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Mastering Binary and Hexadecimal Number Systems for CPU Processing

Learn about binary, sign magnitude, 2s complement, and hexadecimal for CPU operations. Understand conversion methods and storing negative integers. Practice binary addition, subtraction, logical and arithmetic shifts. Explore ASCII and Unicode encoding systems.

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Mastering Binary and Hexadecimal Number Systems for CPU Processing

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  1. Number SystemsChapter 3 • Binary • Sign Magnitude Binary • 2s Complement Binary • Hexadecimal • All used for different purposes within a CPU

  2. Binary • All computer processing is carried out digitally. • Processor handles instructions as binary codes – zeros and ones. • All data on a device is 0’s and 1’s.

  3. Converting binary into positive denary integers • Whole positive denary (base ten) numbers are converted into binary as follows: • 135 from denary into binary 128 + 4 + 2 + 1 = 135 MSB LSB 1 0 0 0 0 1 1 1

  4. The repeated division method A method for converting denary to binary: 98 in denary into binary: 98 divide by 2 = 49 remainder 0 49 divide by 2 = 24 remainder 1 24 divide by 2 = 12 remainder 0 12 divide by 2 = 6 remainder 0 6 divide by 2 = 3 remainder 0 3 divide by 2 = 1 remainder 1 1 divide by 2 = 0 remainder 1 0 divide by 2 = 0 remainder 0 Read the binary code from the remainder from bottom to the top: 01100010 which equals 98 DIV MOD

  5. Hex – Base 16 • Only uses single digits • A = 10 • B = 11 • C = 12 • D = 13 • E = 14 • F = 15

  6. Hex – Base 16 • 75 in denary • is 4B in Hex • 0*0=0 • 4*16=64 • 1*11=11 (B) • 64+11=75 0 4 B 0 64 11

  7. Hex Number • 172 in denary • is AC in Hex • 0*0=0 • 10*16=160 (A) • 1*12=12 (C) • 160+12=172 0 A C 0 160 12

  8. Convert a Byte to Hexadecimal • Split the binary in half • 1100 = 4+8=12 (C) • 1010 = 2+8=10 (A) • AC = Hex value

  9. Convert a Hexadecimal to a Byte • The Hexadecimal number 2E • 1 nibble per hexadecimal column • 0010 = 2 • 1110 = 14 (E)

  10. Storing Negative Integers • 1 method is Sign/Magnitude • 75 • -75 MSB 128 +/- 0 1 1 0 0 1 0 1 1 1 is a Negative, 0 is a Positive

  11. Sign/Magnitude • This method has some limitations • 2 types of data in the same value (MSB is a sign) • Makes calculations difficult by losing 1 bit 127 maximum number +/- 0 1 0 0 1 0 1 1 Sign Value or Magnitude

  12. Storing Negative Integers • Another method is 2s Complement • -75 128 -128 1 0 1 1 0 1 0 1 • -128+32+16+4+1=-75

  13. 2s Complement Conversion • -117 • Stage 1 : work out 117 in binary • Stage 2 : Reverse the 0’s and 1’s 1 0 • Stage 3 : Plus 1

  14. Binary Addition Binary Subtraction

  15. Binary Addition Sums • In binary addition there are 5 possible sums: • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 0, carry 1 • 1 + 1 + 1 = 1, carry 1

  16. Binary Addition Example 1 • 75 + 14 = 89 89 0 1 0 1 1 0 0 1 1 1 1

  17. Binary Addition Example 2 • 79 + 57 = 136 136 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1

  18. Binary Addition Example 3 • 75 + 75 = 150 150 1 0 0 1 0 1 1 0 1 1 1 1

  19. Binary Addition Example 4 • 215 + 138 = 353 (101100001) • (exercise 3c Page 97) 353 0 1 1 0 0 0 0 1 Overflow 1 1 1 1 1

  20. Binary Subtraction • CPUs can’t subtract, they can only add! • 75-14=61 • The same as… • 75+(-14)=61 • Positive + Negative = Negative • To do this we need 2s Complement

  21. Binary Subtraction Example 1 • 75-14=61 • 75+(-14)=61 61 0 0 1 1 1 1 0 1 1 1 1

  22. Binary Subtraction Example 2 • 91-18=73 • 91+(-18)=73 73 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1

  23. Logical Shifts • A left or right logical shift can be performed on a binary number as a method of multiplication or integer division. • This is the same as an // operation in Python • We can perform multiplications or integer divisions in powers of 2

  24. Left Logical Shifts Example 1 • Take this byte – 104 in denary • Shift it left by 1 • 11010000 – 208 in denary (104x2) • 2**1 is 2

  25. Left Logical Shifts Example 2 • Take this byte – 104 in denary • Shift it left by 2 • 110100000 – 416 in denary (104x4) • 2**2 is 4

  26. Left Logical Shifts Example 3(activity 5a page 102) • Take this byte – 58 in denary • Shift it left by 3 - 2**3 (8) • 111010000 – 464 in denary (58x8)

  27. Right Logical Shifts Example 1 • Take this byte – 185 in denary • Shift it right by 1 • 01011100 – 92 in denary (185//2) Integer Division • 2**1 is 2

  28. Right Logical Shifts Example 3Activity 5b Page 102 • Take this byte – 157 in denary • Shift it right by 4 • 00001001 – 9 in denary (157//16) Integer Division • 2**4 is 16

  29. Arithmetic Shifts • A left or right arithmetic shift can be performed on a 2s complement binary number as a method of multiplication or integer division. • Left arithmetic shift – The MSB remains untouched • Right arithmetic shift – use a copy of the MSB as the replacement bits

  30. Left Arithmetic Shifts Example 1(page 102) • Take this byte -36 in denary • (-128+64+16+8+4) • Shift it 1 place left arithmetic shift • 10111000 – -72 (-36x2) • -128+32+16+8

  31. Right Arithmetic Shifts Example 1(page 103) • Take this byte -72 in denary • (-128+32+16+8) • Shift it 2 places right arithmetic shift • 11101110 – -18 (-72//4) (2**2=4) • -128+64+32+8+4+2

  32. Representing characters • There are two main coding systems that provide conversions of keyboard characters into binary: • ASCII • UNICODE

  33. ASCII • ASCII stands for the American Standard Code for Information Interchange. • It has been adopted as the industry standard way of representing English language keyboard characters as binary codes. • Every keyboard character is given a corresponding binary code. • ASCII uses an 7-bit code to provide 128 characters.

  34. UNICODE • UNICODE is the new standard to emerge that is replacing ASCII. • Uses 16 bits per character • It is designed to cover more of the characters that are found in languages across the world. • It has become important due to the increased use of the Internet, as more data is being passed around globally.

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