Lecture notes on number system

1. CHAPTER 2B
NUMBER SYSTEMS,
OPERATIONS AND CODES
Lecture Hours : 3 hours
1

2. Very important to permit the representation of negative numbers in binary.
Binary numbers can also be represented by 1’s complement and 2’s complement.
Example
Find the 1’s complement of 101100102.
Solution
Change all 1 to 0 and all 0 to 1 ,
1 0 1 1 0 0 1 0
0 1 0 0 1 1 0 1
Thus, 1’s complement of 101100102 = 010011012
Binary number :
1’s complement :
1’s AND 2’s COMPLEMENTS
NUMBER OPERATIONS
2

3. Example
Find the 1’s complement of 000110102.
Solution
Change all 1 to 0 and all 0 to 1 ,
0 0 0 1 1 0 1 0
1 1 1 0 0 1 0 1
Thus, 1’s complement of 000110102 = 111001012
Binary number :
1’s complement :
1’s AND 2’s COMPLEMENTS
NUMBER OPERATIONS
3

4. Example
Find the 2’s complement of 101110002.
Solution
1 0 1 1 1 0 0 0
0 1 0 0 0 1 1 1
Binary number :
1’s complement :
2’s complement : 0 1 0 0 1 0 0 0
+ 1
Thus, 2’s complement of 101110002 = 010010002
1’s AND 2’s COMPLEMENTS
NUMBER OPERATIONS
4

5. Example
Find the 2’s complement of 000101102.
Solution
0 0 0 1 0 1 1 0
1 1 1 0 1 0 0 1
Binary number :
1’s complement :
2’s complement : 1 1 1 0 1 0 1 0
+ 1
Thus, 2’s complement of 000101102 = 111010102
1’s AND 2’s COMPLEMENTS
NUMBER OPERATIONS
5

6. Signed numbers can be represented by :
1. Sign-magnitude form
2. 1’s complement form
3. 2’s complement form
SIGNED NUMBERS
NUMBER OPERATIONS
6

7. The leftmost digit is reserved for the sign of the number.
The remaining digits are used to represent the magnitude of the number.
Example
Write the following decimal numbers as an 8-bit binary number in the
sign-magnitude form.
a) +25
b) –25
For positive sign, sign-bit = 0.
For negative sign, sign-bit = 1.
SIGNED MAGNITUDE FORM
NUMBER OPERATIONS
7

8. Example
Write the following decimal numbers as an 8-bit binary number in the
sign-magnitude form.
a) +53
b) –39
SIGNED MAGNITUDE FORM
NUMBER OPERATIONS
8

9. Example
Given, 10010101 is a signed binary number expressed in sign-magnitude. Determine
the decimal value.
Solution :
Binary number : 1 0 0 1 0 1 0 1
(sign-magnitude)
sign magnitude
Decimal : –
(64) (32) (16) (8) (4) (2) (1)
21
SIGNED MAGNITUDE FORM
NUMBER OPERATIONS
9

10. Positive numbers in 1’s complement = positive number in sign-magnitude form.
Example
Write the decimal number +25 as an 8-bit binary number in the 1’s complement
form.
Solution :
+25 in 1’s complement form = +25 in sign-magnitude form
= 00011001
+25 00011001
1’s complement
1’s COMPLEMENT FORM
NUMBER OPERATIONS
10

11. Example
Write the decimal number –39 as an 8-bit binary number in the 1’s complement
form.
Solution
Then, we perform 1’s complement for 00100111
Negative number in 1’s complement form = 1’s complement of its corresponding positive number
= 00100111
= 11011000
First, write +39 in 1’s complement form = +39 in sign-magnitude form
–39 11011000
1’s complement
1’s COMPLEMENT FORM
NUMBER OPERATIONS
11

12. Example
Write the decimal number –19 as an 8-bit binary number in the 1’s complement form.
Solution
Then, we perform 1’s complement for 00010011
= 00010011
= 11101100
First, write +19 in 1’s complement form = +19 in sign-magnitude form
–19 11101100
1’s complement
1’s COMPLEMENT FORM
NUMBER OPERATIONS
12

