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Unit 4: The Number Systems and Data Representation
Problem-solving using “computers” ,[object Object],A Problem Describing  The Problem in Math. “ Computing” The  Corresponding Math.  Problem Returning The Result 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 ,[object Object],[object Object],[object Object],…which counts 10 states
The Needs of Number Systems ,[object Object],[object Object],[object Object],[object Object],[object Object]
 
Egyptian numerals (10-based) Source: http://www-gap.dcs.st-and.ac.uk/~history
Babylonians numerals (60-based) Source: http://www-gap.dcs.st-and.ac.uk/~history
1*60 3  + 57*60 2  + 46*60 + 40*60 0  =  424000 Source: http://www-gap.dcs.st-and.ac.uk/~history
Chinese numerals (10-based) 4359 45698 Source: http://www-gap.dcs.st-and.ac.uk/~history 壹貳參肆伍陸柒捌玖拾佰仟萬億兆京…
The Number Systems ,[object Object],[object Object],[object Object],[object Object]
K-based Number System ,[object Object],[object Object],[object Object],[object Object],[object Object]
K-based Number System ,[object Object],[object Object],State-1 State-2 State-3 State-4 State-5 State-6 State-7 State-8 State-9 State-10 State-11 State-12 State-13 State-14 State-15 State-16
K-based Number System ,[object Object],[object Object],Why not “ 1 0 ” 2 digits
More States ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
More States ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
The Number of Cases ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
K-based vs The Decimal system (10-based) ,[object Object],[object Object],[object Object],[object Object],[object Object]
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Using octal and hex numbers ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
The binary number system ,[object Object],[object Object],[object Object],01111 00011 10010 15 3 + 10 + 2 18
The binary number system ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Data Processing in Computers with the Binary System ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Part I: Numeric Data ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Sign-Magnitude ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
1’s Complement ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
2’s Complement ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Comparison: 4-bit representation of integers
Range of integers Suppose that we use a byte (8-bit) to represent an integer (00000000)  2 (00000000)  2 (10000000)  2 =-128 (01111111)  2 =+127 2‘s complement (11111111)  2 (00000000)  2 (10000000)  2 =-127 (01111111) 2 =+127 1‘s Complement (10000000)  2 (00000000)  2 (11111111)  2 =-127 (01111111) 2 =+127 Sign-Magnitude -0 +0 Max. (-) Max. (+) -(2 n-1 )~0 0~2 n-1 -1 2‘s complement -(2 n-1 -1)~0 0~2 n-1 -1 1‘s Complement -(2 n-1 -1)~0 0~2 n-1 -1 Sign-Magnitude Negative numbers Positive numbers
Note ,[object Object],[object Object]
Other Binary Codes for Decimal Numbers ,[object Object],[object Object],[object Object],[object Object],[object Object]
8421 BCD Code ,[object Object],[object Object],[object Object]
4221 BCD Code ,[object Object],[object Object],advantages
Excess-3 Code ,[object Object],[object Object],[object Object],[object Object],[object Object]
84-2-1 Code ,[object Object],[object Object],[object Object],[object Object]
Self-complementing code Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Gray Codes ,[object Object],[object Object],[object Object],[object Object]
Gray Codes ,[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
從反射葛雷碼 -> 十進位數字 Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Floating-Point Representation ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Floating-point literals ,[object Object],[object Object],[object Object],http://www.nuvisionmiami.com/books/asm/workbook/floating_tut.htm
[object Object],Source: 計算機概論 ,  王孝熙 著 ,  東華書局 .
Example
Binary Arithmetic on Numeric Data ,[object Object],[object Object],[object Object],[object Object],[object Object]
Part II: Alphanumeric Data ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
ASCII Code ,[object Object]
Extended ACSII 8-bit for each character, 2 8 =256 combinations for 256 characters 。
Examples N: 4E=00101110  a:  61=01100001 t:  74=01110100 i:  69=01101001 o:  6F=01101111  n:  6E=01101110 a:  61=01100001 l:  6C=01101010
EBCDIC ,[object Object],[object Object],[object Object],http://www.natural-innovations.com/computing/asciiebcdic.html
 
Big5 Code ,[object Object],[object Object],[object Object]
 
Unicode ,[object Object],[object Object],[object Object]
Unicode samples
Binary Operations on Alphanumeric Data ,[object Object],[object Object],[object Object],[object Object],[object Object]
Part III: Extensions ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Extension: Image representation ,[object Object],[object Object],[object Object],Original  7*7 digitizing 14*14 digitizing
For Black/White Images Original  7*7 digitizing 14*14 digitizing 0000000 1111100 … 00000000000000 00000000000000 01111111110000 01111111111000 01111111111110 … 0000 0001 1111 00 … 0000 0000 0000 0000 0000 0000 0000 0111 1111 1100 0001 11111111100001111111111110… (01F0..) 2 (00000007FA1..) 2
For Colored Images
Extension: Sound Representation time AMP
Extension: Sampling Max. AMP/16 t 1 t 2 t 3 t 4 … 16=2 4 t 1 : 8=(1000) 2 t 2 : 9=(1001) 2 t 3 : 7=(0111) 2 t 4 : 2=(0010) 2 t 1 : 2=(0010) 2 … Quality vs sampling rate
Conclusion

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Number Systems