1 Survey of Computer Science CSCI 110, Spring 2011 Lecture 16 Digital Circuits, binary Numbers

2 Church-Markov-Turing Thesis Any non-trivial computer language is apparently capable of computing no more and no fewer functions than all other non-trivial programming languages. So, all languages are equally powerful. (There may be other reasons for choosing one language over another depending on the needs of the application).

3 Information on Computers We use computers to represent many different kinds of information: Numbers Symbols (letters, punctuation, etc.) Pictures Sound Program instructions Etc. We divide this information into different categories. The computer stores everything as 1's and 0's (binary representation).

4 Binary Information Computers represent all information as 0's and 1's (Binary representation). All the pictures, sounds, programs, etc. on your computer are stored as sets of 0's and 1's! Why? Computers are built using digital circuits. The inputs and outputs of digital circuits can only be one of two values: true (high voltage) or false (low voltage). We represent these as 1 and 0. By combining 1's and 0's in different patterns and sequences, we can represent complex information and compute solutions to some complex problems

5 Digital Electronics Each digital component can have only 1 of two states. High Voltage (+ 5V) = 1 Low Voltage (0 V) = 0 Basic components: Logic display (LED) => lights up when voltage high (1) Doesn't light when voltage low(0) Digital switch => Sends high voltage (1) when up Sends low voltage (0) when down

6 AND Gate An AND Gate takes two inputs and produces 1 output. Depending on the values of the inputs, s1 and s2, the LED will either light up or not. Truth Table: s1s2output output = s1 AND s2 = s1  s2

7 Other Gates Other useful gates: OR GateInverter NOR Gate NAND Gate XOR (Exclusive OR) Gate s1 V s2s1 (s1  s2) (s1 V s2)

8 Combining Gates A three input AND: Truth table: a b c z Practice: Draw a three input OR circuit and write out the truth table.

9 Counting in Base 10 In the decimal number system (base 10), we use 10 digits We count until we run out of digits, and then add a new place with value place value = 10place value = 1 We continue to count, adding 1 to the 10's place every 10th number. When we run out of digits for the 10's place, we add a new place with value 10 2 (or 100) place value = 100

10 Counting in Binary With binary numbers, we have only 2 digits to work with (0 and 1), so we add places more frequently. Each new place has a value that is a power of two. DecimalBinary Note: Each 1 or 0 is called a binary digit or bit.

11 Base 10 vs. Base 2 In base 10, each place represents a power of 10: 4173 = ? 4 x x x x 10 0 In base 2, each place represents a power of 2: = ? 1 x x x x x 2 0 = 22 Practice: Convert from binary to decimal.

12 Converting from Decimal to Binary--Method 1 To convert from decimal to binary, we first find the largest power of 2 that divides our decimal number. We divide by that number and put the result in the binary place associated with that power of two. We then repeat with the remainder from the previous division. Example: Convert 25 (base 10) to binary. The largest power of 2 that divides 25 is /16 = 1R = 9 9/8 = 1R = 1 1/4 = 0R = 1 1/2 = 0R = 1 1/1 = 1R = 0 Binary number = 11001

13 Converting from Decimal to Binary--Method 2 Method 2: Divide repeatedly by 2. Place remainders in order from right to left. Example: 25/2 = 12R = 1 12/2 = 6R = 0 6/2 = 3R = 0 3/2 = 1R = 1 1/2 = 0R = 1 Result = Practice: What is 43 written in base 2?

14 Hexadecimal Numbers Hexadecimal numbers use base 16. To count in base 16, we use the letters A - F to stand for decimal equivalents of Base 10Base 16Base 10Base FF A 11B 12C 13D 14E 15F = 256

15 Converting from Binary to Hexadecimal Every 4 bits represents 1 hexadecimal digit. To convert from binary to Hexadecimal, group the bits into sets of four (from right to left) and find the corresponding hexadecimal digit for each set of 4 bits B ? ?