Number Systems

112/04/12 Copyright: [email protected] 1

Unit 4: The Number Systems and Data Representation


Problem-solving using “computers”

• Computers solve “computable” problems

A ProblemA ProblemDescribing

The Problemin Math.

Describing The Problem

in Math.

“Computing”The

CorrespondingMath.

Problem

“Computing”The

CorrespondingMath.

Problem

ReturningThe Result

ReturningThe Result

Solution To The Problem

Solution To The Problem

Human problem-solving v.s. computer-based problem-solving


Transformation

input

think

act

input

compute

act

Binary system


The Number System

• A number system is to enumerate the “states” of something– For example, money, days, month, year,

minutes, hours, …

• In human’s world, we have very good tools

…which counts 10 states


The Needs of Number Systems

• However, there are some states are not Decimal. For example,– Days and weeks– Minutes and hours– Months and year

• We need convenient representations for different “nature states”


http://images.google.com/imgres?imgurl=http://www.asu.edu/cfa/art/gallery/stepgal/sites/rock%2520no.9.jpg&imgrefurl=http://www.asu.edu/cfa/art/gallery/stepgal/sites/&h=240&w=320&sz=18&tbnid=zl1sIOPV3q0J:&tbnh=84&tbnw=112&start=17&prev=/images%3Fq%3Drock%26hl%3Dzh-TW%26lr%3D%26ie%3DUTF-8





http://images.google.com/imgres?imgurl=http://www.china-window.com.cn/wenwu/xihan/images/5-2-13.jpg&imgrefurl=http://www.china-window.com.cn/wenwu/xihan/2-13-1.htm&h=388&w=214&sz=102&tbnid=M3abmDBUgy4J:&tbnh=117&tbnw=65&start=13&prev=/images%3Fq%3D%25E9%258A%2585%25E9%258C%25A2%26hl%3Dzh-TW%26lr%3D%26ie%3DUTF-8

http://images.google.com/imgres?imgurl=http://www.china-window.com.cn/wenwu/tangwudai/images/7-4-10.jpg&imgrefurl=http://www.china-window.com.cn/wenwu/tangwudai/4-10.htm&h=217&w=226&sz=23&tbnid=PYwTIKb2IN0J:&tbnh=98&tbnw=102&start=14&prev=/images%3Fq%3D%25E9%258A%2585%25E9%258C%25A2%26hl%3Dzh-TW%26lr%3D%26ie%3DUTF-8





Egyptian numerals (10-based)

Source: http://www-gap.dcs.st-and.ac.uk/~history


Babylonians numerals (60-based)



1*603 + 57*602 + 46*60 + 40*600 = 424000



Chinese numerals (10-based)

4359

45698


壹貳參肆伍陸柒捌玖拾佰仟萬億兆京…


The Number Systems

• A number system specifies how/what “numbers” represent and operate.

• A K-based number system uses N different symbols to represent N different entities or “states”

• Each symbol is a “digit”

• Multiple digits are used for describing M>N states


K-based Number System

• 2-based: ON/OFF; Yes/No; T/F; 1/0; etc.

• 3-based: A/B/C; True/False/Unknown; 0/1/2; etc.

• …

• 7-based: Mon/Tue/Wes/Thu/Fri/Sat/Sun; 0/1/2/3/4/5/6; etc.

• 8-based: ; 0/1/2/3/4/5/6/7

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K-based Number System

• 10-based: /////////; 0/1/2/3/4/5/6/7/8/9; etc.

• 16-based: ///////////; 0/1/2/3/4/5/6/7/8/9/A/B/C/D/E/F; etc.

State-1State-2

State-3State-4

State-5State-6

State-7State-8

State-9State-10

State-11State-12

State-13State-14

State-15State-16


K-based Number SystemIn the 16-based number system:

0/1/2/3/4/5/6/7/8/9/A/B/C/D/E/F; etc.

