1. Fundamentals of Computer Science
Lecture 7

2. Floating Point Representation
Numbers used in scientific calculations are designed by a sign, by the magnitude of the number, and by the position of the radix point.
The position of the radix point is required to represent fractions, integers, or mixed integer-fraction numbers.
There are two ways of specifying the position of the radix point which are:
Fixed-point representation (which we have studied up till now)
Floating-point representation
There are two ways for positioning the radix point. They are:
Putting the radix point at the extreme left of the number.
Putting the radix point at the extreme right of the number.

3. Floating Point Representation
However, it should be noted that in both cases, the radix point is not actually present in the digital system; rather, its position is implied by the fact that the number is predefined as an integer or fraction.
It is most popular to use floating-point notation for storing fractions in the main memory of the computer.
In decimal system, the floating-point notation ; which is also called “scientific notation”; of any real number can be represented as :
(N)10 = F × 10E
Where,
(N)10 : is the real number in decimal format
F: is a fraction
E: is an exponent

4. Floating Point Representation
Example: The real number can be expressed using floating point notation as:
( )10 = × 10+04
Similar to the decimal system, floating point notation can also be applied to the binary system as: (N)2 = F × 2E
Example: The binary number can be represented with a 8-bit fraction and 6-bit exponent as:
( )2 = × 2+04

5. Common Formats for Floating Point Binary Numbers Representation
There are 4 different formats for representing floating-point binary numbers. They are:
signed-magnitude method
2’s complement method
Excess method
IEEE standard method

6. Floating-Point Numbers using signed-magnitude notation

7. Floating-Point Numbers using signed-magnitude notation
sum of 7-terms geometric series

8. Floating-point numbers using signed-magnitude notation

9. Floating-point numbers using signed-magnitude notation

10. Floating-point numbers using signed-magnitude notation

11. Float-point numbers using signed-2’s complement
In this method, only the exponent part is expressed using 2’s complement notation. There is only one sign bit exists for the mantissa.

12. Float-point numbers using signed-2’s complement

13. Float-point numbers using excess method
In this method, only the exponent part is expressed using excess notation. There is only one sign bit exists for the mantissa.

14. Float-point numbers using excess method