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Number Bases

Number Bases. Informatics INFO I101 February 9, 2004 John C. Paolillo, Instructor. Items for Today. Last week Digital logic, Boolean algebra, and circuits Logic gates and truth tables This Week Numbers and bases Working with binary. Number Base Systems. …. …. b 4. #. b 3. #. b 2.

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Number Bases

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  1. Number Bases Informatics INFO I101 February 9, 2004 John C. Paolillo, Instructor

  2. Items for Today • Last week • Digital logic, Boolean algebra, and circuits • Logic gates and truth tables • This Week • Numbers and bases • Working with binary

  3. Number Base Systems

  4. … … b4 # b3 # b2 # b1 # b0 # b-1 # b-2 # b-3 # b-4 # b-5 # … … The Format of a Base System The number represented is the sum of all the products of the digit values and their respective place values

  5. Common Bases • Decimal (Base 10) • Binary (Base 2) • Octal (Base 8) • Hexadecimal (Base 16)

  6. Conversion to Base 10 • Identify each of the places in the new number base. These will correspond to the powers of the base, for example, with base 2, they are 1, 2, 4, 8, 16, 32, etc. • Multiply the value for each place by the value of the digit appearing there; • Add the results up, and you have the result in decimal Note that if you divide and add correctly, you can reverse this procedure to convert decimal into another base. It’s harder, because you’re not used to using the appropriate addition and multiplication tables.

  7. Try out these examples • What is 10011 Base 2 in decimal? 116+ 08 + 04 + 12 + 11 = 19 • What is 121 Base 8 in decimal? 164 + 28 + 11 = 81 • What is 247 Base 10 in Binary? Here it helps to have a different procedure…

  8. 256 128 64 32 16 8 4 2 1 0 1 1 1 1 0 1 1 1 28 27 26 25 24 23 22 21 20 128 256 16 64 32 2 4 8 1 247 247 119 55 23 1 7 3 7 0 1 0 1 1 1 1 1 1 Converting to Binary What we’re converting

  9. Octal — base 8 • Sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7 7 = 111two = 7eight • Octal values are usually not specially indicated Unix example: chmod 666 myfile.html

  10. Octal Digits Decimal 0 1 2 3 4 5 6 7 Binary 000 001 010 011 100 101 110 111 Octal 0 1 2 3 4 5 6 7

  11. Octal Tips • each octal digit corresponds to three binary digits (bits) • convert binary to octal by parsing each group of three bits into one octal digit • convert octal to binary by translating each digit into three bits • Examples: 764eight = 111101100two 011011101two = 335eight

  12. Hexadecimal — base 16 • Sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 15 = 1111two = Fsixteen • Hexadecimal (“hex”) values are usually indicated by a preceding base marker HTML: #FFFFFF JavaScript, C: 0xF1AD

  13. Hexadecimal Digits Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Hex 0 1 2 3 4 5 6 7 8 9 A B C D E F

  14. Hexadecimal Tips • each hex digit corresponds to four binary digits (bits) • convert binary to hex by parsing each group of four bits into one hex digit • convert hex to binary by translating each digit into four bits • Two hex digits make up one byte, a very common unit of memory

  15. IP Numbers IP = “Internet Protocol” IP Number 129.79.142.114 Host Name dhcp-Memorial–142-114.memorial.indiana.edu A Domain Name Server (DNS) has a database that matches IP and host name

  16. Net Subnet Node The IP Number • Four Fields • 0-255 in each field • This is really base 256, but we use decimal numbers in each digit 129.79.142.114

  17. Binary Addition

  18. Adding in Binary • Zero plus any other values leaves that value Identity value for addition No carry is generated • One plus one leaves zero and causes a carry (one) to the next digit Each successive digit must accept the carry from the previous

  19. 01001101 Base 2 + 00011011 Base 2 01111111 Base 2 + 00000001 Base 2 102 Base 8 + 121 Base 8 01001101 Base 2  00000100 Base 2 0000101 Base 2  0000101 Base 2 01111111 Base 2  00011011 Base 2 Try these calculations