Why not “10” 2 digits


More States

In a 6-based number system : state 1 (empty state) (0) : state 2 (1st) : state 3 (2nd) : state 4 (3rd) : state 5 (4th) : state 6 (5th)

: state 7 (6th) : state 8 (7th) : state 9 (8th) : state 10 (9th) : state 11 (10th) : state 12 (13th)

• …


More States

In a 6-based number system

• 0: state 1 (empty state) (0)

• 1: state 2 (1st)

• 2: state 3 (2nd)

• 3: state 4 (3rd)

• 4: state 5 (4th)

• 5: state 6 (5th)

• 10: state 7 (6th)

• 11: state 8 (7th)

• 12: state 9 (8th)

• 13: state 10 (9th)

• 14: state 11 (10th)

• 15: state 12 (13th)

• 20: state 13 (14th)

• …


The Number of Cases

• Generally, in a K-based number system,– 1 digit describes K1 cases– 2 digits describes K2 cases– 3 digits describes K3 cases– N digits describes KN cases

• For example,– 2 3-based (0,1,2) digits: 00, 01, 02, 10, 11, 12, 20, 21,

22 (32=9 cases)– 3 10-based (0,1,2,..,9) digits: 000, 001, 002, 003, 004,

005, 006,007, 008, 009, 010, 011,…,998, 999 (103=1000 cases)


K-based vs The Decimal system (10-based)

• In general, any K-based number N can be expressed as

• Usually, Nk is denoted as – (Ap-1Ap-2….A1A0.A-1A-2….A-q)k – Ap-1:Most Significant Digit, MSD– A-q :Least Significant Digit, LSD

112/04/12 Copyright: [email protected] 19Source: 計算機概論 , 王孝熙著 , 東華書局 .


Using octal and hex numbers

• Computers use binary, but the numbers are too long and confusing for people

• Translation between binary and octal or hex is easy

– One octal digit equals three binary digits 101101011100101000001011 5 5 3 4 5 0 1 3– One hexadecimal digit equals four binary digits 101101011100101000001011 B 5 C A 0 B








The binary number system

• In modem computers, the binary number system is easier to be implemented.

• Currently, they are implemented in silicon chips (VLSI)– Circuits for computation

01111

0001110010

15

3+10

+2 18


The binary number system

• The binary (base 2 or 2-based) number system uses two “binary digits, ” (abbreviation: bits) -- 0 and 1

• A bit is a single two-valued quantity: yes or no, true or false, on or off, high or low, good or bad

• One bit can distinguish between two cases: T, F (True/False; Yes/No)

• Two bits can distinguish between four cases: TT, TF, FT, FF• Three bits can distinguish between eight cases: TTT, TTF,

TFT, TFF, FTT, FTF, FFT, FFF• In general, n bits can distinguish between 2n cases• A byte is 8 bits, therefore 28 = 256 cases


Data Processing in Computers with the Binary System

• Data processing in binary computers– Numeric data: +2, -5.5, 50000.235698, etc.– Alphanumeric data: characters, name, address,

telephone number, etc.

• Two main tasks: – All data are converted into proper binary

formats (bits)– Rules for counting and computing these bits


Part I: Numeric Data• Numeric data:

– integers, real – positive, negative

• Representation of numeric data– Sign-magnitude– 1’s Complement – 2’s Complement

• Negative/Positive in a n-bit integer I

I=(An-1An-2….A1A0)2

– An-1= “0”, I is positive– An-1=“1”, I is negative


Sign-Magnitude• An n-bit integer I=

– Min: 00000…0 (n-1 0’s)=0– Max: 11111…1 (n-1 1’s)=2n-1-1– +0= (0000)2 vs –0=(1000)2

– +3= (0011)2 vs –3=(1011)2

• Disadvantages– 2 different 0’s (+0/-0 ）– Not easy to be implemented in simple circuits (usually ad

der)

