Transcript lesson3 - USF Computer Science

Numerals and Numbers
How characters and integers are
represented inside a computer
(and in assembly language)
What is ASCII?
• Humans can communicate with machines
• But need a language each understands
• English language employs a finite set of
character-symbols (‘alphabet’) of letters,
plus digits and various punctuation-marks
• Character-symbols are assigned ‘values’
• American Standard Code for Information
Interchange (ASCII) is a standard scheme
Some ASCII illustrations
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‘A’, ‘B’, ‘C’, … , ‘Z’
‘a’, ‘b’, ‘c’, … , ‘z’
‘0’, ‘1’, ‘2’, … , ‘9’
‘,’ (comma)
‘-’ (hyphen)
‘.’ (period)
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65, 66, 67, … , 90
97, 98, 99, … , 122
48, 49, 50, … 57
44
45
46
There are 128 different numbers defined by the ASCII code
(Type the Linux command: ‘man ascii’ to see the complete table)
ASCII ‘control-codes’
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ASCII was devised for typewriter terminals
Some code-values denote ‘typing motions’
The ‘back-space’ motion is 8
The ‘tab’ motion is 9
The ‘new-line’ motion is 10
The ‘carriage-return’ motion is 13
A code exists for the typewriter’s bell: 7
Numerals versus Numbers
• A ‘numeral’ is a digit-character: ‘1’, ‘2’, …
• The digit-character ‘0’ has ASCII value 48
• The digit-character ‘1’ has ASCII value 49
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• The digit-character ‘9’ has ASCII value 57
• So a ‘numeral’ isn’t what it appear to be!
e.g., character ‘7’ isn’t equal to number 7
Communicating in ASCII
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Humans talk to computers using ASCII
Computer says: Give me a number
Human replies: ok, I will say ‘6’
Computer must ‘interpret’ this input
Must ‘convert’ the character’s ASCII-value
into the number-value that was intended
• It’s easy: computer can just do subtraction
‘6’ – ‘0’ = 54 – 48 = 6
Bigger numbers?
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Computer says: Tell me a bigger number
Human thinks: ok, how about ninety-five?
Human types in two digits: ‘9’ and ‘5’
Here the typing-order is important:
because “95” means 9-times-10, plus 5
• Computer sees two ASCII values: 57, then 53
• It must convert 57 into 9, and 53 into 5, and then
do a multiplication (by 10) and an addition step
Positional Notation
• We use the Hindu-Arabic notation system
• It uses ten digit-symbols: ‘0’, ‘1’, ‘2’. … , ‘9’
• For writing numbers bigger than nine, the
digit-symbols are comined in a sequence:
so three hundred sixty-five is written “365”
• The “position” of each digit is significant,
as it affects what that digit really means
• Meanings determined by the number-base
Other number bases?
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An analogy: base two versus base ten
Base-ten uses ten digits: 0, 1, 2, … , 0
Base-two uses two digits: 0, 1
3-digit number using base ten: 365
3-digit number using base two: 101
365 (base 10) means 3x100 + 6x10 + 5x1
101 (base 2) means 1x4 + 0x2 + 1x1
101 (base 10) means 1x100 + 0x10 + 1x1
Conversions
• How do we write twenty-five using base 2?
• Algorithm: use repeated division by 2, but save
the remainders as they’re generated
• 25 divided by 2:
quot=12
rem=1
• 12 divided by 2:
quot=6
rem=0
• 6 divided by 2:
quot=3
rem=0
• 3 divided by 2:
quot=1
rem=1
• 1 divided by 2:
quot=0
rem=1
• So twenty-five (base 2) can be written: 11001
• It means:
1x16 + 1x8 + 0x4 + 0x2 + 1x1
Exercise
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Write forty-nine in base 8 notation?
49 divided by 8: quot=6 rem=1
6 divided by 8: quot=0 rem=6
We stop when quotient equals zero
Answer: 49 (base 8) is written as 61
It means (positional notation): 6x8 + 1x1
Note: remainders used in backward order!