Floating-Point Numbers: Fractions

Let's now figure out what the number from slide 42 is:

+---+   +---+---+---+---+---+   +---+---+---+---+---+
| 0 |   | 0 | 1 | 1 | 1 | 0 |   | 1 | 0 | 0 | 1 | 0 |
+---+   +---+---+---+---+---+   +---+---+---+---+---+

Analysis: (1) the number is positive since the sign bit is $0$. (2) to figure out what the exponent is, $01110$, subtract $15_{10}$ (the bias) from it. $01110$ in binary is $14_{10}$, so $14 - 15$ is $-1$, so the exponent is $-1$. (3) the significand is $10010$, which is $0.10010$ binary. Together, we get: $0.10010 \times 2^{-1} = 0.010010 = 0.01 + 0.00001$, which is $\frac{1}{4} + \frac{1}{32} = 0.28125_{10}$.

Alternatively, note that the difference between the number $100100_2$ (which we figured out is $36_{10}$) and $0.010010_2$ is that the latter is reached from the former by shifting the former $7$ times to the right, which is the same as dividing the number by $2$ seven times. Indeed: (1) $36/2 = 18$; (2) $18/2 = 9$; (3) $9/2 = 4.5$; (4) $4.5/2 = 2.25$; (5) $2.25/2 = 1.125$; (6) $1.125/2 = 0.5625$; and (7) $0.5625/2 = 0.28125_{10}$.