Significant Figures & Rounding

The concept of significant figures, which came up toward the end of the previous post on scientific notation, deserves a bit more of an explanation. when writing a number, particularly a very small or very large number, it is common to write a number of zeros at its start or end, respectively. These leading/following zeros convey no information other than "move along, there's nothing to see here", you might even say that they are rather 'insignificant' when compared to the other digits, which we call significant digits, or significant figures.

One of the attractions of using scientific notation is that it allows us to omit non-significant figures as the only information that they convey (the magnitude of the number — that these places are empty) is given in the exponent.

An important, and related, idea is that of rounding. Much of the mathematics used in physics involves numbers with many of significant digits (the physical constants, for example) or operations which may result in such numbers (like taking roots and the trigonometric functions). When we combine these with other numbers the result will usually have a large number of significant figures, perhaps belying the accuracy of our input numbers. We usually, therefore, round our answers to as many significant figures as the input with the least. Thus our estimate of "3 m or so" does not result in an answer with a dozen digits — near enough it good enough.

For more information on rounding, including the variety of rounding methods, see some of the links below.

More information can be found at the Wikipedia articles on rounding and the Wolfram Mathworld articles on significant digits