Number_Representation - Number Representation Representing...

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Number Representation
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Representing Concepts How do humans represent concepts? At least several basic kinds of information concepts: Annual earnings of a UT faculty member. ( numeric ) Script of Gone with the Wind. ( characters ) Set of fingerprints on file in a police department. ( visual ) John Hartford song. ( audio ) We want a single, common way to represent. Can be encoded as numbers, which in turn can appear as signals at the hardware level.
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Number Representation Any numbering system will consist of a number of symbols with each symbol representing specific elementary (face) values. The Arabic system that we commonly use consists of exactly ten symbols: 0 1 2 3 4 5 6 7 8 9 The exact value of a symbol is determined by the position (place) it occupies. This is the basis of a positional numbering system. The total value of a number is then the sum of face values multiplied by their place values.
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Decimal Numbering System The Arabic numbering system uses ten symbols and is a base ten numbering system. In any numbering system the base is raised to consecutive powers to determine positional (place) values. 10 3 10 2 10 1 10 0 base powers 1000 100 10 1 place values 2 3 1 4 1 3 7 8 6 0
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Alternative Systems Suppose we only had one arm with five fingers. Our numbering (base 5) system might then be based on five symbols (0 1 2 3 4). 5 4 5 3 5 2 5 1 5 0 base powers 625 125 25 5 1 place values 2 1 1 five = _______ ten 4 0 3 five = _______ ten 10 0 1 2 5 56 5 1 5 1 5 2 211 = + + =
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Computer Representation Binary System At the hardware level computers must store numbers using either the presence or absence of a voltage. Then there are only two states (symbols) available for number representation. So base 2 (binary) is a suitable system for computers. 2 6 2 5 2 4 2 3 2 2 2 1 2 0 base powers 64 32 16 8 4 2 1 place values
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Positional Systems Undetermined until the numbering system base is stated. Base 10 place values:10,0001,000 100 10 1 Base 5 place values: 625 125 25 5 1 Base 2 place values: 16 8 4 2 1 1 1 1 0 base x base base 10 10? 1*1000 + 1*100 + 1*10 + 0*1 1,110 5? 2? 1*125 + 1*25 + 1*5 + 0*1 155 1*8 + 1*4 + 1*2 + 0*1 14 What value does 1110 represent?
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Determine symbols (face values) used in base x. Find place values for base x. Start far left as necessary, try largest face value: does it fit? If not, try next smallest face value. If it fits, how many face values can be accommodated. Repeat for next lowest place value. Eg:
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This note was uploaded on 10/28/2011 for the course CHE 210 taught by Professor Swinnea during the Fall '08 term at University of Texas at Austin.

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Number_Representation - Number Representation Representing...

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