1 / 16

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.

harmon
Télécharger la présentation

Discrete Structures

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related