Representing Numbers in the Floating-Point Format

4. Analysis for $-0.25$: (1) it is negative, so the sign bit will be $1$. (2) The binary form for $0.25$ is computed as follows:
   • $0.25 \times 2 = 0.5$. The integer part in the result is $0$, so use $0.5 - 0 = 0.5$ for the next step.
   • $0.5 \times 2 = 1.0$. The integer part in the result is $1$, so use $1.0 - 1 = 0.0$ for the next step.
   • Since $0.0$ is just $0$, we stop here.
   The integer parts from the steps above make up the fraction part of the binary number we are looking for. As such, the binary form for $0.25$, from top to bottom, is $0.$$0$$1$$ = 0.01 \times 2^0 = 0.1 \times 2^{-1}$. The significand will, thus, be $10000$. (3) Finally, for the exponent, we stopped at $2^{-1}$, whose exponent is $-1$. When we add the bias of $15$, we get $-1 + 15 = 14$, which makes our exponent equal to $01110$. The number $-0.25$, therefore, in its floating-point form is:

   +---+   +---+---+---+---+---+   +---+---+---+---+---+
   | 1 |   | 0 | 1 | 1 | 1 | 0 |   | 1 | 0 | 0 | 0 | 0 | = -0.25_{10}
   +---+   +---+---+---+---+---+   +---+---+---+---+---+

   [Note: we may have also noticed that $0.25 = \frac{1}{4} = \frac{1}{2^2} = 2^{-2} = 0.01_2.$.]