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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
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 m ≤ n, 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).