  20. CI 0 0 0 0 1 1 1 1 A 0 1 0 1 0 1 0 1 B 0 0 1 1 0 0 1 1 S CO Addition: Truth Tables S 0 1 1 0 1 0 0 1 CO 0 0 0 1 0 1 1 1

  21. S  + XOR 00 0 0 1 1 01 0 0 00 0 00 0 0 01 1 1 01 1 01 0 1 1 0 10 B A Addition: Half Adder C The half adder sends a carry, but can’t accept one So we need another for the carry bit

  22. Addition in Binary • Two half-adders gives us a full adder • two inputs plus carry • Adders are cascaded to permit adding binary numbers • Eight adders allows adding (0...256) + (0...256) in binary numbers • Overflow can happen (200 + 100) • Binary adders are used to do other computations as well...

  23. Subtraction Complement Representations

  24. – 00 01 00 00 01 01 –01 00 Subtraction • Subtraction is asymmetrical • That makes it harder • We have to borrow sometimes  827 Minuend –223 Subtrahend =604 Difference/remainder

  25. 456 –123 333 999 –123 876 When Subtraction is Easy • Subtraction is easy if you don’t have to borrow • i.e. if all the digits of the minuend are greater than (or equal to) all those of the subtrahend • This will always be true if the minuend is all 9’s: 999, or 999999, or 9999999999 etc.

  26. Using Easy Subtraction • Subtract the subtrahend from 999 (or whatever we need) (easy) • Add the result to the minuend (ordinary addition) • Add 1 (easy) • Subtract 1000 (drop highest digit) Difference = Minuend + 999 – Subtrahend + 1 – 1000 This works for binary as well as decimal

  27. Easy Subtraction in Binary • Subtract the subtrahend from 111 (or whatever we need) (easy) • Add the result to the minuend (ordinary addition) • Add 1 (easy) • Subtract 1000 (drop highest digit) Difference = Minuend + 111 – Subtrahend + 1 – 1000

  28. Binary Subtraction Example 10010101 –01101110 ????????? 11111111 –01101110 10010001 This is the same as inverting each bit 00100111 + 10010101 100100110 Regular addition +1 100100111 Add one –100000000 00100111 Now drop the highest bit (easy: it’s out of range)

  29. Each of these steps is a simple operation we can perform using our logic circuits Subtraction Procedure Bitwise XOR Invert each bit Cascaded Adders Regular addition Add carry bit Add one Drop the highest bit ( it overflows) Now drop the highest bit (easy: it’s out of range)

  30. These steps make the negative of a number in twos-complement notation Negative Numbers Invert each bit Add one • Twos complements can be added to other numbers normally • Positive numbers cannot use the highest bit (the sign bit) • This is the normal representation of negative numbers in binary

  31. Negative Numbers in Binary 00000000 0 00000001 1 00000010 2 00000011 3 00000100 4 00000101 5 00000110 6 00000111 7 00001000 8 00001001 9 etc. 11111111 –1 11111110 –2 11111101 –3 11111100 –4 11111011 –5 11111010 –6 11111001 –7 11111000 –8 11110111 –9 11110110 –10 etc.

  32. Representations • The number representation you use (encoding) affects the way you need to do arithmetic (procedure) • This is true of all codes: encoding (representation) affects procedure (algorithm) • Good binary codes make use of properties of binary numbers and digital logic

  33. A problem A computer program adds 20,000 and 20,000 and instead of 40,000, it reports –25,566 • No errors in encoding, decoding or addition • How? Because the result is a negative number in twos-complement notation (highest bit = sign bit)

  34. How it works • 20,000 base ten is 0100111000100000 binary 010011100010000001001110001000001001110001000000 • Highest bit is set, so number is negative in twos complement notation: subtract one and invert to display 1001110001000000 – 1 = 1001110000111111 0110001111000000 = 25,566

  35. Bottom Line Representations themselves, as we use them, have limits. Interpretation depends on context two procedures (encoding/decoding and addition) may be in and of themselves correct, but conflict in their application to specific examples

  36. MER has landed

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