Question

Use your calculator to find a decimal approximation to THREE decimal places of both 5/√2 and 5√2/2 How similar are they? Why are they either very similar or pretty different?

Answer 1

Answer on Question #85465 – Math – Algebra

Question

Use your calculator to find a decimal approximation to three decimal places of both 5/V2 and 5V2/2. How similar are they? Why are they either very similar or pretty different?

Solution

Let $x_k$ be a decimal approximation of $x$ to $k$ decimal places. Then:

$$\begin{array}{l} a = \sqrt{2} \approx 1.41421356; \\ b = 5/\sqrt{2} \approx 3.53553391; \\ c = 5\sqrt{2}/2 = 2.5\sqrt{2} \approx 3.53553391; \\ a_3 = 1.414; \\ b_3 = 3.536; \\ c_3 = 3.536; \\ b_3 = c_3. \end{array}$$

If we make calculations of these numbers with 3 decimal approximations in any step, we lose accuracy and so we will not obtain the main results with the same approximation:

$$\begin{array}{l} a_3 = 1.414; \\ b'_3 = 5/a_3 = 5/1.414 \approx 3.53606789 \approx 3.536 = b_3; \\ c'_3 = 2.5 * a_3 = 2.5 * 1.414 = 3.535 \neq c_3. \end{array}$$

We see that after all calculations with 3 decimal places numbers $b'_3$ and $c'_3$ are slightly different:

$$\begin{array}{l} b'_3 = 3.536; \\ c'_3 = 3.535; \\ b'_3 \neq c'_3. \end{array}$$

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