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.

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

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.

2nd

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.

211001

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 = 00 + 1 = 11 + 0 = 11 + 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

2n a

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.

16nd

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