Number representation and Range of float data type

In c language the size of float in bytes is four, which is an equal to 32 bits.

These 32 bits are divided in 3 parts:

1.    Sign bit: Single bit to store the sign i.e. 0 will represent positive and 1 will represent negative.

2.    Exponent bits: In float we will be using 8 bits to store the exponents.

3.    Mantissa bits: remaining 23 bits are used to store mantissa

An example of number representation:

Let the number be 5.5 

Now the binary equivalent of the same will be (101.1)₂

In the floating point representation we represent the number between 1 and 2.

This is done by representing in scientific notation as

(1.011)₂ X 2²  

We get this notation similarly we do in math ie 1.011 X 100 which is the equivalent of 4 or 2^2.

Sign à 0

Exponent à 00000010

Mantissaà 0110000000000000.0000 upto 23 bits

here we have not written 1 before fraction as it is understood .

Range of float

We know that we have exponent of 8 bits so the range comes out to be from 0 to 255 or in sign magnitude it can be written as -127 to 128.

0 and 255 are used to store non normalized data (Data with less accuracy).

Now the range of exponent is from -126 to 127.

The smallest Positive number that can be represented:

0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
±	Exponent bits								Mantissa bits

it comes to be 1.0 X 2^-126

1.0 is the mantissa

-126 is the signed decimal representation of exponent bits.

So the smallest number that can be represented will be 2^-126  or it is equal to 1.17549E-38.

The largest  Positive number that can be represented:

0	1	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
±	Exponent bits								Mantissa bits

It comes to be (2.0 – 2^-23 )X 2¹²⁷

Mantissa can be considered as 2, as 2^-23 is very small in magnitude.

127 is the signed decimal representation of exponent bits.

So the smallest number that can be represented will be 2 X 2¹²⁷ or it is equal to 3.40282E+38.

Range of Float comes out to be ±3.40282E+38