Fixed Point Notation
Fixed point notation is a method of representing real numbers in computers where the position
of the decimal (or binary) point is fixed and does not change. In this system, a constant number
of digits (or bits) is reserved for both the integer part and the fractional part of the number. For
example, if you have a fixed-point format with 8 digits to the left and 2 digits to the right of the
decimal point, the number 23953223.49 would fit this format [1] [2] [3] [4] .
The position of the point is predetermined by the format, not by the value of the number.
This method is simple and fast for hardware implementation, making it common in
embedded systems, digital signal processing, and applications where performance and
predictability are critical [3] .
The range of numbers that can be represented is limited by the fixed allocation of bits to the
integer and fractional parts [2] .
Negative numbers can be represented using two’s complement notation [1] [4] .
Example:
If you have 4 bits for the integer part and 4 bits for the fractional part, the binary number 110.1
(which is 6.5 in decimal) would be represented as 0110.1000 in fixed point with this format [1] .
Floating Point Notation
Floating point notation is a more flexible way of representing real numbers, allowing the "point"
to "float"—that is, the position of the decimal (or binary) point can change depending on the
value being represented. This is achieved by expressing numbers in a form similar to scientific
notation:
For example, the decimal number 123.45 can be written as [5] [2] [6] [7] .
In computers, floating point numbers are typically stored using the IEEE 754 standard, which
divides the bits into three fields: sign bit, exponent, and mantissa (fractional part) [6] [7] .
This format allows representation of a much wider range of values (very large or very small)
compared to fixed point, but with variable precision depending on the magnitude of the
number [2] [6] .
The exponent field determines the range, and the mantissa determines the precision [6] [7] .
Floating point is ideal for scientific, engineering, and general-purpose computing where a
wide range of values is needed, but it is more complex and slower to process than fixed
point [2] [3] .
Example:
A 32-bit IEEE 754 floating point number consists of:
1 bit for the sign
8 bits for the exponent (with a bias)
23 bits for the mantissa (fractional part), with an implicit leading 1 [6] [7] .
So, the number 101.1101 in binary could be normalized and stored as , with the
sign, exponent, and mantissa stored in the appropriate fields [5] .
Key Differences
Feature Fixed Point Notation Floating Point Notation
Decimal/Binary Point Fixed position Position can "float" (determined by exponent)
Range Limited (depends on bit allocation) Very wide (depends on exponent size)
Precision Uniform across range Varies with magnitude
Speed/Complexity Faster, simpler Slower, more complex
Typical Use Cases Embedded systems, DSP, finance Scientific, engineering, general computing
Summary:
Fixed point is simple, fast, and predictable but limited in range and flexibility.
Floating point is flexible and covers a wide range, but is more complex and can introduce
rounding errors [2] [6] [7] .
⁂
1. https://cim.mcgill.ca/~langer/273/2-notes.pdf
2. https://pediaa.com/difference-between-fixed-point-and-floating-point/
3. https://en.wikipedia.org/wiki/Fixed-point_arithmetic
4. https://andybargh.com/fixed-and-floating-point-binary/
5. https://users.cs.fiu.edu/~downeyt/cop2400/float.htm
6. https://cse11-iiith.vlabs.ac.in/exp/floating-point-numbers/theory.html
7. https://www.tutorialspoint.com/fixed-point-and-floating-point-number-representations
