Significant digits a.k.a. Significant figures

1
Significant digitsa.k.a. Significant figures

• Objectives
• At the end of this lesson the students will be
  able to
• State why significant digits are important.
• List the rules for significant digits.
• Calculate problems and record answers with
  significant digits.

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Why are significant digits or significant figures
important?

• The science teachers at a Baltimore County
  middle school wished to acquire a steel cube, one
  cubic centimeter in size to use as a visual aid
  to teach the metric system. The machine shop they
  contacted sent them a work order with
  instructions to draw the cube and specify its
  dimensions. On the work order, the science
  supervisor drew a cube and specified each side to
  be 1.000 cm. When the machine shop received this
  job request, they contacted the supervisor to
  double check that each side was to be one
  centimeter to four significant figures. The
  science supervisor, not thinking about the
  "logistics", verified four significant figures.
  When the finished cube arrived approximately one
  month later, it appeared to be a work of art. The
  sides were mirror smooth and the edges razor
  sharp. When they looked at the "bottom line",
  they were shocked to see the cost of the cube to
  be 500! Thinking an error was made in billing,
  they contacted the machine shop to ask if the
  bill was really 5.00, and not 500. At this
  time, the machine shop verified that the cube was
  to be made to four significant figure
  specifications. It was explained to the school,
  that in order to make a cube of such a high
  degree of certainty, many man hours were used.
  The cube had to be ground down, and measured with
  calipers to within a certain tolerance. This
  process was repeated until three sides of the
  cube were successfully completed. The number of
  man hours to prepare the cube cost 500. The
  science budget for the school was wiped out for
  the entire year. This school now has a steel cube
  worth its weight in gold, because it is a very
  certain cubic centimeter in size.

3
Remember

• Significant digits, which are also called
  significant figures, are very important in
  Physics.
• Each recorded measurement has a certain number of
  significant digits.
• Calculations done on these measurements must
  follow the rules for significant digits.
• The significance of a digit has to do with
  whether it represents a true measurement or not.
• Any digit that is actually measured or estimated
  will be considered significant.
• Placeholders, or digits that have not been
  measured or estimated, are not considered
  significant.

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It is very important, however, to know and
understand the precision of measurement that we
use in our daily lives.
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Rules for Significant Digits

• Digits from 1-9 are always significant.
• Zeros between two other significant digits are
  always significant
• One or more additional zeros to the right of both
  the decimal place and another significant digit
  are significant.
• Zeros used solely for spacing the decimal point
  (placeholders) are not significant.

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Examples of Significant Digits
EXAMPLES OF SIG. DIG. COMMENT
453 kg All non-zero digits are always significant.
5057 L Zeros between 2 sig. dig. are significant.
5.00 Additional zeros to the right of decimal and a sig. dig. are significant.
0.007 Placeholders are not sig.
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• Each number that we record as a measurement
  contains a certain number of significant digits,
  which show accurate or estimated digits. When we
  do calculations our answers cannot be more
  accurate than the measurements that they are
  based on. We must be careful to follow the
  following rules whenever we perform calculations
  in Physics class.

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• Multiplying and Dividing
• RULE When multiplying or dividing, your answer
  may only show as many significant digits as the
  measurement showing the least number of
  significant digits.
• Example
• When multiplying 22.37 cm x 3.10 cm x 85.75 cm
  5946.50525 cm3 we look to the original problem
  and check the number of significant digits in
  each of the original measurements
• 22.37 shows 4 significant digits.
• 3.10 shows 3 significant digits.
• 85.75 shows 4 significant digits.
• Our answer can only show 3 significant digits
  because that is the least number of significant
  digits in the original problem.
• Our calculators show the answer as 5946.50525
  with 9 significant digits, we must round to the
  tens place in order to show only 3 significant
  digits. Our final answer becomes 5950 cm3.

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• Adding and Subtracting
• RULE When adding or subtracting your answer can
  only show as many decimal places as the
  measurement having the fewest number of decimal
  places.
• Example
• When we add
• 3.76 g 14.83 g 2.1 g 20.69 g
• We look to the original problem to see the
  number of decimal places shown in each of the
  original measurements. 2.1 shows the least number
  of decimal places. We must round our answer,
  20.69, to one decimal place (the tenth place).
  Our final answer is 20.7 g

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Calculators and Significant Digits

• Many calculators display several additional,
  meaningless digits, some always display only
  two.  Be sure to record your answer with the
  correct number of significant digits.  Calculator
  answers are not rounded to significant digits.
  You will have to round-off the answer to the
  correct number of digits.
• Note that significant digits are only associated
  with measurements there is no uncertainty
  associated with counting.  If you counted four
  laps for a runner and measured the time to be
  2.34 minutes.  The number of laps does not have
  an uncertainty, but the measured time does.

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• When doing multi-step calculations, do not clear
  your calculator.
• Round your final answer to the smallest number of
  significant digits of any of the values in the
  original question.

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• The Two Greatest Sins Regarding Significant
  Digits
• Writing more digits in an answer (intermediate or
  final) than justified by the number of digits in
  the data.
• Rounding-off, say, to two digits in an
  intermediate answer of a multi-step calculation,
  and then writing three digits in the final
  answer.

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Sample problems

• Instructions work the problems. When you are
  ready to check your answers, go to the next page.
• 1.    37.76 3.907 226.4  ...
• 2.    319.15 - 32.614  ...
• 3.    104.630 27.08362 0.61  ...
• 4.    125 - 0.23 4.109  ...
• 5.    2.02 2.5  ...
• 6.    600.0 / 5.2302  ...
• 7.    0.0032 273  ...
• 8.    (5.5)3  ...
• 9.    0.556 (40 - 32.5)  ...
• 10.    45 3.00  ...

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Answers to Sample Problems

• 1.    37.76 3.907 226.4  268.1
•  2.    319.15 - 32.614  286.54
•  3.    104.630 27.08362 0.61  132.32
•  4.    125 - 0.23 4.109  129
•  5.    2.02 2.5  5.1
•  6.    600.0 / 5.2302  114.7
•  7.    0.0032 273  0.87
•  8.    (5.5)3  1.7 x 102
•  9.    0.556 (40 - 32.5)  4.2
• 10.   45 3.00  1.4 x 102