As we saw in the Metric Prefixes chapter, scientists use both numbers and units to record things they’ve measured or calculated.  For example

5.43 cm

The number or value is 5.43, and the unit is cm, short for centimeter.

When scientists record measurements or calculations, they are very careful to specify the amount of uncertainty in their measurement or calculation.  This is done using significant figures (also called significant digits).  Writing numbers with the proper number of significant figures means that you write all the digits known with certainty plus one (and only one) digit that you have guessed on.  This means that when you record a measurement from a device such as a ruler or graduated cylinder, you record all the numbers you know for sure plus one last number that you guess between the tic marks on the measuring device.  You record only one digit for this guess.  Your text or this online text has examples.

Your text also explains how to round the answer from a mathematical calculation to the correct number of significant figures depending on if the math contains addition and subtraction or multiplication and division.  When performing unit conversions, you most often use the multiplication/division rule, which counts the number of significant figures in each number and rounds the answer to match the fewest significant figures.  Here, we want to practice determining the number of significant figures in equalities that will be used for unit conversions.

Exact Numbers

All quantities that have been measured will have a finite number of significant figures, since the last digit recorded will be a guess and will contain uncertainty.  However, if a quantity is exact, it has an infinite number of significant figures.  This is because there is no last digit that was guessed, all digits were known with certainty (and all places further to the right of the number are know with certainty to be zero). So how do you tell if a number is measured or exact?

One kind of exact number is when you count things that can’t come in parts, such as 4 people in the room or 3 cans of diet coke in the refrigerator.  Each of those has an infinite number of significant figures, not one.  That is, we can keep writing zeros after the decimal place and can never get to a digit that was rounded or had uncertainty.  Another example is when you use the number of things you counted to calculate the average of those things.  For example, to calculate the average of three numbers – say 92, 85, and 89 – you would add the numbers up and divide by 3.  The 3 is counted, so it has an infinite number of significant figures and won’t play a factor in determining the number of significant figures in the answer of the calculation.

Other exact numbers come from definitions of units, such as all the metric prefixes or definitions within the English system.  Here are some examples of defined quantities that have an infinite number of significant figures (sf):

60 s = 1 min     (infinite sf)

60 min = 1 hr   (infinite sf)

4 qt = 1 gal        (infinite sf)

2 c = 1 pt           (infinite sf)

3 ft = 1 yd          (infinite sf)

Measured Numbers

If an equality relates the measurement between the metric and English system, then the numbers are measured, not exact.  The exception to this is the inch to centimeter conversion.  The inch was originally defined as the length of three grains of barley.  Since the length of three grains of barley isn’t constant, in the 1930’s it was redefined based on the metric system, thereby making it exact.

1 in = 2.54 cm     (infinite sf)

Two measured English to metric conversions are shown below, rounded to three significant figures each.

2.20 lb = 1 kg    (3 sf)

1.06 qt = 1 L      (3 sf)

Constants, such as

c = 3.00 x 10⁸ m/s

h = 6.63 x 10^-34 kg·m²/s

R = 1.10 x 10⁸ m^-1

are also measured quantities and have a finite number of significant digits.

If you look up the conversions or constants above on the internet, you would find them reported to more than the three significant figures shown here.  That means the conversions and constants written above have been rounded.  Whenever a conversion or constant has been rounded, the number has a finite number of significant figures.  This means that when you use these numbers in calculations, they could limit the number of significant figures you would report for the answer of the calculation.  That is, if you had a precise calculation with lots of significant figures and used it in a calculation with a conversion factor or constant rounded to just a few significant figures, the conversion factor or constant would reduce the precision of your calculation – it would cause more uncertainty because it was rounded.  Therefore, it is a good rule of thumb to make sure that when conversion factors or constants are used in calculations with measured quantities, they have at least as many significant figures as the measured quantities so that they don’t introduce uncertainty.