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# Significant figure

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In science and statistics, values are sometimes rounded and given as approximations, typically because complete precision is not attainable or not required. The number of digits to the left of (and including) the rounding place is the number of significant figures or significant digits.

Take for example the value 4,215.02474. Rounded to two significant figures, we have 4,200; to three significant figures, it is 4,220; to 5 significant figures we have 4,215.0 and to 7 significant digits we get 4,215.025. Values such as these are often expressed in scientific notation: 4.2 × 103, 4.22 × 103, 4.2150 × 103 and 4.215025 × 103. In this notation, the number of significant digits is directly apparent.

Different conventions are used when rounding a number whose last digit is a five. In one convention, such numbers are always rounded up; in another convention, the rounding is performed so that the new last digit becomes even.

Note that because of the rounding, a number to n significant figures is not necessarily the same as the first n digits of that number (as in 4,220 above).

For numbers written with decimals, the number of decimals can be used to indicate the number of significant figures: for example, 4,215.02 is represented to six significant figures. However, from a notation like 4,220 we can not see whether 0 is a significant digit or not; scientific notation would be more informative here.

It is useful to know how the number of significant figures changes when performing various calculations with rounded numbers.

When multiplying a number having n significant figures with a number having m significant figures, and mn, then the result will have m-1 significant figures. For example, a rectangular table has been measured to be 23.2 inches wide (3 significant figures) and 146.5 inches long (4 significant figures). In order to compute the table's area, we use a calculator and find 23.2 × 146.5 = 3398.8 square inches. This result should be properly stated as 3400 square inches with two significant digits.

When squaring or taking the square root of a value, the number of significant figures can decrease by one.

When adding, it is not the number but the position of the significant figures that determines the significant figures of the result: if the first summand has significant digits which are to the right of the significant digits of the second summand, then these digits are insignificant in the result. For example, adding 2103.45 (6 significant digits) to 3.453245 (7 significant digits) on a calculator results in 2106.903245, but this should be stated with 6 significant digits as 2106.90.

When subtracting two numbers that are approximately equal, the number of significant digits drops. For example, 1.75 - 1.72 = 0.03.

When using a calculator, one should keep track of the significant digits of all numbers, but only the final results should be rounded for presentation, not the intermediate values.

In programming languages which contain the floor function, rounding of the number x to the nearest integer can be achieved by calculating floor(x + 0.5).

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