Floating Point Representation

The floating-point representation is a way to encode numbers in a format that can handle very large and very small values. It is based on scientific notation where numbers are represented as a fraction and an exponent. In computing, this representation allows for a trade-off between range and precision.

Format: A floating point number is typically represented as:

Where:

• Sign: Indicates whether the number is positive or negative.
• Significand (Mantissa): Represents the precision bits of the number.
• Base: Usually 2 in binary systems.
• Exponent: Determines the scale of the number.

Need for Floating-Point Representation

The floating-point representation is crucial because:

• Range: It can represent a wide range of values from the very large to very small numbers.
• Precision: It provides a good balance between the precision and range, making it suitable for the scientific computations, graphics and other applications where exact values and wide ranges are necessary.
• Flexibility: It adapts to different scales of numbers allowing for the efficient storage and computation of real numbers in the computer systems.

Number System and Data Representation

• Number Systems: The Floating point representation often uses binary (base-2) systems for the digital computers. Other number systems like decimal (base-10) or hexadecimal (base-16) may be used in the different contexts.
• Data Representation: This includes how numbers are stored in the computer memory involving binary encoding and the representation of the various data types.

Table-Precision Representation

Components of Floating Point Numbers

The three components of floating point numbers are:

• Sign bit: Indicates positive or negative number.
• Exponent: Represents the power to which the base (usually 2) is raised.
• Mantissa (Significand): Represents the significant digits of the number.

Floating Point to Decimal Conversion

To convert the floating point into decimal, we have 3 elements in a 32-bit floating point representation: 

Sign bit is the first bit of the binary representation. '1' implies negative number and '0' implies positive number. Example:

11000001110100000000000000000000

This is negative number.

Exponent is decided by the next 8 bits of binary representation. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where 'k' is the number of bits in exponent field. 
There are 3 exponent bits in 8-bit representation and 8 exponent bits in 32-bit representation.
Thus

bias = 3 for 8 bit conversion (2^3-1 -1 = 4-1 = 3) 
bias = 127 for 32 bit conversion. (2^8-1 -1 = 128-1 = 127) 

Example:

01000001110100000000000000000000 
10000011 = (131)₁₀ 
131-127 = 4 

Hence the exponent of 2 will be 4 i.e. 2⁴ = 16.

Mantissa is calculated from the remaining 23 bits of the binary representation. It consists of '1' and a fractional part which is determined by: 
Example: 

01000001110100000000000000000000 

The fractional part of mantissa is given by: 

1*(1/2) + 0*(1/4) + 1*(1/8) + 0*(1/16) +......... = 0.625 

Thus the mantissa will be

1 + 0.625 = 1.625 

The decimal number hence given as:

Sign*Exponent*Mantissa = (-1)⁰*(16)*(1.625) = 26

Decimal to Floating Point Conversion

To convert the decimal into floating point, we have 3 elements in a 32-bit floating point representation: 
    i) Sign (MSB) 
    ii) Exponent (8 bits after MSB) 
    iii) Mantissa (Remaining 23 bits) 
 

Sign bit is the first bit of the binary representation. '1' implies negative number and '0' implies positive number. 
Example: To convert -17 into 32-bit floating point representation Sign bit = 1

Exponent is decided by the nearest smaller or equal to 2ⁿ number. For 17, 16 is the nearest 2ⁿ. Hence the exponent of 2 will be 4 since 2⁴ = 16. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where 'k' is the number of bits in exponent field. 

Thus bias = 127 for 32 bit. (2^8-1 -1 = 128-1 = 127) 

Now, 127 + 4 = 131

i.e. 10000011 in binary representation.

Mantissa: 17 in binary = 10001.

Move the binary point so that there is only one bit from the left. Adjust the exponent of 2 so that the value does not change. This is normalizing the number. 1.0001 x 2⁴. Now, consider the fractional part and represented as 23 bits by adding zeros.

00010000000000000000000