1. Binary Numbers

2. Why Binary?
Maximal distinction among values  minimal corruption from noise
Imagine taking the same physical attribute of a circuit, e.g. a voltage lying between 0 and 5 volts, to represent a number
The overall range can be divided into any number of regions

3. Don’t sweat the small stuff
For decimal numbers, fluctuations must be less than 0.25 volts
For binary numbers, fluctuations must be less than 1.25 volts
5 volts
0 volts
Binary
Decimal

4. It doesn’t matter ….
Recall the power supply voltage measurements from lab 1
Ideally they should be 5.00 volts and volts
Typically they were 5.14 volts or volts
So what, who cares

5. How to represent big integers
Use positional weighting, same as with decimal numbers
205 = 2  100
= 127 + 126 + 025 + 0 23 + 122 + 021 + 1 = = 205

6. Converting 205 to Binary
205/2 = 102 with a remainder of 1, place the 1 in the least significant digit position
Repeat 102/2 = 51, remainder 0 1 1

7. Iterate 51/2 = 25, remainder 1 25/2 = 12, remainder 11 1

8. Iterate 6/2 = 3, remainder 0 3/2 = 1, remainder 1 1/2 = 0, remainder 11 1 1

9. Recap
127 + 126 + 025 + 024
+ 123 + 122 + 021 + 120
205 1

10. Adding Binary Numbers
Same as decimal; if sum of digits in a given position exceeds the base (10 for decimal, 2 for binary) then there is a carry into the next higher position 1 3 9 + 5 7 4

11. Adding Binary Numbers 1 +

12. Uh oh, overflow
What if you use a byte (8 bits) to represent an integer
A byte may not be enough to represent the sum of two such numbers 1

13. Bigger Numbers You can represent larger numbers by using more words
You just have to keep track of the overflows to know how the lower numbers (less significant words) are affecting the larger numbers (more significant words)

14. Negative numbers
Negative x is that number when added to x gives zero
Ignoring overflow the two eight-bit numbers above sum to zero 1

15. Two’s Complement
Step 1: exchange 1’s and 0’s
Step 2: add 1 1 1 1

16. Riddle Is it 214? Or is it – 42? Or is it …?
It’s a matter of interpretation
How was it declared? 1

17. Hexadecimal Numbers
Even moderately sized decimal numbers end up as long strings in binary
Hexadecimal numbers (base 16) are often used because the strings are shorter and the conversion to binary is easier
There are 16 digits: 0-9 and A-F

18. Decimal  Binary  Hex 0  0000  0 1  0001  1 2  0010  2
3  0011  3
4  0100  4
5  0101  5
6  0110  6
7  0111  7
8  1000  8
9  1001  9
10  1010  A
11  1011  B
12  1100  C
13  1101  D
14  1110  E
15  1111  F

19. Binary to Hex Break a binary string into groups of four bits (nibbles)
Convert each nibble separately 1 E C 9

20. Addresses
With user friendly computers, one rarely encounters binary, but we sometimes see hex, especially with addresses
To enable the computer to distinguish various parts, each is assigned an address, a number
Distinguish among computers on a network
Distinguish keyboard and mouse
Distinguish among files
Distinguish among statements in a program
Distinguish among characters in a string

21. How many?
One bit can have two states and thus distinguish between two things
Two bits can be in four states and …
Three bits can be in eight states, …
N bits can be in 2N states 1 1

22. IP Addresses
An IP address is used to identify a network and a host on the Internet
It is 32 bits long
How many distinct IP addresses are there?

23. Characters We need to represent characters using numbers
ASCII (American Standard Code for Information Interchange) is a common way
A string of eight bits (a byte) is used to correspond to a character
Thus 28=256 possible characters can be represented
Actually ASCII only uses 7 bits, which is 128 characters; the other 128 characters are not “standard”

24. Unicode Unicode uses 16 bits, how many characters can be represented?
Enough for English, Chinese, Arabic and then some.

25. ASCII
0 
1  …
A 
B 
a 
b 

26. Booleans A Boolean variable is something that is true or false
Booleans have two states and could be represented by a single bit (1 for true and 0 for false)
Booleans appearing in a program will take up a whole word in memory

27. Boolean Operators Aka logical operators
Have Boolean input and Boolean output
Standard:
AND OR
NOT
XOR (either or but not both)
NOR = NOT(OR)
NAND = NOT(AND)

28. Venn Diagrams
If inside circle A means A is true, and similarly for circle B, then (A AND B) is the intersection B A

29. Truth Tables
AND
INPUT
OUTPUT A B
A AND B 1

30. Truth Tables (Cont.) OR
INPUT
OUTPUT A B
A OR B 1

31. Truth Tables (Cont.)
XOR (Excluded OR)
INPUT
OUTPUT A B
A XOR B 1

32. So I lied
We said computing was all about numbers, but it’s really all about logic
The adding operation is just a particular combination of logic operations
Possibilities for adding two bits
0+0=0 (with no carry)
0+1=1 (with no carry)
1+0=1 (with no carry)
1+1=0 (with no carry)

33. Addition Truth Table
INPUT
OUTPUT A B
Sum
A XOR B
Carry
A AND B 1

34. All is NAND
Actually you can use one logic gate and a few tricks (like De Morgan’s theorem) to build all of the “combinatorial” circuitry (the circuitry that doesn’t involve memory)
NORs work too
But we tend to think in ANDs, ORs and NOTs

35. Bit manipulation
You can use an AND to select out part of a word (where s is a 1 or 0, etc) s t u v w x y z
AND 1
gives s t u v

36. IP Addresses Revisted
LaSalle’s IP address is what’s called a Class B IP address
Of the 32 bits the first two are 10 (this identifies us as Class B)
The remaining 14 bits of the first two bytes identify us as LaSalle
The remaining 2 bytes are for our internals use (to assign computers within LaSalle)

37. In or Out
To see if an address is local to LaSalle, you would restrict your attention to the first two bytes.
HOW?
AND it with FFFF0000