# 2 Binary Numbers

```2. Binary Numbers and Binary Encoding
2.1 Number Systems
Number systems come in a variety of forms where the numerical
values of things are represented by a set of symbols of some form.
Even colours can be used to represent numerical values, as for
example in the Resistor Colour Code. We are interested in Positional
Number Systems and in particular in the Decimal and Binary Number
Systems. A Positional Number System has three essential properties:
The term Radix is Latin meaning ‘root’. This is the base or foundation
of the system and is essentially the number of individual numerical
symbols or characters which are present in the system. There are ten
symbols in the Decimal system and so it has a radix of 10, while there
are only two symbols in the Binary system and so it has a radix of 2.
Digits:
The digits of a number system are the individual numerical symbols or
characters which make up the system. The term ‘Bit’ is used to refer to
a ‘Binary Digit’.
Decimal Digits:
0
1
Binary Digits:
0
1
2
3
1
4
5
6
7
8
9
Positional Weighting:
In a Positional Number System, several digits are combined to form a
number which has a value. Each digit in the complete set may be
utilised in each position. However, a different weight applies to each
position and therefore the value of the same digit in a different
position is different. In the Decimal and Binary number systems the
weight is assigned positionally from right to left, in ascending order of
exponent powers of the radix. That is, the significance of the digit
increases from right to left as an increasing power of the radix. The
digits also have an assigned order of increment with the increment
between successive digits being of equal value.
Decimal System Weighting:
100
106
…..
105
104
103
102
101
MSD
LSD
Each weighted position is referred to as a Decade. The digit in the right
hand most position is referred to as the Least Significant Digit (LSD),
while that in the left hand most position is referred to as the Most
Significant Digit (MSD).
Binary System Weighting:
20
…..
26
25
24
23
22
21
MSB
LSB
Each weighted position is referred to as a Binate, but this term is
not common in practice and usually the term Bit is used, e.g. the first
bit, the second bit, etc. The digit in the right hand most position is
referred to as the Least Significant Bit (LSB), while that in the left
hand most position is referred to as the Most Significant Bit (MSB).
The term ‘Bit’ is generally used to refer to either the digits ‘0’ and
‘1’ as the characters of the Binary number system or to the weighted
positions or binates of a Binary number.
2
2.2 Counting
When counting in a positional number system, all of the digits are
cycled through in their assigned order from the lowest to the highest
in the least significant position. When the highest digit is reached in
this position, it returns to the lowest digit and the digit in the next
position to the left is incremented. This repeated cycling can continue
indefinitely, with a gradual incrementing of digits in positions further
to the left.
Decimal
Binary
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
11
1011
12
1100
13
1101
14
1110
15
1111
16
10000
17
10001
18
10010
19
10011
20
10100
21
10101
22
10110
3
2.3 Numerical Value
The value of a number is determined by adding up the weighted values
of all of the digits in the number. This is done by multiplying the digit
at each location by the value of the exponent of the radix at the
location in question.
Take as an example the Decimal number 320795:
The Value is obtained as:
105
3
104
2
103
0
102
7
Value = 3 x 105 + 2 x 104 + 0 x 103 + 7 x 102 +
101
9
100
5
9 x 101 + 5 x 100
3 x 105
=
3 x 100,000
=
300,000
2 x 104
=
2x
10,000
=
20,000
0 x 103
=
0x
1,000
=
0,000
7 x 102
=
7x
100
=
700
9 x 101
=
9x
10
=
90
5 x 100
=
5x
1
=
5
_______
320,795
This is relatively straight forward as the value of a Decimal number
has been obtained in Decimal form.
4
2.4 Binary Powers
Before working on the values of Binary numbers it is useful to have a
table of the Decimal values of powers of the radix 2. This will be
convenient when dealing with Binary numbers having many digits. It
will also allow familiarisation with some commonly used binary values,
e.g. 1kbit, 1Mbit, etc.
Binary
Power
20
Decimal Value
Binary Number
1
1
21
2
10
22
4
100
23
8
1000
24
16
10000
25
32
100000
26
64
1000000
27
128
10000000
28
256
100000000
29
512
1000000000
210
1,024
10000000000
1 kb
211
2,048
100000000000
2 kb
212
4,096
1000000000000
4 kb
213
8,192
10000000000000
8 kb
214
16,384
100000000000000
16 kb
215
32,768
1000000000000000
32 kb
216
65,536
10000000000000000
64 kb
217
131,072
100000000000000000
128 kb
218
262,144
1000000000000000000
256 kb
219
524,288
10000000000000000000
524 kb
220
1,048,576
100000000000000000000
1 Mb
230
1,073,741,824
1000000000000000000000000000000
1 Gb
5
2.5 Binary to Decimal Conversion
Converting a Binary number into Decimal form is essentially the same
thing as getting the decimal value of it. The same procedure can then
be used as for finding the value of the Decimal number above except
that the positional weightings applied will be those of a Radix 2
system.
