This document discusses different methods for representing data in computers, including numeric and character representations. It covers representing signed and unsigned integers using methods like sign-magnitude, 1's complement, and 2's complement. It also discusses floating point number representation using the IEEE standard. Finally, it discusses character representation using ASCII and Unicode encoding schemes.
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Number System
Decimal Number System
Binary Number System
Why Binary?
Octal Number System
Hexadecimal Number System
Relationship between Hexadecimal, Octal, Decimal, and Binary
Number Conversions
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
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Number System
Decimal Number System
Binary Number System
Why Binary?
Octal Number System
Hexadecimal Number System
Relationship between Hexadecimal, Octal, Decimal, and Binary
Number Conversions
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
It's part of Computer Organization And Architecture .Data representation is how to data represented in computer by using complements of number , float point ,fix point, so it's
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Analog System, digital system, numbering system, binary number
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Supervised (inductive) learning
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Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
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3. Data Representation in Computers
Data are stored in Registers
Registers are limited in number & size
With a n-bit register
Min value 0
Max value 2n-1
3
n-1 n-2 ...… ... 2 01
n-bits
4. 4
Data Representation
Data
Representation
• Represents quality or
characteristics
• Not proportional to a
value
• Name, NIC no, index no,
Address
Qualitative
Quantitative
• Quantifiable
• Proportional to value α
• No of students, marks for
CS2052, GPA
6. Number Systems
Decimal number system
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Binary number system
0, 1
Octal number system
0, 1, 2, 3, 4, 5, 6, 7
Hexadecimal number system
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
6
8. Signed Integers
We need a way to represent negative values
3 representations
Sign & Magnitude representation (S&M)
Complement method
Bias notation or Excess notation
8
9. 1. Sign & Magnitude Representation
n-bit unsigned magnitude & sign bit (S)
If S
0 – Integer is positive or zero
1 – Integer is negative or zero
Range –(2n-1) to +(2n-1)
9
sign n-1 ...n-2 ... 2 01
Magnitude (n-bits)
10. Example – Sign & Magnitude
If 8-bit register is used what are min & max
numbers?
What are 0000 0000 and 1000 0000 in decimal?
Representation of zero is not unique
10
11. Sign & Magnitude (Cont.)
Advantages
Sign reversal
Finding absolute value |a|
Flip sign bit
Disadvantage
Adding a negative of a number is not the same as
subtraction
e.g., add 2 and -3
Need different operations
Zero is not unique
11
12. 2. Complement Method
Base = Radix
Radix r system means r number of symbols
e.g., binary numbers have symbols 0, 1
2 types
r’s complement
(r – 1)’s complement
Where r is radix (base) of number system
Examples
Decimal 9’s & 10’s complement
Binary 1’s & 2’s complement
12
13. Complement Method – Definition
Given a number m in base/radix r & having n
digits
(r – 1)’s complement of m is
(rn – 1) – m
r ’s complement of n is
(rn – 1) – m + 1 = rn – m
13
14. Example – Complement Method
If m = 5982 & n = 4 digits
9’s complement is
9 9 9 9 - maximum representable no
5 9 8 2 –
4 0 1 7
10’s complement
9 9 9 9 or 1 0 0 0 0
5 9 8 2 – 5 9 8 2 –
4 0 1 7 4 0 1 8
1 +
4 0 1 8
14
15. Example – Complement Method
If m = 382 & n = 3
-n = -382 =
9 9 9 9 9 9
3 8 2 – 3 8 2 -
6 1 7 6 1 7
1 +
6 1 8
-382 = 617 or 618
Depending on which complement we use
These are called complementary pair
15
16. 1’s complement
Calculated by
(2n – 1) – m
If m = 0101
1’s complement of m on a 4-bit system
1 1 1 1
0 1 0 1 –
1 0 1 0
This represents -5 in 1’s complement
16
17. Finding 1’s Complement – Short Cut
Invert each bit of m
Example
m = 0 0 1 0 1 0 1 1
1’s complement of m = 1 1 0 1 0 1 0 0
m = 0 0 0 0 0 0 0 0 = 0
1’s complement of m = 1 1 1 1 1 1 1 1 = -0
Values range from -127 to +127
17
18. Addition with 1’s Complement
If results has a carry add it to LSB (Least
Significant Bit)
