What is the decimal value of the 32bit IEEE single precision

What is the decimal value of the 32-bit

IEEE single precision computer number 10111000111001100110011001100110?

First we have to split up the binary: 1 01110001 and 11001100110011001100110

The first bit defines the sign and 1 corresponds to negative sign.

The next 8 bit define the binary value of the exponent shifted by 127 :

0 × 2^7 + 1 × 2^6 + 1 × 2^5 + 1 × 2^4 + 0 × 2^3 + 0 × 2^2+ 0 × 2^1 + 1 × 2^0 - 127 =113 – 127 = -14

The last 23 bits together with the leading 1 defines the mantissa field :

1+1×2^1+1×2^2 +0×2^3 ... 1×2^21+1×2^22 +0×2^23 = 1.7999999523162842 = 1.8

( This is calculated using geometrical ratio formula )

The required decimal number is about -1.8 * 2^(-14) which is -0.00010986328125.