Question

Based on your answers to 3.32 and 3.33, does

$\left(3.984375\times {10}^{-1}+3.4375\times {10}^{-1}\right)+1.771\times {10}^3=3.984375\times {10}^{-1}+\left(3.4375\times {10}^{-1}+1.771\times {10}^3\right)?$

Short answer

No, there is a difference between the two results.

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 $3.984375\times {10}^{-1}$is equal to $1.1001100000\times 2^{-2}$in the binary number system.

The decimal number $3.4375\times {10}^{-1}$is equal to $1.0110000000\times 2^{-2}$in the binary number system.

The decimal number $1.771\times {10}^3$is equal to $-1.1011101011\times 2^{10}$in the binary number system.

Given numbers are $3.984375\times {10}^{-1}$, $3.4375\times {10}^{-1}$and $1.771\times {10}^3$

Let, $x=3.984375\times {10}^{-1}$, $y=3.4375\times {10}^{-1}$and

$$\begin{gathered} \left(x+y\right)+z=\left(3.984375\times {10}^{-1}+3.4375\times {10}^{-1}\right)+1.771\times {10}^3=1772 \\ x+\left(y+z\right)=3.984375\times {10}^{-1}+\left(3.4375\times {10}^{-1}+1.771\times {10}^3\right)=1771 \end{gathered}$$

Determine the calculation

Given numbers are $3.984375\times {10}^{-1}$, $3.4375\times {10}^{-1}$and $1.771\times {10}^3$.

Let, $x=3.984375\times {10}^{-1}$, $y=3.4375\times {10}^{-1}$and $z=1.771\times {10}^3$

$$\begin{gathered} x=1.1001100000\times 2^{-2} \\ y=+1.0110000000\times 2^{-2} \end{gathered}$$

----------------------------------------------

$\begin{array}{rcl}\left(x+y\right)& =& 10.1111100000\left[Normalized\right]\\ 10.1111100000& =& 1.01111100000\times 2^{-1}\\ \left(x+y\right)& =& 1.0111110000\\ z& =& +1.1011101011[.000000000010111110000]\\ & & \\ & & \end{array}$

----------------------------------------------------

localid="1654950526652" $\begin{array}{rcl}\left(x+y\right)+z& =& 1.1011101011101[Roundedup]\\ 1.1011101100\times 2^{10}& =& 0110101011101100=1772\left[decimal\right]\\ \left(x+y\right)+z& =& \left(3.984375\times {10}^{-1}+3.4375\times {10}^{-1}\right)+1.771\times {10}^3=1772\\ y& =& 0.0000000000010110000000\\ z& =& +1.1011101011\\ & & \\ & & \end{array}$

----------------------------------------------------------

$$\begin{gathered} \left(y+z\right)=+1.1011101011 \\ x=0.0000000000011001100000 \end{gathered}$$

----------------------------------------------------------

$$\begin{gathered} x+\left(y+z\right)=+1.1011101011 \\ +1.1011101011=+1.1011101011\times 2^{10}=0110101011101011=1771[Indecimal] \end{gathered}$$

$x+\left(y+z\right)=3.984375\times {10}^{-1}+\left(3.4375\times {10}^{-1}+1.771\times {10}^3\right)=1771$

Hence, the two results are not same.