This document discusses representation of binary numbers and binary codes. It covers: - Unsigned and signed binary number representation including sign-magnitude and 1's/2's complement forms. - Binary coded decimal (BCD) which assigns a 4-bit code to each decimal digit from 0-9. BCD is useful for digital displays that show decimal numbers. - Excess-3 code which adds 3 to each decimal digit before converting to binary. - Gray code which increments only one bit at a time to avoid errors during counting. Conversion between binary and gray code is covered.

1. TOPIC: Representation of binary numbers
and
Binary Codes
Department of Electronics and Communication Engineering
Chitkara University, Punjab, India
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2. Representation of binary numbers
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3. Unsigned Numbers
• don’t have any sign
• contain only magnitude of the number.
Example-1: Represent decimal number 92 in unsigned binary number.
(92)10
= (1011100)2
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4. Unsigned Numbers
Example-2: Find range of 6 bit unsigned binary
numbers. Also, find minimum and maximum value in
this range.
Sol: Since, range of unsigned binary number is from 0
to (2n-1). Therefore, range of 6 bit unsigned binary
number is from 0 to (26-1) which is equal from
minimum value 0 (i.e., 000000) to maximum value 63
(i.e., 111111).
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5. Signed Numbers
• Unsigned representation can be used for positive integers
• How about negative integers?
– Everything must be represented in binary numbers
– Computers cannot use – or + signs
Signed numbers:
• contain sign flag
• contains both sign bit and magnitude of a number
• this representation distinguish positive and negative numbers
• For negative numbers the sign bit is always 1, and for positive numbers it
is 0 in these three systems
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6. Representation of signed numbers
There are two ways of representing negative binary numbers:
1. Sign Magnitude form
2. Complement Method
- 1’s Complement form
- 2’s Complement form
• Advantage of using complement method for subtraction is
reduction in hardware.
• Instead of having separate circuits for addition and
subtraction, only addition circuits are needed.
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7. Sign-Magnitude form
• For n bit binary number, 1 bit is reserved for sign symbol
• The leftmost bit is the sign bit (0 is + and 1 is - ) and the
remaining bits hold the absolute magnitude of the number
• For 8 bits, we can represent the signed integers –127 to +127
• How about for N bits? -(2n-1-1)to +(2n-1 -1)
• Examples
• -47 = 1 0 1 0 1 1 1 1
• 47 = 0 0 1 0 1 1 1 1
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8. 1’s Complement form
• Replace each 1 by 0 and each 0 by 1
• Example (-6)
– First represent 6 in binary format (00000110)
– Then replace (11111001)
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9. 2’s Complement form
• Find one’s complement
• Add 1
• Example (-6)
– First represent 6 in binary format (00000110)
– One’s complement (11111001)
– Two’s complement (11111010)
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Handy Trick: Leave all of the least significant 0’s and first 1
unchanged, and then “flip” the bits for all other digits.
Eg: 01010100100 -> 10101011100

10. 1’s and 2’s complements
• 1’s complement of 10111001
– 11111111 – 10111001 = 01000110
– Simply replace 1’s and 0’s
• 1’s complement of 10100010
– 01011101
• 2’s complement of 10111001
– 01000110 + 1 = 01000111
– Add 1 to 1’s complement
• 2’s complement of 10100010
– 01011101 + 1 = 01011110
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11. NOTE
• “Humans” normally use sign-magnitude
representation for signed numbers
– Eg: Positive numbers: +N or N
– Negative numbers: -N
• “Computers” generally use two’s-complement
representation for signed numbers
– First bit still indicates positive or negative.
– If the number is negative, take 2’s complement to
determine its magnitude
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12. BINARY CODES
Classification of Binary Codes:
• BCD code
• Excess three code
• Gray code
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13. Human Perception
• We naturally live in a base 10 environment
• Computer exist in a base 2 environment
• So give the computer/digital system the task of doing
the conversions for us.
