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)
1001+1010=? ([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
= 1100101
= 0100001
= 1111111
= 1000101
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
– 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
 NOT
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




 An




XOR gate can give the unit output:
0+00
0+11
1+01
1+10
AND gate can give the carry digit:
0+00
0+10
1+00
1+1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 11+11
Twos (21)
1 1
Ones (20)
1 1
1
1
1
1
1
0
So answer is 110, or 6 in decimal representation
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 halfadders 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