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Discrete Structures

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. Decimal Notation.

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Discrete Structures

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  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: 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. 2.5 Application: Number Systems and Circuits for Addition

  4. 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. 2.5 Application: Number Systems and Circuits for Addition

  5. 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

  6. Converting Binary to Decimal • Represent the integers in decimal notation. 2.5 Application: Number Systems and Circuits for Addition

  7. 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 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10 2.5 Application: Number Systems and Circuits for Addition

  8. Binary Addition Example 1012 +1012 2.5 Application: Number Systems and Circuits for Addition

  9. 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

  10. 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 2.5 Application: Number Systems and Circuits for Addition

  11. 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

  12. 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

  13. 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. 2.5 Application: Number Systems and Circuits for Addition

  14. Example – pg. 95 #39 • Convert the integer from hexadecimal to decimal notation. E0D16 2.5 Application: Number Systems and Circuits for Addition

  15. Example – pg. 95 #42 • Convert the integer from hexadecimal to binary notation. B53DF816 2.5 Application: Number Systems and Circuits for Addition

  16. Example – pg. 95 #42 • Convert the integer from binary to hexadecimal notation. 001011102 2.5 Application: Number Systems and Circuits for Addition

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