Lec2 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Number system
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This document discusses number systems and binary arithmetic. It begins by explaining decimal and binary number representation, including place value and how to derive numbers in different bases. It then covers counting in binary, octal, and base-22 systems. Next, it discusses representing negative numbers using sign-magnitude, one's complement, and two's complement methods. Finally, it demonstrates binary addition and computation for both unsigned and signed numbers using two's complement.
Lec2 Intro to Computer Engineering by Hsien-Hsin Sean Lee Georgia Tech -- Number system
- 1. ECE2030 Introduction to Computer Engineering Lecture 2: Number System Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee School of Electrical and Computer EngineeringSchool of Electrical and Computer Engineering Georgia TechGeorgia Tech
- 2. Decimal Number Representation • Example: 90134 (base-10, used by Homo Sapien) = 90000 + 0 + 100 + 30 + 4 = 9*104 + 0*103 + 1*102 + 3*101 + 4*100 • How did we get it? 901349013410 9013901310 44 90190110 33 909010 11 99 00
- 3. Generic Number Representation • 90134 = 9*104 + 0*103 + 1*102 + 3*101 + 4*100 • A4A3A2A1A0 for base-10 (or radix-10) = A4*104 + A3*103 +A2*102 +A1*101 +A0*100 (A is coefficient; b is base) • Generalize for a given number NN w/ base-bb NN = An-1An-2 …A1A0 NN = An-1*bn-1 + An-2*bn-2 + … +A2*b2 +A0*b0 **Note that A < b**Note that A < b
- 4. Counting numbers with base-bb 00 11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 Base-10 9090 9191 9292 9393 9494 9595 9696 9797 9898 9999 ….. 100100 101101 102102 103103 104104 105105 106106 107107 108108 109109 How about Base-8 00 11 22 33 44 55 66 77 1010 1111 1212 1313 1414 1515 1616 1717 2020 2121 2222 2323 2424 2525 2626 2727 7070 7171 7272 7373 7474 7575 7676 7777 ….. 100100 101101 102102 103103 104104 105105 106106 107107 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929
- 7. How about base-22 00 = 0= 0 11 = 1= 1 1010 = 2= 2 1111 = 3= 3 100100 = 4= 4 101101 = 5= 5 110110 = 6= 6 111111 = 7= 7 10001000 = 8= 8 10011001 = 9= 9 10101010 = 10= 10 10111011 = 11= 11 11001100 = 12= 12 11011101 = 13= 13 11101110 = 14= 14 11111111 = 15= 15 BinaryBinary == DecimalDecimal
- 8. Derive Numbers in Base-2 • Decimal (base-10) – (25)10 • Binary (base-2) – (11001)2 • Exercise 25252 12122 11 662 00 332 00 11 11
- 9. Base-2 • Decimal (base-10) – (982)10 • Binary (base-2) – (1111010110)2 • Exercise
- 10. 0 Base 8 • Decimal (base-10) – (982)10 • Octal (base-8) – (1726)8 • Exercise
- 11. 1 Base 16 • Decimal (base-10) – (982)10 • Hexadecimal (base-16) • Hey, what do we do when we count to 10?? • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 00 11 22 33 44 55 66 77 88 99 aa bb cc dd ee ff
- 12. 2 Base 16 • (982)10= (3d6)16 • (3d6)16 can be written as (0011 1101 0110)2 • We use Base-16 (or Hex) a lot in computer world – Ex: A 32-bit address can be written as 0xfe8a7d200xfe8a7d20 ((0x0x is an abbreviation of Hex)) – Or in binary formOr in binary form 1111_1110_1000_1010_0111_1101_0010_00001111_1110_1000_1010_0111_1101_0010_0000
- 13. 3 Number Examples with Different Bases • Decimal (base-10) – (982)10 • Binary (base-2) – (01111010110)2 • Octal (base-8) – (1726)8 • Hexadecimal (base-16) – (3d6)16 • Others examples: – base-9 = (1321)9 – base-11 = (813)11 – base-17 = (36d)17
