number systems cpsc125
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Number Systems
Spring 2010Dr. David Hyland-WoodUniversity of Mary Washington
Monday, January 11, 2010
Information Encoding
We use encoding schemes to represent and store information.
• Roman Numerals: I, IV, XL, MMIV
• Acronyms: UMW, CPSC-125, SSGN-726
• Postal codes: 22401, M5W 1E6
• Musical Note Notation: ♪♫
Encoding schemes are only useful if the stored information can be retrieved.
Monday, January 11, 2010
Monday, January 11, 2010
Linear A has not been deciphered. This tablet stores information, but it can no longer be retrieved. Think about floppy disks, tape drives, bad handwriting and other forms of “lost” data.
Decimal Notation
• base 10 or radix 10 ... uses 10 symbols
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Position represents powers of 10
• 547310 or 5473
(5 * 103) + (4 * 102) + (7 * 101) + (3 * 100)
• 73.1910 or 73.19
(7 * 101) + (3 * 100) + (1 * 10-1) + (9 * 10-2)
Monday, January 11, 2010
Monday, January 11, 2010
There is an obvious reason to use base 10.
Monday, January 11, 2010
The ancient Sumerians used base 12, which is why we inherited a 12-hour clock.
Binary Notation
• base 2 ... uses only 2 symbols
0, 1
• Position represents powers of 2
• 110102
(1 * 24) + (1 * 23) + (0 * 22) + (1 * 21) + (0 * 20)
• 10.112
(1 * 21) + (0 * 20) + (1 * 2-1) + (1 * 2-2)Monday, January 11, 2010
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
“On” or “Off”
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
“On” or “Off”
time
voltage threshold
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
“On” or “Off”
time
voltage threshold“On”
“Off”
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
“On” or “Off”
time
voltage threshold“On”
“Off”
10
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
“On” or “Off”
time
voltage threshold“On”
“Off”
10
True
False
Monday, January 11, 2010
Computers are binary all the way down. Electrical signals are “on” or “off” based on voltage. Magnetic storage systems such as hard disc drives are based on magnetism. Optical storage systems are based on depth.
Octal Notation
• base 8 ... uses 8 symbols
0, 1, 2, 3, 4, 5, 6, 7
• Position represents power of 8
• 15238
(1 * 83) + (5 * 82) + (2 * 81) + (3 * 80)
• 56.728
(5 * 81) + (6 * 80) + (7 * 8-1) + (2 * 8-2)
Monday, January 11, 2010
Hexadecimal Notation
• base 16 or ‘hex’ ... uses 16 symbols
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Position represents powers of 16
• B65F16 or 0xB65F
(11 * 163) + (6 * 162) + (5 * 161) + (15 * 160)
• F3.A916 or 0xF3.A9
(15 * 161) + (3 * 160) + (10 * 16-1) + (9 * 16-2)Monday, January 11, 2010
Octal and Hexadecimal
• Humans are accustomed to decimal
• Computers use binary
• So why ever use octal or hexadecimal?
• Binary numbers can be long with a lot of digits
• Longer number make for more confusion
• Copying longer numbers allows greater chance for error
• Octal represents binary in 1/3 the number of digits
• Hexadecimal represents binary in 1/4 the number of digits
Monday, January 11, 2010
Base Conversion
Binary to Decimal
• 110102
• (1*24)+(1*23)+(0*22)+(1*21)+(0*20)
• 16 + 8 + 0 + 2 + 0
• 26
Monday, January 11, 2010
Base Conversion
Octal to Decimal
• 15238
• (1*83)+(5*82)+(2*81)+(3*80)
• 512+320+16+3
• 851
Monday, January 11, 2010
Base Conversion
Hexadecimal to Decimal
• B65F16
• (11 * 163) + (6 * 162) + (5 * 161) + (15 * 160)
• 45056+1536+80+15
• 46687
Monday, January 11, 2010
Base ConversionDecimal to Binary
• 82310
• 823 / 2 = 411 R1(12)
• 411 / 2 = 205 R 1 (112)
• 205 / 2 = 102 R 1 (1112)
• 102 / 2 = 51 R 0 (01112)
• 51 / 2 = 25 R 1 (101112)
• 25 / 2 = 12 R 1 (1101112)
• 12 / 2 = 6 R 0 (01101112)
• 6 / 2 = 3 R 0 (001101112)
• 3 / 2 = 1 R 1 (1001101112)
• 1 / 2 = 0 R 1 (11001101112)
• 11001101112
Monday, January 11, 2010
Base Conversion
Decimal to Octal
• 823
• 823 / 8 = 102 R 7 (78)
• 102 / 8 = 12 R 6 (678)
• 12 / 8 = 1 R 4 (4678)
• 1 / 8 = 0 R 1 (14678)
• 14678
Monday, January 11, 2010
Base ConversionDecimal to Hexadecimal
• 82310
• 823 / 16 = 51 R 7 (716)
• 51 / 16 = 3 R 3 (3716)
• 3 / 16 = 0 R 3 (33716)
• 33716
• 250010
• 2500 / 16 = 156 R 4 (416)
• 156 / 16 = 9 R 12 (C416)
• 9 / 16 = 0 R 9 (9C416)
• 9C416
Monday, January 11, 2010
Base ConversionBinary to Octal
• 11001101112
• (12) (1002) (1102) (1112)
• (18) (48) (68) (78)
• 14678
Octal to Binary
• 14678
• (18) (48) (68) (78)
• (0012) (1002) (1102) (1112)
• 11001101112
Monday, January 11, 2010
Base ConversionBinary to Hexadecimal
• 10110110010111112
• (10112) (01102) (01012) (11112)
• (B16) (616) (516) (F16)
• B65F16
Hexadecimal to Binary
• B65F16
• (B16) (616) (516) (F16)
• (10112) (01102) (01012) (11112)
• 10110110010111112
Monday, January 11, 2010
Computing Systems
• Computers were originally designed for the primary purpose of doing numerical calculations.
