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Approximation and estimation

For measurements or mental calculations, we often approximate numbers to estimate the solution.

You want to estimate the surface of a square with sides of length $212$ metres, with no calculator.

To make it simple, you can round $\Tred{2}12$ to $\Tred{2}00$ $$\Tred{2}00\times \Tred{2}00 = \Tred{2}\times 100\times \Tred{2}\times 100 = 4\times 10000 = 40000.$$ The exact solution is $44944$, so the leading digit is correct.

If you use $\Tblue{21}0$ in the calculation instead, it is more difficult but your result is closer to the exact solution, with two leading digits correct. $$\Tblue{21}0\times \Tblue{21}0 = \Tblue{21}\times 10\times \Tblue{21}\times 10 = 441\times 100 = 44100.$$

Approximation simplifies a number to its most important part. Estimation is the operation using the approximated numbers. The operation is simpler, but the result is not exact. The approximation error is the difference between the result and the estimate.

The error when we approximate $212$ by $210$ is $844$ square metres.