number systems cpsc125

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.



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)



There is an obvious reason to use base 10.


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


• 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


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




time

voltage threshold“On”

“Off”




time


“Off”

10




time


“Off”

10

True

False



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)


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


• 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


Base Conversion

Binary to Decimal

• 110102

• (1*24)+(1*23)+(0*22)+(1*21)+(0*20)

• 16 + 8 + 0 + 2 + 0

• 26


Base Conversion

Octal to Decimal

• 15238

• (1*83)+(5*82)+(2*81)+(3*80)

• 512+320+16+3

• 851


Base Conversion

Hexadecimal to Decimal

• B65F16

• (11 * 163) + (6 * 162) + (5 * 161) + (15 * 160)

• 45056+1536+80+15

• 46687


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


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


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


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


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


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


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


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


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


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


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


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


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


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

+


• 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


• 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


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


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


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.

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


http://www.flickr.com/photos/pitmanra/1184492148/

http://www.flickr.com/photos/pitmanra/1184492148/

http://www.flickr.com/photos/faraz27989/537051865/

http://www.flickr.com/photos/faraz27989/537051865/

http://www.flickr.com/photos/zoutedrop/2317065892/

http://www.flickr.com/photos/zoutedrop/2317065892/

http://www.historywiz.com/images/greece/lineara.jpg

http://www.historywiz.com/images/greece/lineara.jpg

http://www.thinkgeek.com/tshirts-apparel/unisex/frustrations/5aa9/zoom/

http://www.thinkgeek.com/tshirts-apparel/unisex/frustrations/5aa9/zoom/

http://dl.maximumpc.com/galleries/x86/8086.png

http://dl.maximumpc.com/galleries/x86/8086.png

number systems cpsc125

Education

base conversion binary

binary notation base

hexadecimal binary

magnetic storage systems

optical storage systems

binary numbers

base conversion decimal

hexadecimal notation