## 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. $$\displaystyle -54.320999$$