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Understanding 2's Complement Notation for Binary Integers

This article explains the 2's complement notation used for representing integers in binary format. Since binary consists of only 0s and 1s, negative integers cannot use a negative sign; instead, the Most Significant Bit (MSB) indicates the sign. The 2's complement method allows us to represent both positive and negative values using a fixed length of bits. We look at the highest and lowest values representable in an 8-bit system and explore how to calculate the 2's complement when converting negative integers into binary.

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Understanding 2's Complement Notation for Binary Integers

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  1. Integers

  2. Integer Storage Since Binary consists only of 0s and 1s, we can’t 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?

  3. Integers 2’s Complement Notation

  4. 2’s Complement Notation(examples in 8 bits to save space) • Fixed length notation system. • Uses 1 to represent negative values. Since 1 is always greater than 0, • the largest non-negative value: 01111111 • the smallest non-negative value: 00000000 • the largest negative value: 11111111 • the smallest negative value: 10000000

  5. 2’s Complement Notation(examples in 8 bits to save space) What is the decimal equivalent of these? • The largest non-negative value: 01111111 • The smallest non-negative value: 00000000 • The largest negative value: 11111111 • The smallest negative value: 10000000

  6. 2’s Complement Notation(examples in 8 bits to save space) What is the decimal equivalent of these? • The largest non-negative value: 01111111 +127 • The smallest non-negative value: 00000000 • The largest negative value: 11111111 • The smallest negative value: 10000000

  7. 2’s Complement Notation(examples in 8 bits to save space) What is the decimal equivalent of these? • The largest non-negative value: 01111111 +127 • The smallest non-negative value: 00000000 +0 • The largest negative value: 11111111 • The smallest negative value: 10000000

  8. 2’s Complement Notation(examples in 8 bits to save space) What is the decimal equivalent of these? • The largest non-negative value: 01111111 +127 • The smallest non-negative value: 00000000 +0 • The largest negative value: 11111111 -1 • The smallest negative value: 10000000

  9. 2’s Complement Notation(examples in 8 bits to save space) What is the decimal equivalent of these? • The largest non-negative value: 01111111 +127 • The smallest non-negative value: 00000000 +0 • The largest negative value: 11111111 -1 • The smallest negative value: 10000000 -128

  10. 2’s Complement Notation The representations of non-negative integers in 2’s Complement look the same as they do for Natural numbers. However, negative values look VERY different than we might expect.

  11. 2’s 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. • 2’s Complement is a Fixed Length system. There are no signs, so to find a number’s complement, another technique is needed.

  12. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1.

  13. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  14. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  15. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  16. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  17. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  18. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  19. 2’s Complement Notation One such technique is to simply change each bit to its opposite, and then add 1. To find the 2’s complement notation for -5:

  20. Complementary numbers sum to 0. 2’s Complement Notation

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

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

  23. 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 2’s Complement Notation

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