Transcript Binary and Decimal Numbers - Prof. Yitz Rosenthal

Binary and Decimal Numbers
Prof. Rosenthal
2
What is a binary number?
• A binary number is a number that includes only ones and
zeroes.
• The number could be of any length
• The following are all examples of binary numbers
0
1
10
01
111000
10101
0101010
1011110101
0110101110
000111
• Another name for binary is base-2 (pronounced "base
two")
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What is a decimal number?
• The numbers that we are used to seeing are called
decimal numbers.
• decimal numbers consist of the digits from 0 (zero)
through 9.
• The following are examples of decimal #'rs
3
15
890
76
32423234
53
• Another name for decimal numbers are base-10
(pronounced "base ten") numbers.
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Equivalence of Binary and Decimal
•
Every Binary number has a corresponding Decimal
value (and vice versa)
•
Examples:
Binary Number
1
10
11
…
1010111
Decimal Equivalent
1
2
3
…
87
5
The value of a binary number
• Even though they look exactly the same, the value of the
binary number, 101, is different from the value of the
decimal number, 101.
– The value of the binary number, 101, is equal to the decimal
number five (i.e. 5)
– The value of the decimal number, 101, is equal to one hundred
and one
• When you see a number that consists of only ones and
zeroes, you must be told if it is a binary number or a
decimal number.
Computers store information
using binary numbers
All information on computers is stored as
numbers
• All information that is processed by computers is converted in
one way or another into a sequence of numbers. This includes
– numeric information
– textual information and
– Picutures
(We’ll see later how text and pictures can be converted into simple
numbers … for now just take our word for it.)
• Therefore, if we can derive a way to store and retrieve numbers
electronically this method can be used by computers to store and
retieve any type of information.
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How does a CD store information?
• For detailed information about a CD works, see
the following URL:
http://electronics.howstuffworks.com/cd.htm
• This presentation will only focus on the very
basics.
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Information is stored using binary numbers
• A binary number is simply a bunch of 1’s and 0’s
• CD’s that are created in a factory (we’re not talking
about CD-R’s yet) may look perfectly flat. However,
there are many microscopic “bumps” on the surface of
the CD.
• The bumps are laid out in a spiral form on the surface of
the CD.
•
•
How a computer stores
information
Binary Numbers are at the heart of how a
computer stores all information
• Computers Store ALL information using Binary Numbers
• Computers use binary numbers in different ways to store different types of
information.
• Common types of information that are stored by computers are :
– Whole numbers (i.e. Integers).
Examples: 8 97 -732 0
-5 etc
– Numbers with decimal points.
Examples: 3.5 -1.234 0.765 999.001 etc
– Textual information (including letters, symbols and digits)
• Keep reading …
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Integers
• Integers (e.g. 87)
A computer stores integer numbers (i.e. “whole”
numbers) simply as the equivalent binary value
for that number.
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Numbers with Decimal Points
• Numbers with decimal points (e.g. 87.123)
– Internally, a computer stores a number with a decimal point as
two different integer numbers (each stored using binary). To
get the actual value, the computer performs a mathematical
calculation using the two integers to derive the number.
– We will NOT discuss here the actual mathematical calculation
nor how the computer breaks a number with a decimal point
into two integers.
[NOTE: The two integers are NOT the whole number part and
fractional part.]
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Letters and symbols
• Letters and symbols
– To store letters and symbols, the computer assigns
every character on the keyboard a numerical value.
– Computers remember letters and other symbols by
storing the binary number for the symbol.
– For this system to work a standard numbering system
needs to be defined and consistently used for all
symbols that the computer needs to process.
– See the following slide …
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ASCII (Americal Standard Code for Information Interchange)
•
ASCII (Americal
Standard Code for
Information
Interchange) is the
standard numbering
given to all characters
on a standard
keyboard.
•
“ASCII values” range
in number from 1 to
128. Some “ASCII
values” and their
associated symbols
are listed to the right.
•
Note that EVERY
symbol on a standard
keyboard has an
ASCII value. Even
the digits 0,1,2,…9
have ASCII values.
(see next slide)
Some ASCII values (values 1-31 and 128 are not shown)
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32 = Space
33 = !
34 = “
35 = #
36 = $
37 = %
38 = &
39 = `
40 = (
41 = )
42 = *
43 = +
44 = ,
45 = 46 = .
47 = /
48 = 0
49 = 1
50 = 2
51 = 3
52 = 4
53 = 5
54 = 6
55 = 7
56 = 8
57 = 9
58 = :
59 = ;
60 = <
61 = =
62 = >
63 = ?
