Question

Question: Based on your answer to 3.38 and 3.39, is $(1.666015625\times {10}^0\times 1.9760\times {10}^4)+\left(1.666015625\times {10}^0\times -1.9744\times {10}^4\right)=1.666015625\times {10}^0\times \left(1.9760\times {10}^4+-1.9744\times {10}^4\right)$

Short answer

Based on the answers of 3.38 and 3.39, the given equations are not equal.

Step-by-step solution

Determine the floating-point multiplication

The biased exponents of two numbers will be added as the first step of the floating-point multiplication. Then, the significands will be multiplied to get the product. Check the product for normalization if the necessary product will be normalized by shifting right and incrementing the exponent. Then it will be checked for overflow or underflow. If no underflow or overflow occurs, the product will be rounded off. Finally, the sign of the products will be assigned.

Determine whether the equations are equal.

The given equation is

$\left(1.666015625\times {10}^0\times 1.9760\times {10}^4\right)+\left(1.666015625\times {10}^0\times -1.9744\times {10}^4\right)=1.666015625\times {10}^0\times (1.9760\times {10}^4+-1.9722\times {10}^4)$

Based on the answer of 3.38 and 3.39,

$$\begin{gathered} \left(1.666015625\times {10}^0\times 1.9760\times {10}^4\right)+\left(1.666015625\times {10}^0\times -1.9744\times {10}^4\right)=1.0000000000\times 2^{15} \\ 1.666015625\times {10}^0(1.9760\times {10}^4+-1.9744\times {10}^4)=1.1010101010\times 2^4 \end{gathered}$$

Converting the values into decimal values, we have

$$\begin{gathered} 1.1010101010\times 2^4=26.65625 \\ 1.0000000000\times 2^{15}=32 \end{gathered}$$

Since the values are different, it is clear that the two sides of the equation are not equal.