## Exercises16.6Exercises

###### 1.

Using IEEE 754 32-bit format, what decimal number would the bit pattern $\hex{00000000}_{16}$ represent, ignoring the special case of “zero value”.

Solution
1. Compute $s\text{,}$ $e+127\text{,}$ and $f\text{.}$

\begin{align*} s &= \binary{0}\\ e + 127 &= \binary{00000000}_{2}\\ e &= -127_{10}\\ f &= \binary{00000000000000000000000} \end{align*}
2. Finally, plug these values into Equation (16.5.1). (Remember to add the hidden bit.)

\begin{align*} (-1)^0 \times 1.00\dots 00 \times 2^{-127} &= \mbox{a very small number}\\ &\ne 0.0 \end{align*}

so we do need the special case.

###### 2.

Convert the following decimal numbers to 32-bit IEEE 754 format by hand:

1. $\displaystyle 1.0$

2. $\displaystyle -0.1$

3. $\displaystyle 2016.0$

4. $\displaystyle 0.00390625$

5. $\displaystyle -3125.3125$

6. $\displaystyle 0.33$

7. $\displaystyle -0.67$

8. $\displaystyle 3.14$

1. $\displaystyle \hex{3f80 0000}$

2. $\displaystyle \hex{bdcc cccd}$

3. $\displaystyle \hex{44fc 0000}$

4. $\displaystyle \hex{3b80 0000}$

5. $\displaystyle \hex{c543 5500}$

6. $\displaystyle \hex{3ea8 f5c3}$

7. $\displaystyle \hex{bf2b 851f}$

8. $\displaystyle \hex{4048 f5c3}$

###### 3.

Convert the following hexadecimal numbers to decimal by hand using the 32-bit IEEE 754 format:

1. $\displaystyle \hex{4000 0000}$

2. $\displaystyle \hex{bf80 0000}$

3. $\displaystyle \hex{3d80 0000}$

4. $\displaystyle \hex{c180 4000}$

5. $\displaystyle \hex{42c8 1000}$

6. $\displaystyle \hex{3f99 999a}$

7. $\displaystyle \hex{42f6 e666}$

8. $\displaystyle \hex{c259 48b4}$

1. $\displaystyle +2.0$

2. $\displaystyle -1.0$

3. $\displaystyle +0.0625$

4. $\displaystyle -16.03125$

5. $\displaystyle 100.03125$

6. $\displaystyle 1.2$

7. $\displaystyle 123.449997$

8. $\displaystyle -54.320999$