1. 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
Another name for binary is base-2 (pronounced "base two")

3. 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 76
Another name for decimal numbers are base-10 (pronounced "base ten") numbers.

4. Equivalence of Binary and Decimal
Every Binary number has a corresponding Decimal value (and vice versa)
Examples:
Binary Number Decimal Equivalent
… …

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.

6. Computers store information using binary numbers

7. 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.

8. How does a CD store information?
For detailed information about a CD works, see the following URL:
This presentation will only focus on the very basics.

9. 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.

10. How a computer stores information

11. 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: etc
Numbers with decimal points. Examples: etc
Textual information (including letters, symbols and digits)
Keep reading …

12. Integers
Integers (e.g. 87) A computer stores integer numbers (i.e. “whole” numbers) simply as the equivalent binary value for that number.

13. Numbers with Decimal Points
Numbers with decimal points (e.g )
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.]

14. 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 …

15. ASCII (Americal Standard Code for Information Interchange)
Some ASCII values (values 1-31 and 128 are not shown)
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)
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 = ? 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 = _
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 = ~

16. 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
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

17. 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 correspond to the EXACT SAME symbols as ASCII 1-128

18. How to Convert from Binary to Decimal

19. Converting from binary to decimal
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:
1 X 1 = 1
0 X 2 = X 4 = 4
---- 5
position values
add these up
binary #
digit X position value
result

20. What about a longer number?
In general, the "position values" in a binary number are the powers of two.
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

21. Example
The value of binary is decimal 105. This is worked out below:
1 X 1 = 1
0 X 2 = X 4 = 0
1 X = 8
0 X = 0
1 X = 32
1 X = 64
0 X = 0
----
Answer:

22. Another example
The value of binary is decimal 156. This is worked out below:
0 X 1 = 0
0 X 2 = X 4 = 4
1 X = 8
1 X = 16
0 X = 0
0 X = 0
1 X = 128
----
Answer:

23. 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.

24. How many different numbers?
There are two different binary numbers with one bit: 1
There are four different binary numbers with two bits:
00 (i.e. decimal 0)
01 (i.e. decimal 1)
10 (i.e. decimal 2)
11 (i.e. decimal 3)

25. How many different numbers?
There are eight different binary numbers with three bits:
000 (i.e. decimal 0)
001 (i.e. decimal 1)
010 (i.e. decimal 2)
011 (i.e. decimal 3)
100 (i.e. decimal 4)
101 (i.e. decimal 5)
110 (i.e. decimal 6)
111 (i.e. decimal 7)

26. # different numbers - General Rule
For n bits there are 2n different binary numbers:
# of bits # of different binary numbers
1 bit: = 2
2 bits: = 4
3 bits: = 8
4 bits: = 16
5 bits: = 32
6 bits: = 64
7 bits: = 128
8 bits: = 256
9 bits: = 512
10 bits: = 1024
etc.

27. 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:

28. 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 smallest binary # decimal value 1 bit:
2 bits:
3 bits:
4 bits:
5 bits:
6 bits:
7 bits:
8 bits: etc.

29. 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

30. 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.

31. 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)

32. 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 #
? ? ? ? ? ? ?
position values
binary #

33. 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 rest of the binary digits zero
you are done

34. 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.

35. 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
? ? ? ? ? ? ?
Step 3: Check leftmost position value (i.e. 64)
64 is less than 105, therefore
the first binary digit is 1
? ? ? ? ? ?
subtract : = 42

36. Example (continued-2) Step 4: Check next position value (i.e. 32)
32 is less than 42, therefore
the next binary digit is 1
? ? ? ? ?
subtract : = 10
Step 4(continued): Check next position value (i.e. 16)
16 is greater than 10, therefore
the next binary digit is 0
? ? ? ?

37. 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
? ? ?
subtract : = 2
Step 4(continued): Check next position value (i.e. 4)
4 is greater than 2, therefore
the next binary digit is 0
? ?

38. 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 ?
Since the position value was equal to the number (i.e. 2) the rest of the binary digits are all zeros
Answer:

39. 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 etc. )

40. Hexadecimal (AKA “Hex”) numbers

41. 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: A B C D E F
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 …

42. Hexadecimal numbers The following are all valid hexadecimal nubmers
9 (yes, a hexadecimal number does not HAVE TO contain letters)
1001 (yes, a hexadecimal number does not HAVE TO contain letters)
9C5
BFE
Etc.
To understand what a specific hexadecimal number means, you can convert it into an equivalent decimal number. (see next slide)

43. Converting a Hexadecimal number to Decimal
The value of hexadecimal A12F is decimal 41,263. See below:
4096 (i.e 163) (i.e 162) 16 (i.e 161) (i.e 160)
A F
15 X 1 = 15
2 X = X = 256
10 X = 40,960
----
Answer: 41,263

44. 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 …

45. The numbers from decimal 0 through 15 in hex and binary
Draw a table here with appropriate info.

46. Converting from Hex to Binary

47. Converting from Binary to Hex

48. How a computer stores information (i.e. decimal vs. analog)

49. 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

50. Terms (bit, byte, etc) BIT BYTE NYBLE
definition: a single Binary digIT (i.e. BIT)
BYTE
definition: 8 bits
NYBLE
definition: 4 bits

51. Prefixes Prefixes Kilo: one thousand Mega: one millioin
Giga: one billion
Tera: one trillion
Peta: one quadrillion
Exa: one quintillion
… etc.

52. Data sizes Data sizes Kilobyte (KB) Megabyte (MB) Gigabyte (GB)
"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

53. Data Sizes – bytes vs bits
MB = one Mega Byte
Mb = one Mega Bit

54. Speeds
MBPS = one MegaByte per second
MbPS = one Mega Bit per second

55. 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

56. Organization of Computer Memory

57. CPU

58. Passing information on wires

60. Hexadecimal - a shorthand for binary

61. Hexadecimal ("Hex") / Base-16
Uses digits:
0,1,2,3,4,5,6,7,8,9,a,b,c,d,e,f
A - F
a is 10
b is 11
c is 12
d is 13
e is 14
f is 15

62. Converting from Hex to Decimal

63. Converting from Decimal to Hex

64. Applications
HTML : color codes
Shorthand for binary
See hex editor