1. Lecture 2: Logic
Bryan Burlingame
31 Aug 2016

2. Ref: xkcd:

3. Announcements Homework posted Homework 1 due next week
Lab starts this week

4. The Plan for Today Binary mathematics Discuss Boolean logic
Briefly introduce flowcharts
For flowcharts, use a proper graphics program.
Visio is the standard, and is in labs
Gliffy is online, free, and good enough (
Don’t use Powerpoint or Word. They are time wasters and SUPER ugly.

5. Binary Representations
Computers only contain two values
On/Off (High/Low, 0/1, etc.)
Binary number only have two value 0/1
Base 2 numbering system
A bit: A number in base 2 (binary digit)
How computers do what they do?
Humans are the translators
The machine just presents information
Think, what is a letter?
Just a squiggle, humans give meaning

6. Binary Representations
Everything in a PC is encoded in binary
Computers are finite, so are their representations
RGB for colors
24 bit color (8 bits for red, 8 for green, 8 for blue)
TIFF
Music
Midi – Instrument + note
MP3 – Sound waves compressed
Cruise Control
Voltage level going to an actuator

7. Binary to Decimal Binary number only have two value 0/1
Base 2 numbering system
A bit: A number in base 2 (binary digit)
Decimal numbers have ten values 0 – 9
Base ten numbering system
Think scientific notation
6.02E23 = 6 * * * 1021
= 8510=
1 * * * * * 20
Range?
Largest 8 bit value: = 25510
Largest 16 bit value: =

8. Negative Values Recall: Sign bit 1’s Complement
The computer only deals in 0/1
Humans provide context
Sign bit
The first bit represents sign
= -213, = 213
1’s Complement
Flip the bits (change 1’s into 0’s)
= - 213, = 213

9. Negative Values (Cont)
2’s Complement
Take 1’s complement, add one
Start (8410 in this example)
Flip bits
Add one (notice the carries)

10. Binary to Hexadecimal
Hexadecimal (base 16) is a convenient numbering system
0 – 9, A – F (A = 1010, B = 1110, etc
Observe 24 = 16, allows four bits (a nyble) to be grouped
11002 = 1210 = 0xC16

11. Binary to Hexadecimal
Convert Binary to Hex b

12. Binary to Hexadecimal Convert Binary to Hex
b (grouping 4 bits) d
C F h
C39Fh
12 * * * * 160 = 50,079

13. Representations Decimal (default) Binary Hexadecimal 108 0’01101100
0x6C
108d b
6Ch
10810
6C16
108 dec
bin
6C hex
Group by 3s with a comma
Group by 4s with a space or an underbar
Generally don’t group
Spoken short form

14. Decimal to binary Many ways, this is how I do it
120
120 < 128, 0 in 7th position
120 > 64, 1 in 6th position
120 – 64 = 56
56 > 32, 1 in 5th position
56 – 32 = 24
24 > 16, 1 in 4th position
24 – 16 = 8
8 = 8, 1 in 3rd position
8 – 8 = 0
Once we hit zero, all other values are zero
Why do I do it this way? It is quick when I don’t have a calculator. Your mileage may vary
Binary
Decimal 1 10 2
100 4
1000 8
10000 16
100000 32 64
128

15. Decimal to binary Many ways, this is how I do it
120
120 < 128, 0 in 7th position
120 > 64, 1 in 6th position
120 – 64 = 56
56 > 32, 1 in 5th position
56 – 32 = 24
24 > 16, 1 in 4th position
24 – 16 = 8
8 = 8, 1 in 3rd position
8 – 8 = 0
Once we hit zero, all other values are zero
Why do I do it this way? It is quick when I don’t have a calculator. Your mileage may vary
Binary
Decimal 1 10 2
100 4
1000 8
10000 16
100000 32 64
128

16. Bitshifts (<< Left Shift, >> Right Shift)
A bit shift shifts bits in a direction
Left shift shifts bits to the left and simulates a multiply of 2
Right shift shifts bits to the right and simulates an integer division of 2
Bits outside the range are lost
23 << 3 = 184 (left shift by 3)
shifted =
23 >> 2 = 5 (right shift by 2)
= 5 (note the last two bits are lost)

17. Representations Decimal (default) Binary 108 0’01101100 108d
10810
108 dec
bin
Group by 3s with a comma
Group by 4s with a space or an underbar

18. Bitwise arithmetic Mimics digital hardware Based on Boolean logic
Operates on each bit within a number
(i.e. operates bitwise)
Many, many hardware peripherals take advantage of bitwise operations
Implemented directly in hardware
Very fast

19. True and False Generally, false is zero and true is not-zero
Usually, all comparisons which are true generate a 1
(23 > 4) = 1
This comes up a lot!

20. Binary (Bitwise) And - &1 X
194 & 225 = 227
Binary And is commonly used with “bitmasks”
A binary & used with a bitmask can ensure that a given bit is turned “off”
& is shift + 7
(194)
& (225)
(192)

21. Binary (Bitwise) Or - | Y | 1 X 194 | 225 = 2271 X
194 | 225 = 227
Binary OR, is, also, commonly used with “bitmasks”
A binary | used with a bitmask can ensure that a given bit is turned “on”
| is shift + \
(194)
| (225)
(227)

22. Binary (Bitwise) Xor - ⊕1 X
194 ^ 225 = 35
Xors are commonly used to switch the values of selected bits or to test for inequivalence
C uses ^ for Xor
^ is shift + 6
(194)
⊕ (225)
(35)

23. Binary (Bitwise) Not (complement) - ~1 X
194 ^ 225 = 35
Complements simply flip the bits (turn 0’s into 1’s and 1’s into 0’s)
C uses ~ for complement
~ is shift + ` (next to 1)
(194) ~
(61)

24. Order of operations – C & Matlab
Operators
* / %
Multiplication, division, modulo (integer) division
+ -
Addition, subtraction
<< >>
Left bitshift, right bitshift
&
Bitwise and ^
Bitwise xor |
Bitwise or

25. Boolean Logic And && False True Exclusive Or (Xor) ^^ False True Or ||

26. Boolean Logic - Examples
( 6 < 14 ) && ( 9 > 5 )
5 || (19 > 45) && (57 < 108)
(My name is Bryan) && (This is ME30)
(name == “Bryan”) && (class == “ME30”)
(( ) == 61) ^^ ((49) == (7 * 7)) 42

27. Flowcharts - 1 Flowcharts
A graphical tool that diagrammatically depicts the steps and structure of an algorithm or program
Symbol
Name/Meaning
Meaning
Process – Any type of internal operation: data transformation, data movement, logic operation, etc.
Connector – connects sections of the flowchart, so that the diagram can maintain a smooth, linear flow
Input/Output – input or output of data
Terminal – indicates start or end of the program or algorithm
Decision – evaluates a condition or statement and branches depending on whether the evaluation is true or false
Flow lines – arrows that indicate the direction of the progression of the program

28. Flow control
These Boolean operations are used along with flow control (or branching) statements to control the flow of a program
Decisions
True
False

29. References
Nicholas, P. (1994). Digital Design. Minneapolis/St. Paul: West Publishing Company