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64 = @
65 = A
66 = B
67 = C
68 = D
69 = E
70 = F
71 = G
72 = H
73 = I
74 = J
75 = K
76 = L
77 = M
78 = N
79 = O
80 = P
81 = Q
82 = R
83 = S
84 = T
85 = U
86 = V
87 = W
88 = X
89 = Y
90 = Z
91 = [
92 = \
93 = ]
94 = ^
95 = _
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96 = `
97 = a
98 = b
99 = c
100 = d
101 = e
102 = f
103 = g
104 = h
105 = i
106 = j
107 = k
108 = l
109 = m
110 = n
111 = o
112 = p
113 = q
114 = r
115 = s
116 = t
117 = u
118 = v
119 = w
120 = x
121 = y
122 = z
123 = {
124 = |
125 = }
126 = ~
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Why do 0 through 9 have ASCII values?
• Numbers that are used in mathematical calculations
– If a computer needs to do math with a number it will store that number using the
appropriate binary representation of the number.
– This makes it easier for the computer to perform mathematical calculations with the
number.
– Example: 5 would be stored as
00000101
• Numbers that are NOT used in mathematical calculations
– If the computer does NOT need to do math with the number (e.g. a zip code) then it
will generally store the number using the ASCII values of the digits.
– In this case using the ASCII value is more efficient (for reasons we will not explain
here).
– Example 5 would be stored using its ASCII value of 53 which is represented in
binary as
00110101
Other numbering systems (Unicode and
EBCDIC)
•
ASCII
– ASCII was the standard numbering system for many years and is still used widely today.
•
EBCDIC
– Is a different numbering system used by IBM Mainframe computers.
– It is very similar to ASCII but uses different numbers to represent the symbols.
– EBCDIC stands for “Extended Binary Coded Decimal Interchange Code”
•
Unicode
– ASCII and EBCDIC are limited to just the basic English letters and common symbols.
– Today computers use many different symbols including letters from languages that don’t use
English letters (e.g. Hebrew, Chinese, etc.) and international symbols (e.g. the English pound
sign)
– Unicode defines a unique number for every symbol in all known languages (e.g. Hebrew,
Chinese, etc.) and commonly used non-letter symbols (e.g. English pound sign, copyright
symbol, etc).
– Modern programs are moving towards using Unicode to store letters and symbols.
– It should be noted that Unicode numbers 1-128 correspond to the EXACT SAME symbols as
ASCII 1-128
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How to Convert from Binary to
Decimal
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Converting from binary to decimal
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Each position for a binary number has a value.
For each digit, multiply the digit by its position value
Add up all of the products to get the final result
The decimal value of binary 101 is computed below:
4
2
1
--------------------------------------
1
0
1
1X1
1X4
0X2
=1
=0
=4
---5
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What about a longer number?
• In general, the "position values" in a binary number are
the powers of two.
–
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–
–
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The first position value is 20 , i.e. one
The 2nd position value is 21 , i.e. two
The 2nd position value is 22 , i.e. four
The 2nd position value is 23 , i.e. eight
The 2nd position value is 24 , i.e. sixteen
etc.
• Example on next slide
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Example
• The value of binary 01100001 is decimal 105. This is worked out below:
128
64
32
16
8
4
2
1
----------------------------------------------------------------------
0
1
1
0
1
0
0
1
1X1
0X4
1X8
0 X 16
1 X 32
1 X 64
0 X 128
0X2
=1
=0
=0
=8
=0
= 32
= 64
=0
---Answer: 105
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Another example
• The value of binary 10011100 is decimal 156. This is worked out below:
128
64
32
16
8
4
2
1
----------------------------------------------------------------------
1
0
0
1
1
1
0
0
0X1
1X4
1X8
1 X 16
0 X 32
0 X 64
1 X 128
0X2
=0
=0
=4
=8
= 16
=0
=0
= 128
---Answer: 156
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Some Terminology
• The following are some terms that are used in the
computer field
– Each digit of a binary number is called a bit.
– A binary number with eight bits (i.e. digits) is called a
byte.
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How many different numbers?
• There are two different binary numbers with one bit:
– 0
– 1
• There are four different binary numbers with two bits:
–
–
–
–
00
01
10
11
(i.e. decimal 0)
(i.e. decimal 1)
(i.e. decimal 2)
(i.e. decimal 3)
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How many different numbers?