(An-1An-2….A1A0)2

sign:0(+), 1(-) magnitude


1’s Complement• An n-bit integer I=

– For positive integers: the same as that in sign-magnitude

– For negative integers: complement their positive representations

– For example: 4-bit integers: • +3=(0011)2， -3=(1100)2

• +0=(0000)2， -0=(1111)2

• Disadvantages– 2 different 0’s (+0/-0）– Easy to be implemented in simple circuits (usually adder) but not

as efficient as 2’s complement

(An-1An-2….A1A0)2



2’s Complement• An n-bit integer I=

– For positive integers: the same as that in sign-magnitude

– For negative integers: find their 1’s complement + 1• For example: 4-bit integers:

• +3 = (0011)2 ， -3 = (1100)2+1 = (1101)2

• +0 = (0000)2， -0 = (1111)2 + 1 = (0000)2

• There is only one “0”

(An-1An-2….A1A0)2



Comparison: 4-bit representation of integers


Range of integersSuppose that we use a byte (8-bit) to represent an integer

Max. (+) Max. (-) +0 -0

Sign-Magnitude

(01111111)2=+127 (11111111) 2=-127 (00000000) 2 (10000000) 2

1‘s Complement

(01111111)2=+127 (10000000) 2=-127 (00000000) 2 (11111111) 2

2‘s complement

(01111111) 2=+127 (10000000) 2=-128 (00000000) 2 (00000000) 2

Positive numbers Negative numbers

Sign-Magnitude 0~2n-1-1 -(2n-1-1)~01‘s Complement 0~2n-1-1 -(2n-1-1)~02‘s complement 0~2n-1-1 -(2n-1)~0


Note

• Representation level

• Implementation level


Other Binary Codes for Decimal Numbers

• BCD code

• 2421 code

• Excess-3 code

• 84-2-1 code

• 4-bit for a number (0~9)


8421 BCD Code• In the 8421 Binary Coded Decimal (BCD) representation

each decimal digit is converted to its 4-bit pure binary equivalent.

• Each decimal number maps to four bits and is weighted by its bit-position (each bit represents a number, 1, 2, 4, 8)

• BCD code is also called as 8421 code


4221 BCD Code

• In 4221 BCD code, each bit is weighted by 4, 2, 2 and 1 respectively.

• Unlike BCD coding there are no invalid representations.

Decimal 42211's

complement0 0000 11111 0001 11102 0010 11013 0011 11004 1000 01115 0111 10006 1100 00117 1101 00108 1110 00019 1111 0000

advantages


Excess-3 Code• Add 3 for each binary number

• Examples– Excess-3 code of “210” = (0010)2 +(0011)2

=(0101)2

– Excess-3 code of “510” = (0101)2+(0011)2=(1000)2

• Advantages


84-2-1 Code• 4-bit for each number (0~9) weighted (left to righ

t) as 8, 4, -2, and -1 。• Example

– 84-2-1 code of “3” = 0101 (0+4+0+(-1)=3)

– 84-2-1 code of “5” =1011 (8+0+(-2)+(-1)=5)


Self-complementing code

Source: 計算機概論 , 王孝熙著 , 東華書局 .


Gray Codes

• In pure binary coding or 8421 BCD, counting from 7 (0111) to 8 (1000) requires 4 bits to be changed simultaneously. If this does not happen then various numbers could be momentarily generated during the transition so creating spurious numbers which could be read.

• In gray coding, only one bit changes between subsequent numbers.

• Generating gray codes: start with all 0s and then proceed by changing the least significant bit (LSB) which will bring about a new state.

• Advantages: fast, relatively free from errors.


