introduction to Information
Technology
Lecture #4
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.
“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)
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)
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)
15 bit code
140.00º F
139.99º F
139.98º F
139.97º F
100011001010000
100011001001111
100011001001110
100011001001101
0.00º F
000111110100000
-39.98º F
-39.99º F
-40.00º F
000000000000010
000000000000001
000000000000000
Thermometer Coding (Another
solution)
0.04º F
0.03º F
0.02º F
0.01º F
0.00º F
-0.01º F
-0.02º F
-0.03º F
000000000000100
000000000000011
000000000000010
000000000000001
000000000000000
111111111111111
111111111111110
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:
Positive number
0110
+1110
Negative number
overflow
1 0100

The result is not 0, it is: 410 with a 1 overflow



Step1
Step2
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:
Positive number
0110
+1010
Negative number
1 0000
overflow

– The result is 0, with a 1 overflow
Example:

Express the decimal number -5 in 5-bit binary notation.


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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
(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.
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

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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
Start here

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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
Hexadecimal Numbering
system

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
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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
Start here


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
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


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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:


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Two examples of hex to decimal coversion:
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778 converted to decimal form: 7x81+7x80 =6310
26358 converted to decimal form: 2x83+ 6x82+3x81+5x80=143710
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)

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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:

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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

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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…)
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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 :
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Y:
8910
o:
11110
u:
11710
Space: 3210
&:
3810
Space: 3210
I:
7310
,:
4410
5916
10110012
6F16
11011112
7516
11101012
2016
01000002
2616
01001102
2016
01000002
4916
10010012
2C16
01011002

You & I, in Hex:


You & I, in decimal:
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
59 6F 75 20 26 20 49 2C
89 111 117 32 38 32 73 44
You & I, in binary:

1011001 1101111 1110101 0100000
0100110 0100000 1001001 0101100
Other Text Codes
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Extended Binary Coded
Decimal Interchange
Code (EBCDIC) used by
IBM-- 8 bit (28 bits) 256
symbols
Unicode is 16 bit (216)
65,536 symbols

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
World Wide Web
supports many languages
Unicode supports Latin,
Russian, Cherokee and
other alphabet
representations
www.unicode.org