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Significant Digits

The number of significant digits, or significant figures, in a given
number is the number of digits after the given number has been put into
scientific notation. For example, 820 (8.2×10^{2}) has 2 significant
digits (8 and 2), and 0.820 (8.20×10^{-1}) has 3 significant digits (8,
2, and 0). There are three ways to determine the number of significant digits
in a number--use whichever method is easiest for you:

*Method I.* Put the number in scientific notation and count the
digits.

*Method II.* Count the digits in a number, starting with the first
non-zero digit and ending with the last non-zero digit (the zeros in the middle
count as digits). If the number is a whole number, do not count any remaining
zeros. If the number is a decimal, count all zeros at the end of the
number.

*Method III.* Add the following:

(a) The number of non-zero digits

(b) The number of zeros in the middle of the number (between the non-zero
digits)

(c) If the number is a decimal, the number of zeros at the end of the number

*Examples*:

7.957 has 4 significant digits.

79.57 has 4 significant digits.

0.7957 has 4 significant digits.

0.07957 has 4 significant digits.

0.79570 has 5 significant digits.

7,957 has 4 significant digits.

79,570 has 4 significant digits.

79,057 has 5 significant digits.

70,905,007 has 8 significant digits.

709,050,070 has 8 significant digits.

70,905,007.0 has 9 significant digits.

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Significant Digits in Measurement

When we measure something, we do not get a precise measurement. For example, on
a ruler marked with meters and centimeters, the object that we are measuring
might fall between two centimeter lines. We have to estimate how far it falls
between the two lines--0.4 cm.? 0.5 cm.? We know that the object measured is
117 cm. plus a little more; maybe it is 117.4 cm., maybe it is 117.5 cm.
Because there is a limit to the number of digits we can know precisely, we write
down all digits known precisely plus one digit that is estimated. Thus, the
number of significant digits in a measurement is the number known precisely plus
1. In our example, one could write down 117.4 cm. (4 significant digits). It
would be incorrect, however, to write down 117 cm. or 117.45 cm.-- 117 has too
few significant digits, while 117.45 has too many significant digits.

If the ruler included only measurements to the nearest 10 centimeters, we would
know the 10 centimeter place precisely and would estimate in the centimeter
place: we would write down 117 cm. If the ruler measured only meters (1 m. =
100 cm.), we would know the 100 centimeter place precisely and would estimate in
the 10 centimeter place: we would write down 120 cm.

When a measurement is known to more places than another measurement, it is said
to be more precise. 117.4 cm. is more precise than 117 cm., and 117 cm. is
more precise than 120 cm.