Consider the Binary number 101011
The Decimal value is obtained as:
25
1
24
0
23
1
Value = 1 x 25 + 0 x 24 + 1 x 23 +
22
0
21
1
0 x 22
+ 1 x 21
20
1
+
1 x 20
1 x 25
=
1 x 32
=
32
0 x 24
=
0 x 16
=
0
1 x 23
=
1x8
=
8
0 x 22
=
0x4
=
0
1 x 21
=
1x2
=
2
1 x 20
=
1x1
=
1
___
43 Decimal
With experience of this process the zero digit terms can be omitted for
brevity. This means that only the powers of the radix which have a
digit of ‘1’ entered in that position need to be summed.
6
Examples:
1.
Binary 10111001
Value =
1 x 27
+
1 x 25
1
1
1
1
1
2.
x
x
x
x
x
+
27
25
24
23
20
1 x 24
=
=
=
=
=
+
128
32
16
8
1
___
185
1 x 23
+
1 x 20
+
20
Decimal
Binary 11101100101
Value
=
210
+
29
+
28
+
210
29
28
26
25
22
20
26
=
=
=
=
=
=
=
7
+
25
1024
512
256
64
32
4
1
____
1893
+
22
Decimal
2.6 Decimal to Binary Conversion
A Decimal number is converted to Binary form by continuously dividing
it by the Binary radix 2 and saving the remainder. This process
identifies the powers of the radix which are contained in the number
and the remainder of each division operation determines the value of
the bit in one weighted position at a time. The division process is
continued until nothing remains to be divided into. The repeated
division process yields the value of the digits of the number in Binary
form but in reverse order and consequently the result of the process
must be read backwards to extract the Binary number.
Consider the decimal number 237:
2│237
2│118 1
2│59 0
2│29 1
2│14 1
2│7 0
2│3 1
2│1 1
0 1
11101101
Binary
The resulting Binary number is read upwards through the remainders
of the division process to obtain the Binary number reading from left
to right.
The result can
form:
Value
=
=
=
be checked by converting the number back to decimal
27
128
237
+ 26 + 25 +
+ 64 + 32 +
Decimal
8
23
8
+ 22
+ 4
+ 20
+ 1
Example:
Consider the Decimal number 953:
2│953
2│476 1
2│238 0
2│119 0
2│59 1
2│29 1
2│14 1
2│7 0
2│3 1
2│1 1
0 1
1110111001
Binary
Checking the result by converting back to Decimal:
Value
=
29
+
28
+
27
=
512
+ 256 + 128
=
953
Decimal
9
+
25
+ 32
+
24
+ 16
+
+
23
8
+
+
20
1
2.7 Binary Encoding:
Returning again to the analogue signals, it can be seen that if the
analogue signal is transmitted over a medium where it will accumulate
noise then the accuracy and resolution of the original measurement
will be lost.
If instead of transmitting the analogue signal we can transmit digital
pulses then the effects of noise can be overcome. This is done by
taking samples of the signal at particular points of interest, more
usually at regular intervals in time, and encoding these values into
digital pulse form for storage or transmission.
Fig. 2.1
The Effect of Noise on Resolution
Value at t = 250 ms is V = 54.3 mV
with a resolution of 0.1 mV
The encoding can be done in two popular ways. The first is to take the
samples of the analogue signal at regular intervals and convert these
into Binary form. This is the most common form of analogue-to-digital
data conversion.
In this case the value of 54.3 will be treated as 543 and converted to
Binary form
543 Decimal = 1000011111
10
Binary
Some form of scaling convention can be used to deal with the decimal
point in the original sampled value. The binary number can then be
transmitted as a series of digital pulses as indicated in Fig. 2.1:
V
1
0
0
0
0
1
1
1
1
1
t
Another common way to encode the sample is to encode each of the
digits in the decimal number system into its equivalent binary form as
shown below:
Decimal
Number
0
1
2
3
4
5
6
7
8
9
Binary
Number
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
Each digit of a decimal value can then be coded as a 4-bit binary
number. The decimal value can then be transmitted as a sequence of
digital pulses. For example the decimal value
54.3
would be transmitted as
0101 0100
0111
Again, a convention can be used to identify the decimal point.
The transmitted or stored pulses may become noisy but can be
regenerated when received then decoded to recover the precise
decimal value transmitted without error.
11
```
Number theory

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