Floating Point Representation (original) (raw)
Last Updated : 30 Aug, 2025
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:
Value = Sign × Significand × BaseExponent
**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
| Precision | Base | Sign | Exponent | Significant |
|---|---|---|---|---|
| Single precision | 2 | 1 | 8 | 23+1 |
| Double precision | 2 | 1 | 11 | 52+1 |
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 2k-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 (23-1 -1 = 4-1 = 3)
bias = 127 for 32 bit conversion. (28-1 -1 = 128-1 = 127)
**Example:
01000001110100000000000000000000
10000011 = (131)10
131-127 = 4Hence 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 2k-1 -1 where 'k' is the number of bits in exponent field.
Thus bias = 127 for 32 bit. (28-1 -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