Boolean algebra - In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebra where the values of the variables are numbers, and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction and denoted as ∧, the disjunction or denoted as ∨, and the negation not denoted as ¬. It is thus a formalism for describing logical relations in the same way that elementary algebra describes numeric relations.
NOT
True = false
false = true
BOOLEAN GATE
electrode in-current
|
/
control wire (input - on) -----|
\ output - on
electrode in-current
|
/
control wire (input - off) -----|
\ output - off
NOT ▶●
electrode in-current
|------ output (off)
/
control wire (input - on) -----|
\
|
V
ground
electrode in-current
|------> output (on)
/
control wire (input - off) -----|
\
|
ground
AND ⍾
True + True = True
False + False = False
False + True = False
True + False = False
current
|
/
Transistor (input A on) -----|
\
|
/
Transistor (input B on) -----|
\
|Output
V
current
|
/
Transistor (input A off) -----|
X
|
/
Transistor (input B on) -----|
\
|Output
V
current
|
/
Transistor (input A on) -----|
\
|
/
Transistor (input B off) -----|
X
|Output
V
OR ⍲
current
_____|_____
/ |
Transistor (input A on) -----| |
\ |
| |
| /
Transistor (input B on) -------(-------------|
| \
|__________|
|
V
current
____|______
/ |
Transistor (input A off) -----| |
X |
| |
| /
Transistor (input B on) -------(-------------|
| \
|__________|
|
V
current
____|______
/ |
Transistor (input A on) -----| |
\ |
| |
| /
Transistor (input B off) -------(-------------|
| X
|__________|
|
V
current
____|______
/ |
Transistor (input A off) -----| |
X |
| |
| /
Transistor (input B off) -------(-------------|
| X
|__________|
|
X
XOR GATE IS NOT COVERED
Converting between base 10 and 2
BASE 10
263
100 X 2
+
10 X 6
+
1 X 3
BASE 2
101
1 X 1
+
2 X 0
+
4 X 1 = 5
another example
11011
1 X 1
+
2 X 0
+
4 X 1
+
8 X 1 = 27
8Bit size = Byte
1 2 3 4 5 6 7 8
_________________________________________________________
| 128's | 64's | 32's | 16's | 8's | 4's | 2's | 1's |
|________|______|______|______|_____|_____|_____|_____|
| 1 1 1 1 1 1 1 1 = 255 in UTF8 is ÿ
|______________________________________________________
_________________________________________________________
| 128's | 64's | 32's | 16's | 8's | 4's | 2's | 1's |
|________|______|______|______|_____|_____|_____|_____|
| 0 1 1 1 1 1 1 1 = 127 in ASCII is DEL
|______________________________________________________
_________________________________________________________
| 128's | 64's | 32's | 16's | 8's | 4's | 2's | 1's |
|________|______|______|______|_____|_____|_____|_____|
| 0 1 1 1 0 1 1 1 = 119 in ASCII is w
|______________________________________________________
_________________________________________________________
| 128's | 64's | 32's | 16's | 8's | 4's | 2's | 1's |
|________|______|______|______|_____|_____|_____|_____|
| 0 1 1 0 0 0 1 1 = 99 in ASCII is c
|______________________________________________________
16bit encoding came out later.
32bit & 64bit are the standard today.
TAKE NOTE
This is just like in decimal count
The number 1,654 is:
__________________________________
| 1000's | 100's | 10's | 1's |
|_________|________|______|_____|
| 1 6 5 4
|_________________________________
A number at each place is just a multiplacation of 10.
Representing floating numbers:
A floating number, lets say 625.9 can be represented in the following way
0.6259 * 10^3
|___| ^
| | exponent
Segnificant
Floating numbers are represented in a 32bit number storage.
______________________________________________________________________________________________________________________________________________________________
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |
|__||_______________________________||______________________________________________________________________________________________________________________|
^ ^ ^
| | |
sign exponent Segnificant
(-/+) 8bits 23bits
1bit