Gray Codes

• Gray code is not unique, there are many possibilities to generate gray codes

• For example, – G1={0=00， 1=01， 2=11， 3=10}

– G2={00=10， 1=11， 2=01， 3=00}

• Unique gray code: reflected Gray code


從十進位 → 反射葛雷碼



從反射葛雷碼 → 十進位數字



Floating-Point Representation

• Integers are fixed-point numbers in binary computers• Floating-point literals are written with a decimal point:

8.5 -7.923 5.000 • Real numbers are represented as “floating-point” numbers

in binary system (all computers)• The representation of floating-point numbers are various in

different CPU• In Intel 80486 CPU

– Single Precision: 32 bits– Double Precision: 64 bits– Extended Precision: 80bits


Floating-point literals

• Floating-point numbers may also be written in “scientific notation”– times a power of 10

• We use E to represent “times 10 to the”• Example: 4.32E5 means 4.32 x 105

http://www.nuvisionmiami.com/books/asm/workbook/floating_tut.htm


• 範例



Example

Binary ValueBiased

Exponent Sign, Exponent, Mantissa

- 1.11 127 1 01111111 11000000000000000000000

+1101.101 130 0 10000010 10110100000000000000000

- .00101 124 1 01111100 01000000000000000000000

+100111.0 132 0 10000100 00111000000000000000000

+.0000001101011 120 0 01111000 10101100000000000000000


Binary Arithmetic on Numeric Data

• Binary addition

• Binary subtraction

• Binary multiplication

• Binary division

To be discussed in Unit 5


Part II: Alphanumeric Data• Alphanumeric data: character, letter, symbol, digit)• Not for calculation, but for representation some “mea

nings”• Coding

– ASCII(as-kee):America Standard Code for Information Interchange

– EBCDIC(eb-ce-dick):Extended Binary Coded Decimal Interchange Code (used by IBM, UNIVAC mainframes)

– BIG-5: for Chinese characters

– Uni-code


ASCII Code• 7-bit for each character, 27=128 combinations for 128 characters


Extended ACSII8-bit for each character, 28=256 combinations for 256 characters 。


Examples

N: 4E=00101110 a: 61=01100001t: 74=01110100i: 69=01101001o: 6F=01101111 n: 6E=01101110a: 61=01100001l: 6C=01101010


EBCDIC

• 8-bit for each character – 4-bit Zone bits: identifying the code is for char

acter, (un)signed number, or symbols– 4-bit Digit bits: for numbers 0~9 。

http://www.natural-innovations.com/computing/asciiebcdic.html



Big5 Code

• Used for Chinese characters

• 1 character = 2 bytes (16 bits)

• Example



Unicode

• ASCII was very simplistic, and so was extended by adding 'extended' sets by various manufacturers. Apart from being confusing this was still restricted to 256 characters.

• Now computers are more widely established around the world the need to show other characters such as Japanese and Chinese languages along with various symbols became more important.

• 2 bytes (16-bit) for each unicode


Unicode samples


Binary Operations on Alphanumeric Data

• Binary addition?

• Binary subtraction?

• Binary multiplication?

• Binary division?

To be discussed in Unit 5


Part III: Extensions

• Everything in the computer is stored as a pattern of bits– Binary distinctions are easy for hardware to work with

• Numbers are stored as a pattern of bits– Computers use the binary number system

• Characters are stored as a pattern of bits– One byte (8 bits) can represent one of 256 characters

• So, is everything in the computer stored as a number?– No it isn’t, it’s stored as a bit pattern

– There are many ways to interpret a bit pattern


Extension: Image representation

• Nature creations are analog, not digital• Digitizing (sampling)• Resolution: 800*600, 1026*768

Original 7*7 digitizing 14*14 digitizing


For Black/White Images

Original 7*7 digitizing 14*14 digitizing

00000001111100…

0000000000000000000000000000011111111100000111111111100001111111111110…

00000001111100…

0000000000000000000000000000011111111100000111111111100001111111111110…

(01F0..)2

(00000007FA1..)2


For Colored Images


Extension: Sound Representation

time

AMP


Extension: Sampling

Max. AMP/16

t1 t2 t3 t4 …

16=24

t1: 8=(1000)2

t2: 9=(1001)2

t3: 7=(0111)2

t4: 2=(0010)2

t1: 2=(0010)2

…

Quality vs sampling rate


Conclusion

Technology

Number Systems