1. Chapter 2: The Logic of Compound Statements 2.5 Application: Number Systems and Circuits for Addition 1 Counting in binary is just like counting in decimal if you are all thumbs. – Glaser and Way

2. 2.5 Application: Number Systems and Circuits for Addition 2 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. 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 3

4. 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 4

5. Represent the decimal integers in binary notation. 2. 55 3. 287 2.5 Application: Number Systems and Circuits for Addition 5

6. Represent the integers in decimal notation. 2.5 Application: Number Systems and Circuits for Addition 6

7. 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 7

8. 101 2 +101 2 2.5 Application: Number Systems and Circuits for Addition 8

9. 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. 111 2 1010 2 - 10 2 - 110 2 2.5 Application: Number Systems and Circuits for Addition 9

10. 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 10

11. 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 11

12. Find the 8-bit two’s compliment for the integer below. 67 2.5 Application: Number Systems and Circuits for Addition 12

13. 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 13

14. Convert the integer from hexadecimal to decimal notation. E0D 16 2.5 Application: Number Systems and Circuits for Addition 14

15. Convert the integer from hexadecimal to binary notation. B53DF8 16 2.5 Application: Number Systems and Circuits for Addition 15

16. Convert the integer from binary to hexadecimal notation. 00101110 2 2.5 Application: Number Systems and Circuits for Addition 16