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Integers

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Integer Storage Since Binary consists only of 0s and 1s, we cant use a negative sign ( - ) for integers. Instead, the Most Significant Bit is used to represent the sign. This way, half the combinations in a fixed length of bits can be used to represent negative values. But which value of the sign bit (0 or 1) will represent a negative number?

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Integers 2s Complement Notation

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2s Complement Notation (examples in 8 bits to save space) Fixed length notation system. Uses 1 to represent negative values. The largest non-negative value: 01111111 The smallest non-negative value: 00000000 The largest negative value is: 11111111 The smallest negative value is: 10000000

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2s Complement Notation The representations of non-negative integers in 2s Complement look the same as they do for Natural numbers. However, negative values look VERY different than we might expect.

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2s Complement Notation Complementary numbers sum to 0. Decimal is a Signed Magnitude system so complements have the same magnitude but different signs: 5 and -5, for example. 2s Complement is a Fixed Length system. There are no signs, so to find a numbers complement, another technique is needed.

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1.

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5:

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length00000101

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length00000101 flip the bits ( 1 0, 0 1 )

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length00000101 flip the bits ( 1 0, 0 1 ) 11111010

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length00000101 flip the bits ( 1 0, 0 1 ) 11111010 add 1 to the new pattern+1

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2s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2s complement notation for -5: Represent +5 in fixed length00000101 flip the bits ( 1 0, 0 1 ) 11111010 add 1 to the new pattern+1 to produce -511111011

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2s Complement Notation Complementary numbers sum to 0.

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2s Complement Notation Complementary numbers sum to 0. So if to +5 00000101

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2s Complement Notation Complementary numbers sum to 0. So if to +5 we add -5 00000101 +11111011

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2s Complement Notation Complementary numbers sum to 0. So if to +5 we add -5 we should get 00000101 +11111011 1 00000000 discard the carry bit to retain the fixed length

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Two’s Complement 1.As an action: (Assume the starting value is 1011) 1.Flip the bits from the starting value. 1011 => 0100 2.Add one to get the answer.

Two’s Complement 1.As an action: (Assume the starting value is 1011) 1.Flip the bits from the starting value. 1011 => 0100 2.Add one to get the answer.

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