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Representing Data

Representing Data. By Prof. Hintz, selected and modified by Dr. Gaj Electrical and Computer Engineering. Codes for Numbers. Egyptian ~4000 BC “Sum of Symbols” =1 =10 =100 = (34) 10. Positional Code for Numbers. Babylonians Positional system 2000 BC

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Representing Data

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  1. Representing Data By Prof. Hintz, selected and modified by Dr. Gaj Electrical and Computer Engineering 331_2

  2. Codes for Numbers • Egyptian • ~4000 BC • “Sum of Symbols” =1 =10 =100 = (34)10 331_2

  3. Positional Code for Numbers • Babylonians • Positional system • 2000 BC • Radix 60 • = 1 = 10 331_2

  4. Babylonian Example 1 x 602 20 x 601 56 x 600 = (4,856)10 331_2

  5. Positional Code with Zero • Zero Represented by Space • Partial solution • What about trailing zeros? • Babylonians Introduced New Symbol • or • 4th to 1st Century BC • Zero Allows Representation of Fractions • Fractions started with zero 331_2

  6. Mixed System • Roman Numerals • Sum of all symbols • I = 1 V = 5 X = 10 L=50 C = 100 M = 1000 • Difficult to do arithmetic • e.g., 331_2

  7. Positional Code Decimal System • Hindu • 500 AD • Position of coefficient determines its value • Coefficient in position is multiplied by radix (10) raised to the power determined by its position, e.g., 331_2

  8. Migration of Positional Notation • ~750 AD • Zero spread from India to Arabic countries • ~1250 AD • Zero spread to Europe • Importance of Zero • Ease of arithmetic which leads to improved commerce 331_2

  9. Binary Number System • Binary • Positional number system • Two symbols, B = { 0, 1 } • Easily implemented using switches • Easy to implement in electronic circuitry • Algebra invented by Boole allows easy manipulation of symbols 331_2

  10. Conversion Considerations • Purpose of Binary Representation is to Put Numbers in a Form which can be Manipulated by Digital Logic • Register Size Limits Resolution • Not All Numbers in One Radix Can Be Accurately Represented in Another Radix 331_2

  11. Binary, Octal, Hexadecimal • It Is Easy to Convert Binary Numbers to Other, More Convenient Representations by Grouping Bits Together and Then Converting by Inspection • Octal { 0, 1, ... , 7 } • 3-bit groups • Hexadecimal { 0, ... , 9, A, ... , F } • 4-bit groups 331_2

  12. Binary to Octal Example 331_2

  13. Binary to Hex Example 331_2

  14. Classification of number systems (1) Number system Non-positional Positional Fixed-radix Mixed-radix Conventional Unconventional Binary Decimal Hexadecimal Signed-digit 331_2 Non-redundant Redundant

  15. Classification of number systems (2) Positional wi - weight of the digit xi Fixed-radix r - radix of the number system Conventional fixed-radix r integer, r > 0 xi {0, 1, …, r-1} 331_2

  16. Classification of number systems (3) Unconventional fixed-radix xi {-, …,  } Signed-digit >0  negative digits Non-redundant number of digits =  +  + 1  r Redundant number of digits =  +  + 1 > r 331_2

  17. Range of numbers Xmin Xmax Number system Decimal X = (xk-1xk-2 … x1x0.x-1 … x-l)10 0 10k - 10-l Binary X = (xk-1xk-2 … x1x0.x-1 … x-l)2 2k - 2-l 0 Conventional fixed-radix X = (xk-1xk-2 … x1x0.x-1 … x-l)r 0 rk - r-l Notation: unit in the least significant position unit in the last position ulp = r-l 331_2

  18. Number of digits Number of digits in the integer part necessary to cover the range 0..Xmax Number system Decimal Binary Conventional fixed-radix 331_2

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