Data Representation

Digital computers store and process information in binary form as digital logic has only two values "1" and "0" or in other words "True or False".
This system is called radix 2.
We human generally deal with radix 10 i.e. decimal.
As matter of convenience there are many other representations like Octal (Radix 8), Hexadecimal (Radix 16), Binary coded decimal (BCD), Decimal etc.
Every computer's CPU has a width measured in terms of bits such as 8 bit CPU, 16 bit CPU, 32 bit CPU etc.
Similarly, each memory location can store a fixed number of bits and is called memory width.
Given the size of the CPU and Memory, it is for the programmer to handle his data representation.

Data Representation in Computers

Information handled by a computer is classified as intruction and data.
A broad overview of the internal representation of the information ↓
No matter whether data is in numeric or non-numeric form or integer, everything is internally represented in Binary.
It is up to the programmer to handle the interpretation of the binary pattern and this interpretation is called Data Representation
Choice of Data representation to be used in a computer is decided by
- The number types to be represented (integer, real, signed, unsigned, etc)
- Range of vlaues likely to be represented (maximum and minimum to be represented)
- The precision of the numbers i.e. maximum accuracy of representation (floating point single precesion, double precision etc)
- If non-numeric i.e. character, character representation standard to be chose.
  - ASCII, EBCDIC, UTF are examples of character representation standards.
- The hardware support in terms of word width, instruction.

The number of bits used in representing the integer also implies the maximum number that can be represented in the system hardware.
However for the efficiency of storage and operations, one may choose to represent the integer with one Byte, two Bytes, Four bytes or more.
This space allocation is translated from the definition used by the programmer while defining a variable as integer short or long and the Instruction Set Architecture.
In addition to the bit length definition for integers, we also have a choice to represent them as below:
- Unsigned Integer
  - A positive number including zero can be represented in this format.
  - All the alloted bits are utilised in defining the number.
  - So if one is using 8 bits to represent the unsigned integer, the range of values that can be represented is 28 i.e. "0" to "255".
  - If 16 bits are used for representing then the range is 216 i.e. "0 to 65535".
- Signed Integer
  - In this format negative numbers, zero and positive numbers can be prepresented.
  - A sign bit indicates the magnitude direction as positive or negative.
  - There are three posibble representations signed integer and these are Sign Magitude format, 1's Compliment format and 2's Compliment format.

Sign Magnitude Format

Most Significant Bit (MSB) is reserved for indicating the direction of the magnitude (value).
A "0" on MSB means a positive number and a "1" on MSB means a negative number.
If n bits are used for representation, n-1 bits indicate the absolute value of the number.
Example for n = 8
- 0010 1111 = + 47 Decimal (Positive number)
- 1010 1111 = - 47 Decimal (Negative Number)
- 0111 1110 = +126 (Positive number)
- 1111 1110 = -126 (Negative Number)
- 0000 0000 = + 0 (Postive Number)
- 1000 0000 = - 0 (Negative Number)
Although this method is easy to understand, Sign Magnitude representation has several shortcomings like
- Zero can be represented in two ways causing redundancy and confusion.
- The total range for magnitude representation is limited to 2n-1, although n bits were accounted.
- The separate sign bit makes the addition and subtraction more complicated. Also comparing two numbers is not straigt forward.

1's Complement Format

In this format too, MSB is reserved as the sign bit.
But the difference is in representing the Magnitude part of the value for negative numbers (magnitude) is inversed and hence called 1's Complement form.
The positive numbers are represented as it is in binary.
Examples for n = 8
- 0010 1111 = + 47 Decimal (Positive number)
- 1101 0000 = - 47 Decimal (Negative Number)
- 0111 1110 = +126 (Positive number)
- 1000 0001 = -126 (Negative Number)
- 0000 0000 = + 0 (Postive Number)
- 1111 1111 = - 0 (Negative Number)

notes

Converting a given binary number to its 2's complement form

Step 1 → -x = x' + 1 where x' is the one's complement of x.
Step 2 → Extend the data width of the number, fill up with sign extension i.e. MSB bit is used to fill the bits.
Example : -47 decimal over 8bit representation
- Binary equivalent of + 47 is 0010 1111
- Binary equivalent of - 47 is 1010 1111 (Sign Magnitude Form)
- 1's complement equivalent is 1101 0000
- 2’s complement equivalent is 1101 0001
As you can see zero is not getting represented with redundance.
There is only one way of representing zero.
The other problem of the comlexity of the arithmetic operation is also eliminated in 2's complement representation.
Subraction is done as Addition.

more notes

The maximum number at best represented as a whole number is 2ⁿ.
In Scientific world, we do come across numbers like Mass of an Electron is 9.10939 x 10^-31 Kg. Velocity of light is 2.99792458 x 10⁸ m/s.
It makes no sense to write a number in non-readable form or non-processible form.
Hence we write such large or small numbers using exponent and mantissa.
This is said to be Floating Point representation or real number representation.
Representationin computer
- Unlike the two's compliment representation for integer numbers, Floating Point number uses Sign and Magnitude representation for both mantissa and exponent,
- In the number 9.10939 x 10^-31, in decimal form, -31 is Exponent, 9.10939 is known as Fraction. Mantissa, Significand and fraction are synonymously used terms.

Floating Point Representation

We have two standard known as Single Precision and Double Precision from IEEE.
These standards enable portability among different computers.
Single Precision uses 32bit format.
- Single Precision can represent exponent in the range -127 to +127.
- It is possible as a result of arithmetic operations the resultinf exponent may not fit in. This situation is called overflow in case of positive exponent and underflow in the case of negative exponent.
Double precision uses 64 bits word length
- Double precision can represent fractions with larger accuracy.
- The Double Precision format has 11 bits for exponent meaning a number as large as -1023 to 1023 can be represented.
In both the cases, MSB is sign bit for mantissa part, followed by Exponent and Mantissa. The exponent part has its sign bit.
The Floating Point operations on the regular CPU is very very slow.
- Generally, a special purpose CPU known as Co-processor is used.
- This Co-processor works in tandem with the main CPU.
- The programmer shoulb be using the float declaration only if his data is in real number form. float declaration is not to be used generously.