Question #049da

Significant figures express to the reader how precise the measurement is. They correspond directly to the amount of precision one can get from using a particular measurement tool.

The better the precision, the less the uncertainty, and the more sure you are of your answer.

For instance, an average ruler, which can generally measure with an uncertainty in the #pm##"cm"# range, is less precise than a caliper one tends to find in college physics labs, which usually measures with an uncertainty of #pm "0.1 mm"#.

The rules for significant figures in General Chemistry are as follows (it might be different in a more quantitative course, or in Physics):

PROMINENT DEFINITIONS

• All nonzero digits are significant. EX: #color(blue)(21)0#
• A trailing zero goes somewhere after a decimal point, or after all nonzero digits, or both, and no nonzero digits follow it. EX: #5.2color(blue)(000)#
• A zero is sandwiched when there is at least one nonzero digit to the left AND right. All zeros in between two nonzero digits, even if multiple zeros are adjacent, are significant. EX: #5color(blue)(00)5#
• A "placeholder" zero before an explicit decimal point, if a number is less than #1#, is NOT significant. EX: #color(blue)(0).523#
• In scientific notation, for which the number must be represented as #N xx 10^n#, where #1.00 < N < 9.99#, all digits in #\mathbf(N)# are significant. EX: #color(blue)(6.293)xx10^3#

SPECIAL CONDITIONS

• All "placeholder" zeroes AFTER a nonzero digit but BEFORE an implicit or explicit decimal point are subject to the following conditions:

  • If an explicit decimal point is specified, these zeroes are significant if and only if a nonzero digit precedes them. EX: #25color(blue)(00).# vs. #color(red)(0).25#
  • If no explicit decimal point is specified, they are NOT significant. EX: #25color(red)(00)#

• All zeroes AFTER an implicit or explicit decimal point are subject to the following conditions:

  • If the number is GREATER than #1# (if the units digit is #1# or greater), AND the decimal point directly precedes the zero(es), the zero(es) are significant. Otherwise, the zeroes are NOT significant. EX: #1color(blue)(.0)50# vs. #0color(red)(.0)50#
  • If it is sandwiched between two nonzero digits, even if the decimal point directly precedes the zero, it is significant. EX: #5color(blue)(.0)5#

Other examples:

• #\mathbf(5404)# has four sig figs.
• #\mathbf(54)00# has two sig figs.
• #\mathbf(5400.)# has four sig figs.
• #\mathbf(5.400) xx 10^3# has four sig figs.
• #\mathbf(540.0)# has four sig figs.
• #0.00\mathbf(5400)# has four sig figs.
• #0.0\mathbf(5400)# has four sig figs.
• #\mathbf(5.040)# has four sig figs.
• #\mathbf(5.004)# has four sig figs.
• #0\mathbf(.5004)# has four sig figs.