• There are eight different binary numbers with three bits:
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000
001
010
011
100
101
110
111
(i.e. decimal 0)
(i.e. decimal 1)
(i.e. decimal 2)
(i.e. decimal 3)
(i.e. decimal 4)
(i.e. decimal 5)
(i.e. decimal 6)
(i.e. decimal 7)
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# different numbers - General Rule
• For n bits there are 2n different binary numbers:
# of bits
1 bit:
2 bits:
3 bits:
4 bits:
5 bits:
6 bits:
7 bits:
8 bits:
9 bits:
10 bits:
etc.
# of different binary numbers
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256
29 = 512
210 = 1024
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Smallest value for a binary #
• The smallest value for a binary number of any
number of bits is zero.
• This is the case when all bits are zero:
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Smallest value for a binary #
• The smallest value for a binary number with any number of
bits is zero (i.e. when all the bits are zeros)
# of bits
1 bit:
2 bits:
3 bits:
4 bits:
5 bits:
6 bits:
7 bits:
8 bits:
etc.
smallest binary #
decimal value
0
00
000
0000
00000
000000
0000000
00000000
0
0
0
0
0
0
0
0
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Largest value for a binary #
• The largest value for a binary number with a
specific number of bits (i.e. digits) is when all of
the bits are one.
• General rule: for a binary number with n bits,
the largest possible value is : 2n - 1
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Largest numbers
• The following are the largest values for binary numbers
with a specific number of bits:
# of bits
largest binary #
decimal value
1 bit:
2 bits:
3 bits:
4 bits:
5 bits:
6 bits:
7 bits:
8 bits:
etc.
1
11
111
1111
11111
111111
1111111
11111111
1
3
7
15
31
63
127
255
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Converting a decimal# to a binary# (1)
• Step 1: figure out how many bits you will need
(see the chart on the previous slide).
– Example 1: To convert the decimal number 16 to binary,
you will need at least 5 bits.
(With 4 bits you can only store numbers up to 15 but with 5 bits
you can store numbers up to 31)
– Example 2: To convert the decimal number 106 to binary,
you will need at least 7 bits.
(With 6 bits you can only store numbers up to 63 but with 7 bits
you can store numbers up to 127)
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Converting a decimal# to a binary# (2)
• Step 2: Keep a chart of the position values and the digits
for your binary number. At first you will not know what
any of the digits will be.
• Example: convert decimal 106 to binary
– step 1 : figure out that you need 7 bits (see earlier slides)
– step 2 : keep track of position values and bits for binary #
64
32
16
8
4
2
1
--------------------------------------------------------------------------------
?
?
?
?
?
?
?
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Converting a decimal # to binary (3)
• Step 3: starting with the leftmost digit, see if the position
value is greater than, less than or equal to the number
you are trying to convert.
– if the position value is greater than the number then
• make the binary digit in that position zero
– if the position value is less than the number then
• make the binary digit in that position one
• subtract the position value from the decimal # you are trying to convert
– if the position value is equal to the number then
• make the binary digit in that position one
• make the rest of the binary digits zero
• you are done
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Converting a decimal # to binary (4)
• Step 4: do step 3 again with the next digit. Keep
doing this until you've figured out all of the
digits.
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Example
• Example: convert decimal 106 to binary
– Step 1: You need 7 bits (see earlier slides for explination).
– Step 2: keep track of position values for bits
64
32
16
8
4
2
1
--------------------------------------------------------------------------------
?
?
?
?
?
?
– Step 3: Check leftmost position value (i.e. 64)
?
• 64 is less than 105, therefore
• the first binary digit is 1
64
32
16
8
4
2
1
--------------------------------------------------------------------------------
1
?
• subtract : 106 - 64 = 42
?
?
?
?
?
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Example (continued-2)
– Step 4: Check next position value (i.e. 32)
• 32 is less than 42, therefore
• the next binary digit is 1
64
32
16
8
4
2
1
-------------------------------------------------------------------------------
1
1
?
?
?
?
?
• subtract : 42 - 32 = 10
– Step 4(continued): Check next position value (i.e. 16)
• 16 is greater than 10, therefore
• the next binary digit is 0
64
32
16
8
4
2
1
-------------------------------------------------------------------------------
1
1
0
?
?
?
?
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Example (continued-3)
– Step 4(continued): Check next position value (i.e. 8)
• 8 is less than 10, therefore
• the next binary digit is 1
64
32
16
8
4
2
1
-------------------------------------------------------------------------------
1
1
0
1
?
?
?
• subtract : 10 - 8 = 2
– Step 4(continued): Check next position value (i.e. 4)
• 4 is greater than 2, therefore
• the next binary digit is 0
64
32
16
8
4
2
1
-------------------------------------------------------------------------------
1
1
0
1
0
?
?
