Data Representation By Surya and his Team #techwithsurya

Outline

Representing numbers

• Unsigned

• Signed

• Floating point

Representing characters & symbols

• ASCII
• Unicode

Data Representation in Computers

• Data are stored in Registers
• Registers are limited in number & size

• Range of n-bit register for unsigned value is 0 to 2^n-1.
• Range of n-bit register for signed value is -2^n-1+1 to 2^n-1-1.

Data Representation 

Number System:

• A Number system of base  or radix, r is a system that uses distinct symbols for r digits. 
• Decimal number system(Base 10)
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Binary number system(Base 2)
• 0, 1
• Octal number system(Base 8)
• 0, 1, 2, 3, 4, 5, 6, 7
• Hexadecimal number system(Base 16)
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Decimal

749.4=7*10^2+4*10^1+9*10^0+4*10^-1=700+40+9+0.4=749.4

Conversion between number system

• decimal to binary
• binary to decimal

Quantitative Numbers

Integers(Fixed-point Representation)

• Unsigned
• Signed

Non-integers(Floating-point Representation)

• Floating point numbers - 10.25, 3.33333…, 1/8 = 0.125

1.Signed Integers

• We need a way to represent negative values 

• 3 representations

Sign-Magnitude representation (S&M)

1’s Complement method

2’s Complement method

1. Sign-Magnitude Representation

• n-bit unsigned magnitude & sign bit (S)
• If MSB (Most Significant Bit) is-

0 – Integer is positive or zero 

1 – Integer is negative or zero

• Range –2^n-1+1 to +(2n-1-1)

Example – Sign-Magnitude

• If 8-bit register is used what are min & max  numbers?
• What are 0000 0000 and 1000 0000 in decimal?

1. Representation of zero is not unique
2. -  2^(n-1)-1   to   + 2^(n-1) -1
3. --127    to +127(8-bit)
4. -7    to    +7(4-bit)
5.  0,1…7,-0,-1…-7

Advantages

• Sign reversal
• Finding absolute value |a|
• Flip sign bit

Disadvantage

• Overflow of number 
• Example

(a)

                      1010

                     +1110
                     11000 ( where 1 is overflow in binary number system)

(b)

                    010

                  +111

                   1101         

(c)

                      642

                    +321

                    1163( overflow in octal)

• Adding a negative of a number is not the same as  subtraction

e.g., add 2 and -3Need different operations 

• Zero is not unique

2. Complement Method

• Base = Radix

Radix r system means r number of symbols

e.g., binary numbers have symbols 0, 1

• 2 types

1. r’s complement (Radix/Base Component)
2. (r – 1)’s complement (Diminished Radix/Base)
3. Where r is radix (base) of number system

Examples

1.Decimal
2.Binary
3.9’s & 10’s complement
4. 1’s & 2’s complement

Definition

• Given a number m in base/radix r & having n digits
• (r – 1)’s complement of m is

(r^n– 1) – m

• r ’s complement of m is

(r^n– 1) – m + 1= r^n– m

Example:

(a) If m = 5982 & n = 4 digits

• 9’s complement is

    9 9 9 9

 – 5 9 8 2 

   4 0 1 7

• 10’s complement 

                    or                 

1 0 0 0 0

–   5 9 8 2 

4 0 1 8

   9 9 9 9

–5 9 8 2

   4 0 1 7                

  +1 

 4 0 1 8

(b) If m = 382 & n = 3

n = 382 

 999                           1000

-382                           - 382

617                             618

-382 = 617 or 618

Depending on which complement we use

These are called complementary pair

(i) 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

1

1011(2’s Compliment)

This represents -5 in 1’s complement(1010).

Finding 1’s Complement – Short Cut

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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

         110

         100 

       1010

        +1

         011

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

Example

Pending 

Difference B/W 1s and 2s 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  in terms of circuit design

However, multiplication & division is slow.

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

 

Floating Point Numbers

• We needed to represent fractional values &  values beyond 2n – 1
• +3207.23 =3.20723*10^3
• -0.000321=-3.21*10^-4

where -3=Sign

           21=mantissa

           10=Radix

           -4=Exponent 

Formula

N=m.r^e

IEEE Floating Point Standard (FPS)

 

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