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Understanding Binary Numbers and Representation in Computer Science

Introduction to binary numbers, sign/magnitude, and 2s complement in computer science. Learn how binary is converted to denary, including methods like repeated division and Binary Coded Decimal (BCD). Discover ways of storing negative integers like using Sign/Magnitude and 2s Complement. Get insights into coding systems for keyboard characters such as EBCDIC, ASCII, and UNICODE.

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Understanding Binary Numbers and Representation in Computer Science

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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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