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Example (continued-4)
– Step 4(continued): Check next position value (i.e. 2)
• 2 is equal to 2, therefore
• the next binary digit is 1
64
32
16
8
4
2
1
-------------------------------------------------------------------------------
1
1
0
1
0
1
?
• Since the position value was equal to the number (i.e. 2) the rest of the
binary digits are all zeros
64
32
16
8
4
2
1
---------------------------------------------------------------------------
1
– Answer:
1
0
1101010
1
0
1
0
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Why is it called "binary" (or base-2)?
• The prefix "bi" means "two" in Latin
• Binary derives its name from the fact that the
digits in a "Binary" number can only have two
possible values, 0 or 1
• It is also called "base-2" based on the fact that the
column values are the powers of 2.
(i.e. 20 21 22 23 24 25 etc. )
Hexadecimal (AKA “Hex”)
numbers
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What is a hexadecimal or base-16 number.
• A “hexadecimal” number is a number where each digit
may be one of sixteen possible values.
• The possible values for a hexadecimal digit are:
0123456789ABCDEF
• A digit of
“A” stands for the number 10
“B” stands for the number 11
“C” stands for the number 12
“D” stands for the number 13
“E” stands for the number 14
“F” stands for the number 15
• Keep reading …
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Hexadecimal numbers
• The following are all valid hexadecimal nubmers
–
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–
–
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–
A
9
1001
9C5
BFE
Etc.
(yes, a hexadecimal number does not HAVE TO contain letters)
(yes, a hexadecimal number does not HAVE TO contain letters)
• To understand what a specific hexadecimal
number means, you can convert it into an
equivalent decimal number. (see next slide)
Converting a Hexadecimal number to
Decimal
• The value of hexadecimal A12F is decimal 41,263. See below:
1 (i.e 160)
---------------------------------------------------------------------4096 (i.e 163)
256 (i.e 162)
16 (i.e 161)
A
1
2
F
15 X 1
1 X 256
10 X 4096
2 X 16
= 15
= 32
= 256
= 40,960
---Answer: 41,263
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Hex numbers are a “shorthand” for
binary nubmers
• It is very easy to convert between Hex and binary
numbers.
• Each Hex number is 1/4th the length of its
equivalent binary number.
• Therefore Hex is often used as a “shorthand” for
writing an equivalent binary number.
• Keep reading …
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The numbers from decimal 0 through
15 in hex and binary
• Draw a table here with appropriate info.
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Converting from Hex to Binary
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Converting from Binary to Hex
How a computer stores information
(i.e. decimal vs. analog)
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How a computer stores info.
• A computer stores all information as binary
numbers.
• Computer memory simply remembers ones and
zeros
• Computer storage remembers ones and zeros
• Data is passed inside the computer from one
portion of the computer to another (e.g. memory
to CPU to graphics card, etc) by ones and zeros
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Terms (bit, byte, etc)
• BIT
– definition: a single Binary digIT (i.e. BIT)
• BYTE
– definition: 8 bits
• NYBLE
– definition: 4 bits
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Prefixes
• Prefixes
–
–
–
–
–
–
–
Kilo: one thousand
Mega: one millioin
Giga: one billion
Tera: one trillion
Peta: one quadrillion
Exa: one quintillion
… etc.
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Data sizes
• Data sizes
– Kilobyte (KB)
• "about" one thousand bytes
• exactly 210 or 1024 bytes
– Megabyte (MB)
• "about" one million bytes
• exactly 220 or 1,048,576 bytes
– Gigabyte (GB)
• "about" one billion bytes
• exactly 230 or 1,073,741,824 bytes
– Terabyte (TB)
• "about" one trillion bytes
• exactly 240 or 1,099,511,627,776 bytes
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Data Sizes – bytes vs bits
• MB = one Mega Byte
• Mb = one Mega Bit
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Speeds
• MBPS = one MegaByte per second
• MbPS = one Mega Bit per second
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How data is stored using binary
• Integers are stored as a binary number
• A character is stored as the ASCII value (i.e. an
integer) for that character
• A decimal number is stored using two different
integer values - the mantissa and the exponent
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Organization of Computer Memory
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CPU
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Passing information on wires
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Hexadecimal - a shorthand for
binary
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Hexadecimal ("Hex") / Base-16
• Base 16
• Uses digits:
– 0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f
• A-F
–
–
–
–
–
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a is 10
b is 11
c is 12
d is 13
e is 14
f is 15
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Converting from Hex to Decimal
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Converting from Decimal to Hex
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Applications
• HTML : color codes
• Shorthand for binary
– See hex editor