Significant Figures

Patrick D. Williford

A significant figure is any digit in a value that is known with reasonable certainty. All measurements should include one doubtful digit.

A doubtful digit is the estimated digit when making a measurement. For example, if a ruler has markings to read millimeters, then a measurement taken with that ruler should be reported with tenths of a millimeter rather than just the integer value.

Significant Figure Rules

1. Non-zero digits are significant. (The doubtful digit is underlined.)

5,729.7 has five significant figures, 2.356 has four significant figures, 1.13 has three significant figures.

2. Zeroes written between non-zero digits are significant.

66,000.10005 has ten significant figures, 2.0012 has five significant figures, 1.03 has three significant figures.

3. Leading zeroes are not significant (zeroes written to the left of the first non-zero digit).

0.00061093 has five significant figures, 0.3098 has four significant figures, 0.0126 has three significant figures.

4. Trailing zeroes written to the left of an unwritten decimal point are not significant.

902,008,000 has six significant figures, 512,360 has five significant figures, 23,000,000 has two significant figures.

5. Trailing zeroes written to the left of a decimal point are significant.

21,300. has five significant figures, 5,000. has four significant figures, 230. has three significant figures.

6. Trailing zeroes written to the right of a decimal point are significant.

23.0000 has six significant figures, 45.210 has five significant figures, 0.0002300 has four significant figures.

Defined or Empirical Conversion Factors

1. Defined conversion factors (or unity operators) are generally those from within the same measurement system and contain an infinite number of significant figures.

12 in = 1 ft is the exact definition of the number of inches in one foot. One can assume that it is the same as 12.0000. . . in = 1.0000. . . ft, where ". . ." represents an infinite string of zeroes. Similarly, 5280 ft. = 1 mile, 1000 mL = 1 L, 16 oz = 1 lb, 16 fl oz = 1 pt, 1000 m = 1 km, and 1000 g = 1 kg are exact definitions. One noted exception is the redefined inch. Several years ago many scientists decided to accept the convention that 1 in = 2.54 cm (exactly) [unknown reference].

2. Empirical (or experimentally obtained) conversion factors are generally those obtained by using one measurement system to measure an exact value from another measurement system. They are limited to the number of significant figures as indicated by its value when applying the "Significant Figures Rule."

1 lb (infinite number of significant figures) = 454 g (three significant figures)

1 L (infinite number of significant figures) = 1.056718 qt (seven significant figures)

Direct Observation Values

Integer values obtained by direct observation contain an infinite number of significant figures.

A solid cube has exactly 6 sides.
That dog has exactly 4 legs.
A bicycle has exactly 2 wheels.

Rules to Round-Off Numbers

From left to right count and keep as many significant figures as are appropriate for the final value and discard all remaining digits.

If the digit dropped is < 5, do not change the last kept digit.

If the digit dropped is > 5, increase the last kept digit by 1.

If the digit dropped = 5, increase the last kept digit by 1 if it is followed by any non-zero digit.

If the digit dropped = 5 and is followed by nothing or only zeroes increase the last kept digit by 1 if it is odd or leave it alone if it is even (zero is even).

47.08149 rounded to 3 sig. fig.s => 47.1	96.996 rounded to 4 sig. fig.s => 97.00
6.044999 rounded to 3 sig. fig.s => 6.04	0.00765000 rounded to 2 sig. fig.s => 0.0076
0.00335 rounded to 2 sig. fig.s => 0.0034	31.0500 rounded to 3 sig. fig.s => 31.0
0.9996 rounded to 3 sig. fig.s => 1.00	0.00645001 rounded to 2 sig. fig.s => 0.0065
0.00645 rounded to 2 sig. fig.s => 0.0064	31.1500 rounded to 3 sig. fig.s => 21.2

Calculations with Significant Figures

For addition and subtraction round-off each value so that all values have the same number of digits to the right of the decimal point as the value with the least number of significant digits to the right of the decimal point, then add or subtract. Another method is to add or subtract all values then round-off the result as stated above.

225.9913 + 73.76 + 3.089 = 225.99 + 73.76 + 3.09 = 302.84
(method #2: 302.8403 rounds to 302.84)

340.73 - 26.504 = 340.73 - 26.50 = 314.23
(method #2: 314.226 rounds to 314.23)

62.0084 + 0.86 - 5.1 = 62.0 + 0.9 - 5.1 = 57.8
(method #2: 57.7684 rounds to 57.8)

For multiplication and division do the calculation with all the digits you are given then round the result to that of the value with the least number of significant digits.

(125.00)(300.1)(6.0) / (12) = 18,756.25 which rounds to 19,000

(6.022 x 10²³) / (100.) / (35.45)(1.66 x 10^-19) = 28.19892807... which rounds to 28.2

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Last Updated 23 January 1999
Last Accessed - - - Copyright 1997