13. Positive numbers in 2’s complement = positive number in sign-magnitude form.
Example
Write the decimal number +25 as an 8-bit binary number in the 2’s complement
form.
Solution :
+25 in 2’s complement form = +25 in sign-magnitude form
= 00011001
+25 00011001
2’s complement
2’s COMPLEMENT FORM
NUMBER OPERATIONS
13

14. Negative number in 2’s complement form = 2’s complement of its corresponding positive number
Example
Write the decimal number –39 as an 8-bit binary number in the 2’s complement form.
Solution
Then, we perform 1’s complement for 00100111
= 00100111
= 11011000
First, write +39 in 2’s complement form = +39 in sign-magnitude form
–39 11011001
2’s complement
Finally, we perform 2’s complement by + 1
11011001
2’s COMPLEMENT FORM
NUMBER OPERATIONS
14

15. Example
Write the decimal number –19 as an 8-bit binary number in the 2’s complement form.
Solution
Then, we perform 1’s complement for 00010011
= 00010011
= 11101100
First, write +19 in 2’s complement form = +19 in sign-magnitude form
–19 11101101
2’s complement
Finally, we perform 2’s complement by + 1
11101101
2’s COMPLEMENT FORM
NUMBER OPERATIONS
15

16. 1’s complement and 2’complement binary format has the same weight as the true
binary EXCEPT, the most significant digit has negative sign.
–2n ….. 24 23 22 21 20
For 1’s complement, if the sum of weight is a negative value, we must add 1.
SIGNED BINARY NUMBER TO DECIMAL CONVERSION
NUMBER OPERATIONS
16

17. Example
The followings are signed binary numbers expressed in 1’s complement.
Convert to decimal value.
a) 00010111
b) 11101000
c) 11101011
Answer : a) +23 b) –23 c) –20
SIGNED BINARY NUMBER TO DECIMAL CONVERSION
NUMBER OPERATIONS
17

18. Example
The followings are signed binary numbers expressed in 2’s complement.
Convert to decimal value.
a) 01010110
b) 10101010
c) 11010111
Answer : a) +86 b) –86 c) –41
SIGNED BINARY NUMBER TO DECIMAL CONVERSION
NUMBER OPERATIONS
18

19. Exercise
Express the following decimal numbers as an 8-bit binary number in sign-magnitude,
1’s complement and 2’s complement.
a) + 9
b) – 33
c) – 46
NUMBER OPERATIONS
19

20. Decimal Binary BCD
0 0000 0000
1 0001 0001
2 0010 0010
3 0011 0011
4 0100 0100
5 0101 0101
6 0110 0110
7 0111 0111
8 1000 1000
9 1001 1001
10 1010 0001 0000
11 1011 0001 0001
12 1100 0001 0010
13 1101 0001 0011
14 1110 0001 0100
15 1111 0001 0101
20
BINARY CODED DECIMAL
DIGITAL CODES: BCD, GRAY, PARITY
What:
• a way to express each of the decimal digits with a binary
code.
• each decimal digit, 0 through 9, is represented by a 4-bit
binary code
• codes 1010 through 1111 not used
Why:
It is very easy to convert between decimal and BCD. Because
we like to read and write in decimal, the BCD code provides an
excellent interface to binary systems.
How:
Examples - interfaces are keypad inputs and digital readouts

21. 21
BINARY CODED DECIMAL
DIGITAL CODES: BCD, GRAY, PARITY
Y
ou can think of BCD in terms of column weights in groups of four bits. For an 8-bit
BCD number, the column weights are: 80 40 20 10 8 4 2 1.
Note that you could add the column weights where there is a 1 to obtain the decimal number.
For this case:
What are the column weights for the BCD number 1000 0011 0101 1001?
8 4 2 1
80 40 20 10
800 400 200 100
8000 4000 2000 1000
8000 + 200 +100 + 40 + 10 + 8 +1 = 835910

