are as follows; these can be used to compute significant figures for all elementary functions: The fact that the number of significant figures of the function output is equal to the number of significant figures of the function input (function argument) minus the base-10 logarithm of the condition number (which is approximately the order of magnitude/number of digits of the condition number) can be easily derived from first principles: let and be the true values and let and be approximate values with errors and respectively, so that and . Then , and hence . The significant figures of a number is related to the uncertain error of the number by where "significant figures of x" here means the number of significant figures of x. Substituting this into the above equation gives , and thus . Therefore , giving . Rounding rules Because significance arithmetic involves rounding, it is useful to understand a specific rounding rule that is often used when doing scientific calculations: the round-to-even rule (also called banker's rounding). It is especially useful when dealing with large data sets. This rule helps to eliminate the upwards skewing of data when using traditional rounding rules. Whereas traditional rounding always rounds up when the following digit is 5, bankers sometimes round down to eliminate this upwards bias. See the article on rounding for more information on rounding rules and a detailed explanation of the round-to-even rule. Disagreements about importance Significant figures are used extensively in high school and undergraduate courses as a shorthand for the precision with which a measurement is known. However, significant figures are not a perfect representation of uncertainty, and are not meant to be. Instead, they are a useful tool for avoiding expressing more information than the experimenter actually knows, and for avoiding rounding numbers in such a way as to lose precision. For example, here are some important differences between significant figure rules and uncertainty: Uncertainty is not the same as a mistake. If the outcome of a particular experiment is reported as 1.234 ± 0.056 it does not mean the observer made a mistake; it may be that the outcome is inherently statistical, and is best described by the expression indicating a value showing only those digits that are significant, i.e. the known digits plus one uncertain digit, in this case 1.23 ± 0.06. To describe that outcome as 1.234 would be incorrect under these