- 14. 4 Convert between different bases • Convert a number base-x to base-y, e.g. (0100111)2 to (?)6 – First, convert from base-x to base-10 if x ≠ 10 – Then convert from base-10 to base-y 0100111 = 0∗26 + 1∗25 + 0∗24 + 0∗23 + 1∗22 + 1∗21 + 1∗20 = 39 39396 666 33 11 00 ∴ (0100111)2 = (103)6
- 15. Base-b Addition
- 16. 6 Negative Number Representation • Options – Sign-magnitude – One’s Complement – Two’s Complement (we use this in this course)
- 17. 7 Sign-magnitude • Use the most significant bit (MSB) to indicate the sign – 00: positive, 11: negative • Problem – Representing zeros? – Do not work in computation • We will NOT use it in this course ! +0 000 +1 001 +2 010 +3 011 -3 111 -2 110 -1 101 0 100
- 18. 8 One’s Complement • Complement (flip) each bit in a binary number • Problem – Representing zeros? – Do not always work in computation • Ex: 111 + 001 = 000 → Incorrect ! • We will NOT use it in this course ! +0 000 +1 001 +2 010 +3 011 -3 100 -2 101 -1 110 0 111
- 19. 9 Two’s Complement • ComplementComplement (flip) each bit in a binary number and adding 1adding 1, with overflow ignored • Work in computation perfectly • We will use it in this course ! 011 100 One’s complement 3 101 Add 1 -3 010 One’s complement 101-3 011 Add 1 3
- 20. 0 Two’s Complement • ComplementComplement (flip) each bit in a binary number and adding 1adding 1, with overflow ignored • Work in computation perfectly • We will use it in this course ! 0 000 +1 001 -1 111 +2 010 -2 110 +3 011 -3 101 ?? 100 100 011 One’s complement 100 Add 1 The same 100 represents both 4 and -4 which is no good
- 21. 1 Two’s Complement • ComplementComplement (flip) each bit in a binary number and adding 1adding 1, with overflow ignored • Work in computation perfectly • We will use it in this course ! 0 000 +1 001 -1 1111 +2 010 -2 1110 +3 011 -3 1101 --4 1100 100 011 One’s complement 100 Add 1 MSB = 1 for negative Number, thus 100 represents -4
- 22. 2 Range of Numbers • An N-bit number – Unsigned: 0 .. (2 N -1) – Signed: -2 N-1 .. (2 N-1 -1) • Example: 4-bit 1110 (-8) 0111 (7) Signed numbers 0000 (0) 1111 (15)Unsigned numbers
- 23. 3 Binary Computation 010001 (17=16+1) 001011 (11=8+2+1) --------------- 011100 (28=16+8+4) Unsigned arithmetic 010001 (17=16+1) 101011 (43=32+8+2+1) --------------- 111100 (60=32+16+8+4) Signed arithmetic (w/ 2’s complement) 010001 (17=16+1) 101011 (-21: 2’s complement=010101=21) --------------- 111100 (2’s complement=000100=4, i.e. -4)
- 24. 4 Binary Computation Unsigned arithmetic 101111 (47) 011111 (31) --------------- 001110 (78?? Due to overflow, note that 62 cannot be represented by a 6-bit unsigned number) The carry is discarded Signed arithmetic (w/ 2’s complement) 101111 (-17 since 2’s complement=010001) 011111 (31) --------------- 001110 (14) The carry is discarded
- 25. BACKUP
- 26. 6 Application of Two’s Complement • The first Pocket CalculatorPocket Calculator “Curta” used Two’s complement method for subtractionsubtraction • First complement the subtrahend – Fill the left digits to be the same length of the minuend – Complemented number = (9 – digit) • 4’s complement = 5 • 7’s complement = 2 • 0’s complement = 9 • Add 1 to the complemented number • Perform an addition with the minuend
- 27. 7 Examples • 13 – 7 – Two’s complement of 07 = 92 + 1 = 93 – 13 + 93 = 06 (ignore the leftmost carry digit) • 817 – 123 – Two’s complement of 123 = 876 + 1 = 877 – 817 + 877 = 694 (ignore the leftmost carry digit) • 78291 – 4982 – Two’s complement of 04982 = 95017 + 1 = 95018 – 78291 + 95018 = 73309 (ignore the leftmost carry digit)