Let's first see an example showing that computer arithmetic really is an approximation. First, consider the linear system

\begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp \frac12y \amp {}+{} \amp \frac13z \amp {}={} \amp 1 \\ \frac12x \amp {}+{} \amp \frac13y \amp {}+{} \amp \frac14z \amp {}={} \amp 0 \\ \frac13x \amp {}+{} \amp \frac14y \amp {}+{} \amp \frac15z \amp {}={} \amp 0 \\ \end{alignedat} \end{equation*}

If the coefficients are entered into Sage as fractions, Sage will find the exact reduced row echelon form. Find the exact solution to this linear system.

Now let's ask Sage to compute with real numbers. We can do this by representing one of the coefficients as a decimal. For instance, the same linear system can be represented as

\begin{equation*} \begin{alignedat}{4} x \amp {}+{} \amp 0.5y \amp {}+{} \amp \frac13z \amp {}={} \amp 1 \\ \frac12x \amp {}+{} \amp \frac13y \amp {}+{} \amp \frac14z \amp {}={} \amp 0 \\ \frac13x \amp {}+{} \amp \frac14y \amp {}+{} \amp \frac15z \amp {}={} \amp 0 \\ \end{alignedat} \end{equation*}

Most computers do arithmetic using either 32 or 64 bits. To magnify the problem so that we can see it better, we will ask Sage to do arithmetic using only 10 bits as follows.

What does Sage give for the solution now? Compare this to the exact solution that you found previously.