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14. Binary Codes
14
 A binary code represents text, computer processor
instructions, or any other data using a two-symbol
system.
The two-symbol system used is often "0" and "1"
from the binary number system.
 The binary code assigns a pattern of binary digits,
also known as bits, to each character, instruction, etc.
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15. Classification of Binary Codes
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16. Defination of BCD..Binary Coded
Decimal
•Binary coded decimal (BCD) is a system of writing numbers that
assigns a four-digit binary code to each digit 0 through 9 in a decimal
(base-10) numeral.
• The four-bit BCD code for any particular single base-10 digit is its
representation in binary notation, as follows:
0 = 0000
1 = 0001
2 = 0010
3 = 0011
4 = 0100
5 = 0101
6 = 0110
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17.  7 = 0111
 8 = 1000
 9 = 1001
Numbers larger than 9, having two or more digits in the decimal
system, are expressed digit by digit. For example, the BCD
rendition of the base-10 number 1895 is
1 8 9 5
0001 1000 1001 0101
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18. Why BCD is Used...
• It is easy to encode and decode decimals into BCD and
vice versa. • It is also simple to implement a hardware
algorithm for the BCD converter.
• It is very useful in digital systems whenever decimal
information is given eitheras inputs or displayed as
outputs.
• Digital voltmeters, frequency converters and digital
clocks all use BCD as they display output information in
decimal
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19. 8421 BCD Code
• Valid BCD code are : 0000 to 1001
• Invalid BCD code are :1010 to 1111
Example:
Decimal number 4926 4 9 2 6
BCD coded number 0100 1001 0010 0110
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20. Contd.
Convert the BCD coded number 1000 0111 0001
into decimal.
BCD Coded Number 1000 0111 0001
Decimal Number 8 7 1
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21. Convert the decimal number 350 to its
BCD equivalent.
Decimal Number 3 5 0
BCD Coded Number 0011 0101 0000
Contd.
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22. Excess 3 Code
• Add 3 to each digit of decimal and convert it to 4-bit binary form.
• Valid excess-3 code : 0011 to 1100
• Invalid excess-3 code : [0000 to 0010] and [1101 to 1111]
Decimal Binary +3 Excess-3
0 0000 0011 0011
1 0001 0011 0100
2 0010 0011 0101
3 0011 0011 0110
4 0100 0011 0111
5 0101 0011 1000
6 0110 0011 1001
7 0111 0011 1010
8 1000 0011 1011
9 1001 0011 1100
Decimal 3 5 9
Sample Problem:
Excess-3 0110 1000 1100
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23. Gray Code
• The Gray code’s most important
characteristic is that only one
digit changes as you increment or
decrement the count. (unit
distance code)
• The Gray code is NOT a BCD
code.
Decimal Gray code
0 00000
1 00001
2 00011
3 00010
4 00110
5 00111
6 00101
7 00100
8 01100
9 01101
10 01111
11 01110
12 01010
13 01011
14 01001
15 01000
16 11000
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24. Binary to Gray Code Conversion
• The MSB in the Gray code is the same as corresponding MSB in the binary
number.
• Going from left to right, add each adjacent pair of binary code bits to get the
next Gray code bit. Discard carries.
• Example 1:
• Example 2:
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25. Activity 1:
Convert the following binary codes to gray codes:
1. 10010111
2. 10001001
3. 01101010
Convert the following to Gray codes:
4. (527)8 - 1111 11100
5. (3A7)16 - 0
010 0111 0100
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26. Gray to Binary Conversion
• The MSB in the Binary code is the same as corresponding MSB in the Gray
number.
• Going from bottom to top, add each pair of gray code bits to get the next
binary code bit. Discard carries.
• Example 1:
• Example 2:
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27. Activity 2:
Convert the following gray codes to binary codes:
1. 11001100
10001000
1. 00110011
2. 00100010
3. 11111000
4. 10101111
11111000
10101111
11111000
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28. Thank You
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