Significant digitsa.k.a. Significant figures

ObjectivesAt 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.

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.

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.

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

Rules for Significant Digits

1. Digits from 1-9 are always significant.

2. Zeros between two other significant digits are always significant

3. One or more additional zeros to the right of both the decimal place and another significant digit are significant.

4. Zeros used solely for spacing the decimal point (placeholders) are not significant.

Examples of Significant Digits

EXAMPLES # OF SIG. DIG. COMMENT

453 kgAll non-zero digits are always

significant.

5057 LZeros 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.

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.

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.

Adding and SubtractingRULE: 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 gWe 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

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.

• 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.

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.

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 = ...

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) = 4.210.   45 × 3.00 = 1.4 x 102