[Math Lair] Accuracy and Precision

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Physical measurements necessarily yield approximations. Although we can measure some lengths with astonishing accuracy, we can't measure any perfectly. Some inexactness, if only one part in thousands, remains. If we try to get too fine with determining length, say of an iron rod, our object-level notion of rigid boundaries fails. We reach a level of atomic structure where the "iron rod" is in flux without exact boundaries.

Refers to the number of significant digits and/or decimal places that can be reliably determined with a given instrument or technique.
Refers to how close a measured value is to the "true" or correct value.
Significant figures
Those figures in a measurement that are known with reasonable accuracy.

The following rules apply to determine how many significant figures a measurement has:

Here are some examples of significant figures:
327three significant digits
2.09three significant digits
0.000000382three significant digits
16.83four significant digits
3.2two significant digits
3.20three significant digits
150,000,000not clear how many significant digits. Use scientific notation to clarify.
1.5 × 108two significant digits
1.50 × 108three significant digits

When rounding off measurements: