1. 1 Ethics of Computing MONT 113G, Spring 2012 Session 1 Digital Circuits, binary Numbers Course webpage: http://mathcs.holycross.edu/~croyden/mont113G

2. 2 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).

3. 3 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

4. 4 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

5. 5 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: s1s2output0 0101 10101 output = s1 AND s2 = s1  s2

6. 6 Other Gates Other useful gates: OR GateInverter NOR Gate NAND Gate XOR (Exclusive OR) Gate s1 V s2s1 (s1  s2) (s1 V s2)

7. 7 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.

8. 8 Counting in Base 10 In the decimal number system (base 10), we use 10 digits 0 - 9. We count until we run out of digits, and then add a new place with value 10. 0 1 2 3 4 5 6 7 8 9 10 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).... 98 99 100 place value = 100

9. 9 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 0 1 2 10 3 11 4100 5101 6110 7111 8 1000 Note: Each 1 or 0 is called a binary digit or bit.

10. 10 Base 10 vs. Base 2 In base 10, each place represents a power of 10: 4173 = ? 4 x 10 3 + 1 x 10 2 + 7 x 10 + 3 x 10 0 In base 2, each place represents a power of 2: 10110 = ? 1 x 2 4 + 0 x 2 3 + 1 x 2 2 + 1 x 2 + 0 x 2 0 = 22 Practice: Convert 110110 from binary to decimal.

11. 11 Converting from Decimal to Binary--Method 1 To convert from decimal to binary, we first find the largest power of 2 that is less than or equal to 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. 25/16 = 1R = 9 9/8 = 1R = 1 1/4 = 0R = 1 1/2 = 0R = 1 1/1 = 1R = 0 Binary number = 11001

12. 12 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 = 11001 Practice: What is 43 written in base 2?