# Converting a 32-bit bit pattern into floating point number

In The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. i have read something like that,

Single precision or 32-bit floating point forms the floating point number, $v$, by the following relation: $$v = (-1)^S \times M \times 2^{E-127}$$

The term: $(-1)^S$, simply means that the sign bit, $S$, is $0$ for a positive number and $1$ for a negative number. The variable, $E$, is the number between $0$ and $255$ represented by the eight exponent bits. Subtracting $127$ from this number allows the exponent term to run from to In other words, the exponent is stored in offset binary with an offset of $127$.

Everything I have understood but this,

Subtracting 127 from this number allows the exponent term to run from 128 to −127. In other words, the exponent is stored in offset binary with an offset of 127.

I need an easy to understand explanation of subtracting 127 from $E$ and binary offset.

## 1 Answer

Suppose you didn't include that shift of $127$, i.e. your number $\nu$ would have the form $\nu = (-1)^S \cdot M \cdot 2^x$ with $x$ being the exponent.

With $8$ bits for the exponent $x$ can have a value from $0$ to $255$. This means you can represent numbers from $\nu = (-1)^S \cdot M \cdot 2^0$ to $\nu = (-1)^S \cdot M \cdot 2^{255}$ .

However, what if you wanted $\nu$ to be $2^{-10}$? It would not be possible.

For this to be possible you want to allow negative exponents $x$. Instead of $x$ ranging from $0$ to $255$ you introduce a shift of 127 such that $x$ ranges from $-127$ to $+128$ .

Now you can represent numbers $\nu$ that range from $(-1)^S \cdot M \cdot 2^{-127}$ to $(-1)^S \cdot M \cdot 2^{+128}$ .