22. 22
GRAY CODE
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
10 1010
11 1011
12 1100
13 1101
14 1110
15 1111
Decimal Binary Gray code
0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000
What:
Gray code is an unweighted code that has a single bit
change between one code word and the next in a
sequence.
Why:
Gray code is used to avoid problems in systems
where an error can occur if more than one bit
changes at a time.
DIGITAL CODES: BCD, GRAY, PARITY

23. 23
• Binary to Gray code
• MSB of Gray is set to the MSB of binary
• Going from left to right, add each adjacent pair of binary code bits. Discard carries.
• Example. Convert 10110 to Gray Code
MSB of binary number is 1, so set the MSB of the Gray Code to 1.
1 0 1 1 0 Binary
1 Gray code
Add adjacent pairs of binary numbers
1+0
1
0+1
1
1+0
1
1+1
0
GRAY CODE CONVERSION
DIGITAL CODES: BCD, GRAY, PARITY

24. 24
• Gray code to Binary
• MSB of binary is set to the MSB of Gray code
• Going from left to right, add the binary bit to the next Gray code bit. Discard carries.
• Example. Convert 11011 to binary
MSB of Gray Code is 1, so set the MSB of the binary number to 1.
1 1 0 1 1 Gray Code
1 Binary
+
0
+
0
+
1
+
0
GRAY CODE CONVERSION
DIGITAL CODES: BCD, GRAY, PARITY

25. 25
A simplified illustration of how the Gray code solves the error problem in shaft position encoders.
GRAY CODE APPLICATION
Ashaft encoder is a typical
application. Three IR
emitter/detectors are used
to encode the position of
the shaft. The encoder on
the left uses binary and can
have three bits change
together, creating a
potential error. The encoder
on the right uses gray code
and only 1-bit changes,
eliminating potential errors.
DIGITAL CODES: BCD, GRAY, PARITY

26. 26
SUMMARY
Decimal Binary BCD Gray Code
0 0000 0000 0000
1 0001 0001 0001
2 0010 0010 0011
3 0011 0011 0010
4 0100 0100 0110
5 0101 0101 0111
6 0110 0110 0101
7 0111 0111 0100
8 1000 1000 1100
9 1001 1001 1101
10 1010 0001 0000 1111
11 1011 0001 0001 1110
12 1100 0001 0010 1010
13 1101 0001 0011 1011
14 1110 0001 0100 1001
15 1111 0001 0101 1000
2.3 DIGITAL CODES: BCD, GRAY, PARITY
DIGITAL CODES: BCD, GRAY, PARITY

27. Sender Receiver
Transmission error happens
0101 0101 

1101 1100
The parity method is a method of error detection for simple
transmission errors involving one bit (or an odd number of bits).
A parity bit is an “extra” bit attached to a group of bits to force the
number of 1’s to be either even (even parity) or odd (odd parity).
ERROR DETECTION: PARITY METHOD
DIGITAL CODES: BCD, GRAY, PARITY
27

28. Sender Receiver
Even parity
01010 01010 
11011 11001 
Parity bit
Odd parity
01011 01011 
11010 11000 
ERROR DETECTION: PARITY METHOD
DIGITAL CODES: BCD, GRAY, PARITY
28

29. 29
SELECTED KEY TERMS
Byte
Floating-point
number
Hexadecimal
Octal
BCD
Agroup of eight bits
A number representation based on scientific
notation in which the number consists of an
exponent and a mantissa.
Anumber system with a base of 16.
Anumber system with a base of 8.
Binary coded decimal; a digital code in which each
of the decimal digits, 0 through 9, is represented by
a group of four bits.

30. 30
SELECTED KEY TERMS
Alphanumeric
Parity
Consisting of numerals, letters, and other
characters
In relation to binary codes, the condition of
evenness or oddness in the number of 1s in a code
group.