How computers calculate How binary operations yield complex capabilities

Activity Part 1 Answer the questions in the Before You Start section

Why Digital Not Analog?  What is difference?  Digital is discrete number; analog varies continuously  Analog signals prone to distortion because of large range  Analog signals lose information with every replication and over time

Discussion Question: What are some applications for which you would prefer an analog device? When would a digital device be preferable?

Why Binary not Decimal?  Two options is a lot easier to manage than 10  Switch can be on/off  Less confusion between different settings  Faster

What is Binary?  Only uses digits 1 and 0  So 1+1=10(1+1=2) 10+1=11(2+1=3) 11+1=100(3+1=4) =?([1+8]+[2+8]=19) 10011(1+2+16)

Activity Part 2 Hook up the equipment as shown by your instructor, and answer the first 3 questions in the What type of signal is it? section

What Good is Binary?  Can use numbers to represent letters  Standard code ASCII has 7 binary digits (bits) per character:  e=  !=  Del=  E=

Activity Part 3 Answer questions 4-8 in the What are the properties of the signal? section

Activity Part 4 Answer questions 9-15 in the How do signals from different keys compare? section

Boolean Logic  NOT – the simplest: outputs the opposite of the input.  AND – outputs a 1 only if both of its two inputs are 1.  NAND – outputs a 1 unless both of its two inputs are 1.  OR – outputs 1 if either input is 1.  XOR – outputs 1 if only one input is 1

So how do these do math?  Can create a half-adder and a full adder  Half-adders add two digits and output the unit output and the carry digit  Full -adders add two digits and a carried digit and give the two outputs

Creating a half-adder  An XOR gate can give the unit output:   0   1   1   0  An AND gate can give the carry digit:   0   0   0   1

Creating a full-adder  Combine 2 half-adders and 1 OR:  First half-adder adds two input digits, giving unit and carry digit  Second adds new unit digit with carry input, giving final unit output and carry digit  OR gives carry digit if either half-adder has a carry digit (you will never have carry digits from both)

Adding Ones (2 0 )Twos (2 1 ) So answer is 110, or 6 in decimal representation 1 0 1

Adding large numbers  Start with right-most digits and work left  Number of gates needed grows quickly

An exercise  How many gates needed to add any possible 2-digit decimal integers?  99=1*64+1*32+0*16+0*8+0*4+1*2+1*1  So need to add 2 7-digit binary numbers  Need one half-adder and 6 full-adders  Half-adder has 1 XOR and 1 AND  Full-adders have 2 XOR, 2 AND, 1 OR  Total is 13 XOR, 13 AND, 6 OR

Discussion Question: Think about different mathematical functions you are familiar with. Could you perform each of them using only adders and half-adders? How would you do it?

The Rest of Mathematics  Subtraction, Multiplication, and Division can be performed by combos of Addition and data shifting  Mathematical features such as Taylor sums can be used to express complicated functions using addition.  Data processing boils down to nothing but addition (and storage and shifting).

So what have we learned today?  Computers use digital binary numbers to communicate and calculate  One convention for expressing text as binary numbers is ASCII  Keyboards do not use ASCII, but use a quite different format  Boolean operators can process binary information  AND, NAND, and XOR can be used to create half- adders and full-adders  All mathematical functions can be performed by a series of addition and data shifting

Evaluation Log on to WebCT and answer the 6 questions in the Act02eval quiz