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Introduction to Number Representation

Introduction to Number Representation. Binary Numbers Sign/Magnitude 2s Complement. A Level Computer Science. Binary. Binary. All computer processing is carried out digitally. This means that the processor handles instructions as binary codes – zeros and ones.

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Introduction to Number Representation

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  1. Introduction to Number Representation • Binary Numbers • Sign/Magnitude • 2s Complement A Level Computer Science

  2. Binary

  3. Binary • All computer processing is carried out digitally. • This means that the processor handles instructions as binary codes – zeros and ones. • All data on a PC is essentially 0’s and 1’s.

  4. 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

  5. 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

  6. Binary Coded Decimal (BCD) • BCD represents denary integers using blocks of four binary digits. • Each block of four is converted and the denary values are then read off: • Therefore 1001 0011 1000 in BCD = 938 in denary.

  7. Uses of BCD • BCD enables fast conversions from denary to binary for applications such as pocket calculators. • Each digit on a calculator corresponds directly to a four-bit block in BCD.

  8. 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

  9. Sign/Magnitude • This method has some limitations • Makes calculations difficult by losing 1 bit 127 maximum number +/- 0 1 0 0 1 0 1 1 Sign Value or Magnitude

  10. 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

  11. 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

  12. Representing characters • There are three main coding systems that provide conversions of keyboard characters into binary: • EBCDIC • ASCII • UNICODE

  13. EBCDIC • EBCDIC stands for Extended Binary Coded Decimal Interchange Code. • It is an extension of BCD which includes non-numeric characters, including all the keyboard characters and special characters. • It is commonly used to encode data onto magnetic tape.

  14. ASCII • ASCII stands for the American Standard Code for Information Interchange. • It has been adopted as the industry-standard way of representing keyboard characters as binary codes. • Every keyboard character is given a corresponding binary code. • ASCII uses an 8-bit code to provide 256 characters.

  15. UNICODE • UNICODE is the new standard to emerge that is replacing ASCII. • It has been adopted by many of the big businesses in the computing industry. • 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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