Binary System
Digital circuits process signals that contain just two voltage levels or states,
labeled logic "0" and logic "1".
These discrete voltage levels are commonly known as Binary digits and are
normally referred to as BITS.
Because there are only two valid Boolean values for representing either logic "1"
or logic "0", the Binary Numbering system is ideal for use in digital or logic circuits
and systems.
The Binary Numbers system is a Base-2 system which follows the same rules in
mathematics as the common decimal system meaning instead of powers of ten it
uses a power of two,
1, 2, 4, 8, 16, 32 etc.
Phys 4330 Digital Electronics
Binary System
Basic Concepts
In the decimal system, things are organized into columns:
H|T|O
9|2|4
102 | 101 | 100
9 |2 |4
Each column has a value equals to its weight multiply by its frequency
The frequency of each column can be 0 or 1 or 2 or ………. or 9 (ten possible values)
The column weight is a power of ten;
1 (or Ones), 101 (or Tens), 102 (or Hundreds), and so on
The number 924 is really {(9*102)+(2*101)+(4*100)}.
Phys 4330 Digital Electronics
Binary System
The binary system works under the exact same principles as the decimal
system, only it operates in base 2 rather than base 10.
But because we have only two states or two frequency; 0 and 1, the
weight of consecutive columns is a power of two instead of tens
22 | 21 | 20
1 |0 |1
Each column has a value equals to its weight multiply by its frequency
The frequency of each column can be 0 or 1 (two possible values)
The column weight is a power of 2;
1 , 2, 4, 8, 16, 32, 64 and so on
The binary number 101 is really equivalent to the decimal number
{(1*1)+(2*0)+(4*1)} = 1+0+4 =5.
Phys 4330 Digital Electronics
Binary System
Least Significant Number LSN and Highest Significant Number HSN
As in decimal the LSN is the one bit to the most right of the digital
number while the HSN is the bit to the most left of the digital
number
Example: What would the binary number 10, 111, 10101 and 11110 be in
Decimal notation?
Binary
24
23
22
21
20
Decimal
1
0
1*2+0*1=2
1
1
1
1*1+1*2+1*4=7
10
111
10101
1
0
1
0
1
1*16+0*8+1*4+0*2+1*1=21
11110
1
1
1
1
0
1*16+1*8+1*4+1*2+1*0=30
Phys 4330 Digital Electronics
Binary System
The same approach applies to non-integral
numbers.
23
22
21
20
.
2-1
2-2
2-3
Example:
110.101 = 1*22+1*21+0*20+1*2-1+0*2-2+1*2-3 = 4 + 2 + 0 + 0.5 + 0 + 0.125 = 6.625
Convert the following number from binary
form to decimal form:
100011
100000
111111
101010
101.101
110.011
Phys 4330 Digital Electronics
Binary System
Binary Addition
In decimal system we add the ones together and tens together and hundred together and
so on. If the sum exceeds the bit value (9 fold), we carry the exceeded to the left bit.
Example:
273
465
----738
In this example when we add the tens (7 tens + 6 tens) we get 13 tens. We put the SUM,
3 tens in tens bit and add the CARRY 10 tens as 1 to the hundred bit.
In adding binary number we do the same, we put the SUM and take the CARRY to left
bit.
Because 1 is the maximum value in any binary bits, the CARRY is 1 only if both bits is
1.
Phys 4330 Digital Electronics
Binary System
Adding rules:
0
0
1
1
+ 0
+ 1
+ 0
+ 1
----SUM
0
1
1
1
CARRY 0
0
0
1
---------------------------------------------------------Result 0
0
0
10
The process is the same for multiple-bit binary numbers:
Number 1
1
1
0
1
Number 2
1
1
1
0
SUM
0
0
1
1
CARRY
1
1
0
0
Total
1
1
0
1
Phys 4330 Digital Electronics
1
Binary System
Try the following examples of binary addition:
11101 10111 10011
+11010 +11100 +11100
_____ ______ ______
Phys 4330 Digital Electronics
Binary System
Decimal to Binary conversion
Method one:
To convert a decimal number to binary, first subtract the largest possible power of
two, and keep subtracting the next largest possible power form the remainder,
marking 1s in each column where this is possible and 0s where it is not.
Remember, the power of two is: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024,
2048, 4096, ………..
Example: Convert the decimal number 44 into
its corresponding binary number.
44
- 32
____
12
-8
____
4
-4
____
0
Phys 4330 Digital Electronics
32
16
8
4
2
1
1
0
1
1
0
0
Binary System
Method two:
Keep dividing the decimal number by two; put
0 if there is no remaining and put 1 if there is
until you done
Example: convert 44 into binary
44/2
22
0
22/2
11
0
11/2
5
1
5/2
2
1
2/2
1
0
1/2
0
1
Phys 4330 Digital Electronics
LSN
1
HSN
Binary System
0
1
1
0
0