Example
Add 6 and -3 on a 3-bit system
6 = 1 1 0
-3 = 1 0 0 +
= 1 0 1 0
1 +
0 1 1
18
19. 2’s Complement
Doesn’t require end-around carry operation as in
1’s complement
2’s complement is formed by
Finding 1’s complement
Add 1 to LSB
New range is from -128 to +127
-128 because of +1 to negative value
19
20. Example – 2’s Complement
Find 2’s complement of 0101011
m = 0 1 0 1 0 1 1
= 1 0 1 0 1 0 0
________ 1 +
2’s = 1 0 1 0 1 0 1
Short-cut
1. Search for the 1st bit with value 1 starting from LSB
2. Inver all bits after 1st one
20
21. Example – 2’s Complement (Cont.)
Add 6 and -5 on a 4-bit system
5 = 0101
-5 = 1011
6 = 0 1 1 0
-5 = 1 0 1 1+
1 0 0 0 1 0 0 0 1
21
Discard
22. 1’s vs. 2’s Complement
1’s complement has 2 zeros (+0, -0)
Value range is less than 2’s complement
2’s complement only has a single zero
Value range is unequal
No need of a separate subtract circuit
Doing a NOT operation is much more cost effective
interms of circuit design
However, multiplication & division is slow
22
25. Detecting Negative Numbers &
Overflow
Check for MSB
To find magnitude
1’s complement
Flip all bits
2’s complement
Flip all bits + 1
Rules to detect overflows in 2’s complement
If sum of 2 positive numbers yields a negative result,
sum has overflowed
If sum of 2 negative numbers yields a positive result,
sum has overflowed
Else, no overflow 25
26. 3. Bias Notation
Number range to be represented is given a
positive bias so that smallest unsigned value is
equal to -B
If bias is B
Value B in number range becomes zero
If bias is 127 for 8-bit number range is
-127 (0 - 127) to 128 (255 - 127) 26
n
i
i
i Bb
0
2
31. IEEE Floating Point Standard (Cont.)
E – 127 = e
E = e + 127
What is stored is E
31
s31 e30 ...… e23 … m0m1m22 …mn-2
MantissaExponent
127
2].1[)1(
Es
mN
32. Single Precision
23-bit mantissa
m ranges from 1 to (2 – 2-23 )
8-bit exponent
E ranges from 1 to 254
0 & 255 reserved to represent -∞, +∞, NaN
e ranges from -126 to +127
Maximum representable number
2 x 2127 = 2128
32
33. Review – Binary Fractions
What is 101.11012 in decimal?
22 x 1 + 21 x 0 + 20 x 1 + 2-1 x 1 + 2-2 x 1 + 2-3 x 0 + 2-4 x 1
= 4 x 1 + 2 x 0 + 1 x 1 + 0.5 x 1 + 0.25 x 1 + 0.125 x 0 + 0.0625 x 1
= 4 + 1 + 0.5 + 0.25 + 0.0625
= 5.8125
What is 5.812510 in binary?
Integer – 5 = 101
Fraction – Keep multiplying fraction until answer is 1
0.8125 x 2 = 1.625
0.625 x 2 = 1.25
0.25 x 2 = 0.5
0.5 x 2 = 1.0
0.812510 = .11012
33
34. Example – Single Precision
IEEE Single Precision Representation of 17.1510
Find 17.1510 in binary
17.1510 = 10001.00100112
= 1.00010010011 x 24
E = e + 127
E = 4 + 127 = 131
34
0 1000 0011 0001 0010 0110 0000 ……..
MantissaExponent
35. Character Representation
Belong to category of qualitative data
Represent quality or characteristics
Includes
Letters a-z, A-Z
Digits 0-9
Symbols !, @ , *, /, &, #, $
Control characters <CR>, <BEL>, <ESC>, <LF>
35
36. Character Representation (Cont.)
With a single byte (8-bits) 256 characters can be
represented
Standards
ASCII – American Standard Code for Information
Interchange
EBCDIC – Extended Binary-Coded Decimal
Interchange Code
Unicode
36
37. ASCII
De facto world-wide standard
Used to represent
Upper & lower-case Latin letters
Numbers
Punctuations
Control characters
There are 128 standard ASCII codes
Can be represented by a 7 digit binary number
000 0000 through 111 1111
Plus parity bit
37
40. ASCII – Things to Note
ASCII codes for digits aren’t equal to numeric
value
Uppercase & lowercase alphabetic codes differ
by 0x20
Shift key clears bit 5
0x20 = 32 = 0010 0000
Example:
A = 65 = 0100 0001
a = 97 = 0110 0001
Most languages need more than 128 characters
40
41. Unicode (www.unicode.org)
Designed to overcome limitation of number of
characters
Assigns unique character codes to characters in
a wide range of languages
A 16-bit character set
UCS-2
UCS-4 is 32-bit
65,536 (216) distinct Unicode characters
Unicode provides a unique number for every character,
no matter what the platform,
no matter what the program,
no matter what the language
41