Number Systems

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]

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

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]

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]

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

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

Number Systems

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Number Systems