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2.5 Application: Number Systems and Circuits for Addition
1
Discrete Structures
Chapter 2: The Logic of Compound Statements
2.5 Application: Number Systems and Circuits for Addition
Counting in binary is just like counting in decimal if you are all thumbs.
– Glaser and Way
2.5 Application: Number Systems and Circuits for Addition
2
Decimal Notation
• Decimal notation (base 10) expresses a number as a string of digits in which each digit’s position indicates the power of 10 by which it is multiplied.
• For example:
3 2 1 0
5280 5 1000 2 100 8 10 0 1
5 10 2 10 8 10 0 10
2.5 Application: Number Systems and Circuits for Addition
3
Decimal Notation
• Decimal notation is based on the fact that any positive integer can be written uniquely as a sum of products of the form
where n is a nonnegative integer and each d is one of the decimal digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9.
10nd
2.5 Application: Number Systems and Circuits for Addition
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Binary Notation
• In computer science, base 2 notation, or binary notation, is of special importance because the signals used in modern electronics are always in one of only two states.
• Any integer can be represented uniquely as a sum of products of the form
where each n is an integer and each d is one of the binary digits 0 or 1.
2nd
2.5 Application: Number Systems and Circuits for Addition
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Example – pg. 94 # 2 & 3
• Represent the decimal integers in binary notation.
2. 55
3. 287
2.5 Application: Number Systems and Circuits for Addition
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Converting Binary to Decimal
• Represent the integers in decimal notation.
211001
2.5 Application: Number Systems and Circuits for Addition
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Adding in Binary Notation
• Addition in binary notation is similar to addition in decimal notation, except that only 0's and 1's can be used, instead of the whole spectrum of 0-9. This actually makes binary addition much simpler than decimal addition, as we only need to remember the following:
0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 10
2.5 Application: Number Systems and Circuits for Addition
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Binary Subtraction
• Binary subtraction is simplified as well, as long as we remember how subtraction and the base 2 number system. Let's look at two examples.
1112 10102
- 102 - 1102
2.5 Application: Number Systems and Circuits for Addition
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Compliments
• Given a positive integer a, the two’s compliment of a relative to a fixed bit length n is the n-bit binary representation of
2n a
2.5 Application: Number Systems and Circuits for Addition
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Finding a Two’s Complement
• To find the 8-bit two’s complement of a positive integer a that is at most 255:–Write the 8-bit binary representation for a.– Flip the bits (switch all the 1’s to 0’s and 0’s to
1’s).– Add 1 in binary notation.
2.5 Application: Number Systems and Circuits for Addition
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Example – pg. 94 # 24
• Find the 8-bit two’s compliment for the integer below.
67
2.5 Application: Number Systems and Circuits for Addition
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Hexadecimal Notation
• Base 16 notation, or hexadecimal notation can be represented uniquely as a sum of products of the form
where each n is an integer and each d is one of the integers 0 to 15. 10 through 15 are represented by A, B, C, D, E and F.
16nd
2.5 Application: Number Systems and Circuits for Addition
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Example – pg. 95 #39
• Convert the integer from hexadecimal to decimal notation.
E0D16
2.5 Application: Number Systems and Circuits for Addition
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Example – pg. 95 #42
• Convert the integer from hexadecimal to binary notation.
B53DF816