Abacus, counting machines, calculators
• We still do numerical operations, but not necessarily as a primary function.
• Computers are now “information processors” and manipulate primarily nonnumeric data.
Text, GUIs, sounds, information structures, pointers to data addresses, device drivers
Monday, January 11, 2010
Codes
• A code is a scheme for representing information.
• Computers use codes to store types of information.
• Examples of codes:
• Alphabet
• DNA (biological coding scheme)
• Musical Score
• Mathematical Equations
Monday, January 11, 2010
Codes
• Two Elements (always)
• A group of symbols
• A set of rules for interpreting these symbols
• Place-Value System (common)
• Information varies based on location
• Notes of a musical staff
• Bits in a binary number
Monday, January 11, 2010
Computers and Codes
• Computers are built from two-state electronics.
• Each memory location (electrical, magnetic, optic) has two states (On or Off)
• Computers must represent all information using only two symbols
• On (1)
• Off (0)
• Computers rely on binary coding schemesMonday, January 11, 2010
Quantum computing? Tertiary-state machines? Maybe, but not yet and/or not common. Probably coming, though.
Computers and Binary
• Decimal
109
• Binary
11011012
• Computer 16-bit word size
0000 0000 0110 11012
• Computer 32-bit word size
0000 0000 0000 0000 0000 0000 0110 11012
Monday, January 11, 2010
Many modern computers now use a 64-bit word size in their CPUs. 231 ~ 4 billion. 263 ~ 9.2 quintillion (~1019). Also IP address sizes (v4 and v6).
Binary Addition
• 4502 + 1234
• 10001100101102 + 100110100102
• 10110011010002
• 5736
1 1 1 1 1
10001100101102
00100110100102
10110011010002
Monday, January 11, 2010
Negative Numbers
Sign-Magnitude Format
• Uses the highest order bit as a ‘sign’ bit. All other bits are used to store the absolute value (magnitude)
• Negative numbers have the sign bit set
• Reduces the range of values that can be stored.
• -10910 in 16-bit representation using a sign bit 1000 0000 0110 11012
Monday, January 11, 2010
Negative Numbers
One’s Complement Format
• Exact opposite of the sequence for the positive value. Each bit is “complemented” or “flipped”
• 109 in 16-bit representation0000 0000 0110 11012
• -109 in 16-bit One’s Complement1111 1111 1001 00102
• This makes mathematical operations difficult
Monday, January 11, 2010
Negative Numbers
Two’s Complement Format
• Add 1 to One’s Complement
• 109 in 16-bit representation0000 0000 0110 11012
• -109 in 16-bit One’s Complement1111 1111 1001 00102
• -109 in 16-bit Two’s Complement1111 1111 1001 00112
Monday, January 11, 2010
Negative Numbers
• -109 in 16-bit One’s Complement
1111 1111 1001 00102
• -109 in 16-bit Two’s Complement
1111 1111 1001 00102
0000 0000 0000 00012
1111 1111 1001 00112
+
Monday, January 11, 2010
• 109 in 16-bit representation
0000 0000 0110 11012
• -109 in 16-bit Two’s Complement:
• Copy everything left of the first ‘1’ including the first ‘1’0000 0000 0110 11012
• Complement (flip) all other bits
1111 1111 1001 00112
Two’s Complement Shortcut
Monday, January 11, 2010
• 10656 in 16-bit representation
0010 1001 1010 00002
• -10656 in 16-bit Two’s Complement:
• Copy everything left of the first ‘1’ including the first ‘1’0010 1001 1010 00002
• Complement (flip) all other bits
1101 0110 0110 00002
Two’s Complement Shortcut
Monday, January 11, 2010
Two’s Complement Arithmetic
• 4502 + (-1234)
Convert to binary, 16-bit representations
0001 0001 1001 01102
1111 1011 0010 11102
10000 1100 1100 01002
450210
-123410 +
• This is 17 bits – the highest order bit simply gets dropped!
00001100110001002 = 326810
Monday, January 11, 2010
0 0 0 1 0 0 0 1 1 0 0 1 0 1 1 0
1 1 1 1 1 0 1 1 0 0 1 0 1 1 1 0
0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0
1
Input
Output
450210
-123410
326810
Monday, January 11, 2010
Imagine that you are doing this operation in a silicon chip with only 16 pins on each side. The 17th “bit” has nowhere to go! In actuality, the input would be preceded by an “add” command.
Monday, January 11, 2010
Credits - CC LicensedDecimal numbers on shoji http://www.flickr.com/photos/pitmanra/1184492148/
Hands http://www.flickr.com/photos/faraz27989/537051865/
Clock http://www.flickr.com/photos/zoutedrop/2317065892/
Credits - Fair UseLinear A tablet http://www.historywiz.com/images/greece/lineara.jpg
T-shirt http://www.thinkgeek.com/tshirts-apparel/unisex/frustrations/5aa9/zoom/
8086 chip http://dl.maximumpc.com/galleries/x86/8086.png
Monday, January 11, 2010
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