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Representations

Representations. Example: Numbers 145 CVL 10010001 91. Meaning of Number Representation. Examples: 145 = 1*10 2 + 4*10 1 + 5*10 0 Decimal CVL = 100 – 5 + 50 Roman 10010001=1*2 7 + 1*2 4 + 1*2 0 Binary

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Representations

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  1. Representations • Example: Numbers • 145 • CVL • 10010001 • 91

  2. Meaning of Number Representation • Examples: • 145 = 1*102 + 4*101 + 5*100 Decimal • CVL = 100 – 5 + 50 Roman • 10010001=1*27 + 1*24 + 1*20 Binary • 91 = 9*161 + 1*160 Hexadecimal • = 100 + 10+10+10+10+1+1+1+1+1 Egypt • = 2*60 + 10+10+5 Babylon • =7*20 + 5 Maya

  3. Meaning of Numbers: Convention/Agreement • Any number consists of symbols • The value of a number is defined by a set of rules of how to interpret these symbols • Most systems have a base number • 10 Decimal • 2 Binary • 8 Octal • 16 Hexadecimal

  4. What makes a good representations? • Meet certain constraints on the symbols • Intuitive interpretation • Can express everything you need • !! Support for frequent operations • Efficiency • Space

  5. What do we want to represent: Data Types • Set of objects of the same “kind” • Defined by • a way of representing each object • a group of operations to perform on such objects • Basic data types of computer • integers (unsigned and signed) • plain text characters • bit vectors • (floating point numbers)

  6. Computer Representation • Computer representation: • Symbols: 0,1 • Words = sequence of k symbols (bit’s) • 8 bit =1 byte • notation for an unknown k-bit word: • ak-1a k-2 … a1a0 • ak-1is called the most significant bit • a0 is called the least significant bit • k is always a power of 2: 16 or 32

  7. Unsigned Integer Representation • 145=1*102 + 4*101 + 5*100 Decimal • 10010001ui Binary =1*27 +0*26+0*25 +1*24 +0*23 +0*22 +0*21 +1*20 =1*128+0*64+0*32+1*16+0*8+0*4+0*2+1*0 =145 “Multiply and Add Algorithm”

  8. How good is unsigned integer? • Positive • Uses only 0 and 1 • Easy addition and conversion to decimal • Negative • Limited size (2k) for k-bit word • No negative • Limited subtraction

  9. Signed Integers • need to represent both non-negative and negative integers • need to be able to perform the following operations • addition (using the same rules as before) • negation • subtraction (trivial) • three different representations will be considered • in all three representations words whose most significant bit • is 0 represent the same non-negative integer

  10. Signed Magnitude • Most significant bit determines whether the number is positive (ak-1=0, as before ) or negative (ak-1=1) 1 1 1 0 0 k=4 1 1 1 0 (-6) + 0 1 1 1 (+7) 0 1 0 1 (+5) • + • We now have negative numbers • Easy negation, only change first bit • - • Addition does not work anymore • Does not work!

  11. One’s Compliment • Positive number as before • Negation is performed by inverting all bits • Example: -6 = Inverse (6) • = Inverse (0110) = 1001 • “test” addition by adding 1001 with 0111 1 1 1 1 0 1 0 0 1 (-6) + 0 1 1 1 (+7) 0 0 0 0 (0) • Does not work!

  12. Two’s Compliment • Positive as before • Negation is performed by inverting all of the bits, and then adding 1 (binary) • -6 = Inverse(6)+0001 = Inverse(0110)+0001 = 0110+0001=0111 • “test” addition by adding 1010 with 0111 1 1 1 0 0 1 0 1 0 (-6) + 0 1 1 1 (+7) 0 0 0 1 (+1) • Two’s compliment is useful for representing signed integers

  13. Things you should be able to do • Convert decimal number to binary and vice versa to all 4 forms of binary representation • Addition in unsigned integer • Addition and subtraction for two’s complement • Negation in two’s complement • Recognize the different subscripts: ui, sm, 1c,2c • Understand why 2c is better than sm • Know which representation (of the 4) is used for integer

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