number system and codes

1. G.Vinothini M.sc.,M.phil.,
Department of Information Techonology,
Bon Secours College for Women,
Thanjavur.

3.  The binary number system is a
numbering system that represents
numeric values using two unique digits
(0 and 1).
 The binary number system is also
called base-2 number system.
Ex: (1011001)2

4. The decimal or “denary” counting system
uses the Base-of-10 numbering system
where each digit in a number takes on one
of ten possible values, called “digits”,
from 0 to 9
Ex: 21310

5.  Binary to Decimal Conversion of
numbers uses weighted columns to
identify the order of the digits to
determine the final value of the number
 Conversion of binary to decimal
(base-2 to base-10) numbers

6. binary
number
:
1 1 1 0 0 1
power
of 2:
25 24 23 22 21 20
Example :
Find the decimal value of 1110012:
1110012 = 1(2)5+1(2)4+1(2)3+0(2)2+0(2)1+1(2)0
= 5710

7. Conversion steps:
 Divide the number by 2.
 Get the integer quotient for the next
iteration.
 Get the remainder for the binary digit.
 Repeat the steps until the quotient is
equal to 0.

8.  Convert (43)10 Binary
2 43
2 21 - 1
2 10 - 1
2 5 - 0
2 2 - 1
1 - 0
=(101011)2
 Convert (21.6)10 Binary
2 21 0.6*2=1.2(1)
2 10 - 1 0.2*2=0.4(0)
2 5 - 0 0.4*2=0.8(0)
2 2 - 1 0.8*2=1.6(1)
1 - 0
=(10101 . 1001)2

9. Binary addition is much like your
normal everyday addition (decimal
addition), except that it carries on a
value of 2 instead of a value of 10.
Therefore in binary:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (which is 0 carry 1)

10. 1. 100101 + 10101 = ?.
Answer: 100101
10101 +
= 111010.
2. 1010 + 11 = ?
Answer: 1010
11 +
= 1101.

11. Binary subtraction is also similar to that of decimal
subtraction with the difference that when 1 is
subtracted from 0, it is necessary to borrow 1 from
the next higher order bit and that bit is reduced by
1 (or 1 is added to the next bit of subtrahend) and
the remainder is 1.
Rules:
0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 with a borrow of 1

12. I) 101 from 1001
Solution:
101 from 1001
1 Borrow
1 0 0 1
1 0 1
1 0 0
ii) 1010101.10
from 1111011.11
Solution:
1010101.10 from
1111011.11
1 Borrow
1 1 1 1 0 1 1 . 1 1
1 0 1 0 1 0 1 . 1 0
1 0 0 1 1 0 . 0 1

13. The 1’s complement of a binary number is
the number that results when we change
each 0 to 1 and each 1 to o in other words
1’s complement of 100 is 011.
Ex:
The 1’s complement of 1001 is 0110
The 1’s complement of 1010 is 0101

14. The 1’s complement of the subtrahend is added the
minuend .The last carry (called the end around carry) if any ,
is then added to the partial result to get the final answer.
1.Sub 1101 from 1010
1010
0010 +
--------
1100
Ans : -0011
2. Sub 0110 from 1001
1001
1001 +
--------
10010
1 +
---------
Ans: 0011

15. The 2’s complement of a binary number
is obtained by adding 1 to its 1’s complement .
i.e.,
2’s complement = 1’s complement + 1
Ex:
Number 2’s complement
1110 0001+1 =0010
0001 1110+1 = 1111
10110 01001+1 =01010

16. 1.Sub 101 from 111
111
010 +
--------
1010
Ans : 010
2.Sub 0111 from 0110
0110
1001 +
--------
1111
--------
0000
1 +
--------
Ans: -0001

17. The octal numeral system, or oct for short, is
the base-8 number system, and uses the digits
0 to 7.
5017 in Octal is equivalent to 101 000 001 111 in
binary.
24.3 in Octal is 010 100. 011 in binary.

18. The hexadecimal numeral system, often shortened
to "hex", is a numeral system made up of 16 symbols
symbols (base 16). The standard numeral system is
called decimal (base 10) and uses ten symbols:
0,1,2,3,4,5,6,7,8,9.
The English alphabet are used, specifically A, B, C, D, E
and F. Hexadecimal A = decimal 10, and hexadecimal F =
decimal 15.
Ex:
Convert (0111 1101)2 to hexadecimal
0111 1101 = (7D)16
7 D

19. They are several methods that are used to
express both numbers & letters as binary codes.
It can be classified into following
categories:
 8421 code
 BCD code
 Express – 3

20. It is the non-weighted code and it is not
arithmetic codes. That means there are no
specific weights assigned to the bit position.
Gray code cannot be used for arithmetic
operation.
0+0=0
1+0=1
0+1=1
1+1=0

21.  Another weighted code is 5043210.
This biquinary code is an example of a 7 bit
code with error detection properties. Each
biquinary code consists of 5 zeros and 2 ones
placed in the corresponding weighted column
.
 One or more bits may change value.

22.  Error-detecting codes are a sequence of numbers
generated by specific procedures for detecting errors in data that
has been transmitted over computer networks.
 When bits are transmitted over the computer network,
they are subject to get corrupted due to interference and network
problems. The corrupted bits leads to spurious data being
received by the receiver and are called errors.
 Error – detecting codes ensures messages to be
encoded before they are sent over noisy channels. The encoding
is done in a manner so that the decoder at the receiving end can
detect whether there are errors in the incoming signal with high
probability of success.

23. It is used to detect to errors within the same word. For
example , if 10101010 where transmitted using even
parity and a 10011010 where received , it would
appear as though no error had occurred.
EXAMPLE :
Word A 10110111
Word B 00100010
Sum 11011001

24. ECC (either "error correction [or correcting]
code" or "error checking and correcting") allows
data that is being read or transmitted to be
checked for errors and, when necessary,
corrected on the fly. It differs from parity-
checking in that errors are not only detected but
also corrected. ECC is increasingly being
designed into data storage and transmission
hardware as data rates (and therefore error
rates) increase.

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