fundamentals of engineering supplied-reference handbook - Ventech!

NUMBER SYSTEMS AND CODES
An unsigned number of base-r has a decimal equivalent D
defined by
n
D = ∑ a r
k
k = 0
k
m
−i
i
i=
1
+ ∑ a r
, where
ak = the (k+1) digit to the left of the radix point and
ai = the ith digit to the right of the radix point.
Binary Number System
In digital computers, the base-2, or binary, number system is
normally used. Thus the decimal equivalent, D, of a binary
number is given by
D = ak 2 k + ak–12 k–1 + …+ a0 + a–1 2 –1 + …
Since this number system is so widely used in the design of
digital systems, we use a short-hand notation for some
powers of two:
2 10 = 1,024 is abbreviated "K" or "kilo"
2 20 = 1,048,576 is abbreviated "M" or "mega"
Signed numbers of base-r are often represented by the radix
complement operation. If M is an N-digit value of base-r,
the radix complement R(M) is defined by
R(M) = r N – M
The 2's complement of an N-bit binary integer can be
written
2's Complement (M) = 2 N – M
This operation is equivalent to taking the 1's complement
(inverting each bit of M) and adding one.
The following table contains equivalent codes for a four-bit
binary value.
Binary
Base-2
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
Decimal
Base-10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Hexadecimal
Base-16
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Octal
Base-8
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
BCD
Code
0
1
2
3
4
5
6
7
8
9
---
---
---
---
---
---
Gray
Code
0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000
187
ELECTRICAL AND COMPUTER ENGINEERING (continued)
LOGIC OPERATIONS AND BOOLEAN ALGEBRA
Three basic logic operations are the "AND ( · )," "OR (+),"
and "Exclusive-OR ⊕" functions. The definition of each
function, its logic symbol, and its Boolean expression are
given in the following table.
Function
Inputs
A B C = A·B C = A + B C = A ⊕ B
0 0 0
0
0
0 1 0
1
1
1 0 0
1
1
1 1 1
1
0
As commonly used, A AND B is often written AB or A⋅B.
The not operator inverts the sense of a binary value
(0 → 1, 1 → 0)
NOT OPERATOR
Input Output
A C = Ā
0 1
1 0
DeMorgan's Theorems
first theorem:
A + B = A ⋅ B
second theorem:
A⋅
B = A + B
These theorems define the NAND gate and the NOR gate.
Logic symbols for these gates are shown below.
NAND Gates: A ⋅ B = A + B
NOR Gates: A +
B = A ⋅ B