Question

Calculate by hand $8.625\times {10}^1$divided by $-4.875\times {10}^0$. Show all the steps necessary to achieve your answer. Assume there is a guard, a round bit, and a sticky bit, and use them if necessary. Write the final answer in both the 16-bit floating point format described in Exercise 3.27 and in decimal and compare the decimal result to that which you get if you use a calculator.

Short answer

The required result will be 0.00480769230.

Step-by-step solution

Define the concept

The decimal number system uses base 10 and the binary number system uses base 2.

For example, “12” is also equivalent to the value of the binary number system “1100”.

The decimal number $8.625\times {10}^1$is equal to $1.0101100100\times 2^6$in the binary number system.

The decimal number -4.875 is equal to$-1.0011100000\times 2^2$in the binary number system.

Determine the calculation

Given numbers are $8.625\times {10}^1$and $-4.875\times {10}^0=-4.875\times 1=-4.875$.

It is needed to do$\frac{1.0101100100\times 2^6}{-1.0011100000\times 2^2}$

Hence,

The sign will be negative. [As, the sign of one number is positive and the other one is negative].

The exponent will be $=\left(6-2\right)=4$and$\left(15+4\right)=19$

.

The decimal number 19 is equal to 10011 in the binary number system.

For the division,

Hence,

$$\begin{gathered} 1.0001101100\times 2^4 \\ =1101000001101100 \\ =10001.101100 \\ =\frac{-17.687586.25}{-4.875} \\ =-17.692307692307 \end{gathered}$$

Hence, the required result will be 0.00480769230