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Floating Point Representation of Binary Numbers
Binary Numbers (floating-point representation): In this tutorial, we will learn about the floating-point representation of binary numbers with the help of examples.
By Saurabh Gupta Last updated : May 10, 2023
Prerequisite: Number systems
Binary representation of the floating-point numbers
We all very well know that very small and very large numbers in the decimal number system are represented using scientific notation form by stating a number (mantissa) and an exponent in the power of 10. Some of the examples are 6.27 * 10-27 and 5.21 * 1034. Similarly, Binary numbers can also be represented in the same form by stating a number (mantissa) and an exponent of 2. The format of this representation will be different for different machines.
The 16-bit machine consists of 10 bits as the mantissa and 6 bits for the exponent part whereas 24-bit machine consists of 15 bits for mantissa and 9 bits for exponent.
Format of the 16-bit machine can be represented as:
| Mantissa Part | Exponent Part |
|---|
| 0110011010 | 101010 |
The mantissa is written in 2's complement form, so the MSB of the Mantissa can be thought of as a sign bit. The binary point is assumed to be to the right of this sign bit. The 6-bit of the exponent can be used to represent 0 to 63, however, to express negative exponents a number (32)10 or (100000)2 is added to the desired exponent.
Excess-32 Representation
This is a common system to represent floating-point numbers. In this notation, to represent a negative exponent, we add (32)10 to the given exponent which are given by the 6 bits.
Given table illustrates representation of exponent part.
| Desired Exponent |
2's complement notation |
Excess-32 notation (in 6 bits) |
Binary representation |
| -32 | 100000 | 100000 +100000 = 000000 | 000000 |
| -31 | 100001 | 100001 +100000 = 000001 | 000001 |
| -30 | 100010 | 100010 +100000 = 000010 | 000010 |
| -15 | 110001 | 110001 +100000 = 010001 | 010001 |
| 0 | 000000 | 000000 +100000 = 100000 | 100000 |
| +1 | 000001 | 000001 +100000 = 100001 | 100001 |
| +15 | 001111 | 001111 +100000 = 101111 | 101111 |
| +30 | 011110 | 011110 +100000 = 111110 | 111110 |
| +31 | 011111 | 011111 +100000 = 111111 | 111111 |
| Mantissa Part | Exponent Part |
|---|
| 0110011010 | 101010 |
As given above, the floating-point number given in the above format is:
At the extreme left (MSB) is the sign-bit '0', which represents it is a positive number. Also, just after the sign-bit, we assume a binary point. Thus,
In Mantissa Part: .110011010
In Exponent Part: 101010, In Excess-32 notation,32 is already added. So
Subtracting 100000 001010 (i.e.,10 in decimal, so exponent part is 210)
The number is N
= +(.110011010)2 * 210
= +(1100110100.00)
= +(820)10
Example 1: Express the following decimal number into 16-bit floating point number (45365.125)10
Solution
Binary equivalent of (45365.125)10: 1011000100110101.001
Binary format: .1011000100110101 * 216
Mantissa: + .101100010
Exponent: 010000 (Value of exponent is 16)
Equivalent exponent: 010000 + 100000 = 110000
Since the number is a positive number an additional sign-bit '0' is added in the MSB.
So, the floating-point format will be 0101100010110000
Example 2: What floating point number do the given number 0100101001101011 represents
Solution
At the extreme left (MSB) is the sign-bit '0' which represents it is a positive number. Also, just after the sign-bit we assume a binary point. Thus,
In Mantissa Part: .100101001
In Exponent Part: 10101, In Excess-32 notation,32 is already added. So
Subtracting 100000 001011 (i.e.,11 in decimal, so exponent part is 211)
The number is N
= +(.100101001)2 * 211
= +(10010100100.0)
= +(1188)10