In general, any instrumental measurement involves an "uncertainty" intrinsic to the device used for the measurement; for example, the balance used in a hospital to measure a person's mass cannot, usually, be expected to accurately measure one's mass to the nearest 0.1gram.
For some observations the system itself may have intrinsic uncertainties; for example, one's mass will change by the simple consumption of a glass of water, and thereafter it declines due to water loss via perspiration for instance. Therefore, it does not make any sense to specify one's mass to the nearest 0.1gram because at any other time, before or after, the mass measurement is made, it would simply be WRONG!.
Example: If we had found the average mass (using simple bathroom scales) of all the students in a grade nine class and found it to be 54,694.2 grams, we would prefer to write it down using 3-significant figures, i.e. as 54,600 grams, or better yet, as 54.7 kg. This acknowledges both the uncertainty intrinsic to the individual masses used to calculate the average and the fact that such a balance (accurate to 6 significant figures) for determining a person's mass was not used.
(NOTE: large numbers of figures can sometimes be generated by
simple mathematical processes, such as multiplication or division...the excess figures created should never be considered
significant!)
Numbers with lots of zeros are usually more efficiently handled by using "exponential" notation. If the ONLY purpose of the zeros is to "hold" the decimal place then exponential notation is almost always used.
Example The mean distance to the sun is 150,000,000 km. Assuming this number is really good to three significant figures, it should be written as 1.50x108km
Students should be able to convert directly from seconds to years. You might help them work out a simple conversion constant to do this. The number 3.15x107seconds per year(s/a) is usually accepted. You could have your students work this out individually or perhaps you might do this a class (teacher-led) activity.
Hours per year(h/a) is also a good conversion factor to have on hand for this activity.
In this activity students will get a "feel" for astronomical distances as applied to the solar system by calculating the approximate times required to travel to various solar system bodies. It might be useful to "walk" the class through row 2 (the sun) before letting them complete the chart on their own. Students are usually astonished at their results.
This exercise makes a good prerequisite to any discussions about the distances to stars and galaxies.
The equation is usually written; distance(d)=speed(v) x time(t). In this instance; however,we are actually looking for time; therefore,
In this exercise students will generally get enormous numbers of seconds or hours since the speeds (given in the chart) contain these units (see below). It is important that students recognize this and convert to more appropriate temporal units.
Example: A quick calculation will show that the time to walk to the moon (non-stop of course) is about 46,200 hours. A better way to express this is as 5.28 years.
The process of substituting units into an equation and then reducing the expression to its simplest form is called dimensional analysis
Show students that from the equation for travel-time (t=d/v), the appropriate units of time can be easily determined, as shown below. For this example we assume that the distance is in kilometres, and the speed is in units of kilometres per second, then from...
In this example time would have units of hours