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Chapter 1

Chapter 1. Digital Computers and Information. Digital Computer Basics. Digital values represented as voltage values, e.g. Logic 1 is 4.0-5.0 V Logic 0 is 0.0-1.0 V Why digital? Why binary? Simplest Cheapest. A Generic Computer. Memory = RAM, caches I/O = keyboard, terminal, hard disk

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Chapter 1

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  1. Chapter 1 Digital Computers and Information

  2. Digital Computer Basics • Digital values represented as voltage values, e.g. • Logic 1 is 4.0-5.0 V • Logic 0 is 0.0-1.0 V • Why digital? Why binary? • Simplest • Cheapest CSC 480 – Winter 2002

  3. A Generic Computer • Memory = RAM, caches • I/O = keyboard, terminal, hard disk • CPU • Datapath= performs basic instructions • CPU = supervisor, flow • MMU = memory management • FPU = floating point specialist CSC 480 – Winter 2002

  4. Number Systems • Radix r number represented as: • An-1 ... A1 A0 . A-1 A-2 ... A-m • Value is An-1rn-1 + An-2rn-2 + ... A1r1 + A0r0 + A-1r-1 + ... A–mr-m • Digit range: 0 - (r-1) • An-1 is the “most significant digit” • A-m is the “least significant digit” • Notation: (6342)7 – use parens and subscript to indicate number base, in this case 7 • 4 common bases: 2 (binary), 8 (octal), 10 (decimal), 16 (hex) CSC 480 – Winter 2002

  5. Base R Wishlist We want to be able to “handle” radix R numbers, including: • Decimal shortcut - Convert to decimal a number in any radix • Power of 2 shortcut - Power of 2 conversions between binary, octal, and hex numbers • General radix conversion - convert a number in radix R1 to radix R2 • We also want to be able to perform the standard math operations CSC 480 – Winter 2002

  6. Convert to Decimal • We use decimal (r=10), so conversion from other bases to decimal is common... • Ex: Convert (43.2)5 to decimal • Trick: Just expand each digit. We know how to manipulate decimal numbers which makes this process “easy”. • Ans: 4 * 5 + 3 * 1 + 2 * .2 = 23.4 • Note 1: Conversion from decimal is handled by standard technique, no shortcut • Note 2: We are lucky. We can easily convert to decimal to check our work! CSC 480 – Winter 2002

  7. Convert to Decimal, cont • There is another way (shortcut?) called Horner’s rule (not in the text except as a homework problem W = an-1 for i=n-2 to 0 by –1 w = w*r + ai • Ex: Convert (2504)6 to decimal 2 * 6 + 5 = 17 17 * 6 + 0 = 102 102 * 6 + 4 = 616 • This is actually more efficient as well CSC 480 – Winter 2002

  8. The trick(s) Base 8 (octal) is Base 2 “cubed” 23 = 8 So, 3 bits equals one octal digit Base 16 (hex) is Base 2 to the 4th power 24 = 16 So, 4 bits equals one hex digit Convert to Binary Ex: (743.2)8 Expand each octal digit into its 3-bit value Ans: (111 100 010. 010)2 Convert from Binary Ex: (10010101)2 to hex Trick: Collapse each 4 bit group into its hex value Ans: (95)16 Power of 2 Conversions CSC 480 – Winter 2002

  9. General Conversion, cont. • Don’t forget fractions • Handle fractional parts separately • Similar process as integer part, except multiply and use whole numbers, not remainders • Ex. (0.625)10 to binary 0.625 * 2 = 1.25, use 1 0.25 * 2 = 0.5, use 0 0.5 * 2 = 1.0, use 1 Answer is whole numbers in order: (101)2 Again, you can check work in decimal. CSC 480 – Winter 2002

  10. General Number Conversion • Trick: To convert to radix R, successively divide your number by R and the remainders are your converted value. • Ex: Convert (169)10 to octal 169 / 8 = 21 R1 21 / 8 = 2 R5 2 / 8 = 0 R2 Answer is reverse of remainders (251)8 Check our work: (251)8 = 2*8*8 + 5*8 + 1 = 169 CSC 480 – Winter 2002

  11. Arithmetic Operations We want to add, subtract and multiply numbers of radix R • Good news: Same basic rules as decimal math • Bad news: Your decimal brain may rebel at the non-decimal carries, borrows, etc. • More good news: Convert to decimal to verify your work CSC 480 – Winter 2002

  12. Adding • Let’s add: (3A)16 + (69)16 • Watch the carries! • Carry value is now worth 16 (the base) • Double check with decimal values CSC 480 – Winter 2002

  13. Subtracting • Let’s subtract: (69)16 - (3A)16 • Watch the borrows! • Borrow value is now worth 16 (the base) • Double check with decimal values CSC 480 – Winter 2002

  14. Multiplying • Let’s multiply: (3A)16 * (69)16 • Labor-intensive, error-prone with multiple steps CSC 480 – Winter 2002

  15. BCD = binary-coded decimal “Human” binary coding of decimal numbers Convert number one digit at a time: Ex: 17 = 0001 0111 BCD addition Add each digit separately Carry is “special” If sum for any digit >= 10, add 6 (0110) to sum to get proper BCD result Ex. 8 + 5 = 1000 + 0101 = 1101 (Add 0110!) = 0001 0011 Binary-coded Decimals CSC 480 – Winter 2002

  16. Alphanumeric codes • ASCII – encoding for alphanumeric data • 7 bits long • Covers a-z, A-Z, 0-9, !@#$ and many other special characters • See table on page 21 for complete mapping • 8th bit often added for parity check • Only for English-language characters CSC 480 – Winter 2002

  17. Chapter 1 Summary • Most important: • Generic computer & information • Understand radix R number systems • Radix conversion (general and shortcuts) • Radix math operations • BCD encoding and addition • ASCII code basics CSC 480 – Winter 2002

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