## The Significance of Numbers and Units

I will begin this blog with one of the most important concepts in physics: UNITS.

Why are they so important, you might ask me… well…

When NASA sent the Mars Climate Orbiter to probe the Martian atmosphere in 1999, the probe disintegrated in the upper atmosphere. The cause… non-SI units vs. SI units.

As the group of rebels that formed a nation in defiance of the British government, the US has been very nonconforming to the Système International d’Unités (SI) or International System of Units. That is why we have units such as: feet (ft), inches (in), miles (mi), acres, slugs, pounds (lb), Fahrenheit (ºF), Rankine (ºR), pounds per square inch (PSI), etc. The SI units are: meters (m), kilograms (kg), seconds (s), kelvin (K), Pascal (Pa), etc.

So why is this SI system so much better? Well in the US 12 inches is a foot, and there are 5280 feet in a mile. The counterpart in SI units are 100 centimeters in a meter and 1000 meters in a kilometer. This means there are 63,360 inches in a mile and 100,000 centimeters in a kilometer. To me it’s much easier to multiply 1000 by 100 than 5280 by 12, but I digress…

So if the SI unit was meters, why did I talk about centimeters and kilometers. This is where the powers of ten come in. Since there is a standard unit for all the measurements in the SI system, it was agreed upon that the easiest way to divide measurements was by using a factor of ten. Therefore, a meter cut by ten becomes a decimeter, a tenth of decimeter is a centimeter, a tenth of a centimeter is a millimeter, etc. In fact all measurements in SI units follow this system, (prefix)(unit), where the prefixes are yotta (Y, 1024), zetta (Z, 1021), exa (E, 1018), peta (P, 1015), tera (T, 1012), giga (G, 109), mega (M, 106), kilo (k, 103), (deca (D, 10), deci (d, 10-1), centi (c, 10-2)), milli (m, 10-3), micro (μ or u, 10-6), nano (n, 10-9), pico (p, 10-12), femto (f, 10-15), atto (a, 10-18), zepto (z, 10-21), yocto (y, 10-24). There is an exception: kg. This SI unit has the kilo prefix making the base unit grams (g), but kg is the SI unit.

Now that units are covered, we must talk about what makes a measurement scientific. To start off we must discuss that not all digits in a number are as important as others. We, therefore, will define what makes a digit significant. Taking one measurement is not good enough for statistics, in fact you need at least three measurements to get any real information. This is what we will base this system off of.

Lets assume you have a small cube that you want to measure. You have two instruments that you will use for this task, a ruler and a tape measure. The ruler is in inches with 1/4 inch divisions (0.25 in), while the tape measure is in inches with 1/16 inch divisions (0.0625 in). You measure with the ruler and get measurements of: 3.25 in, 3.50 in, and 3.25 in. The statistics say that the average measurement is 3.3333333 in ± 0.117851 in. We are going to let 0.117851 in become 0.1 inches as we believe that, with certainty, we can gather another set of measurements and get near the same standard deviation (stdev). Since the stdev was truncated to 0.1 inches, we will match the decimal place on our average to make 3.3333333 in become 3.3 inches. Therefore, our average has two significant figures (sig figs) for this ruler. Now you measure with the tape measure and get measurements of: 3.3750 in, 3.4375 in, and 3.3750 in. The statistics make the measurement 3.395833 in ± 0.029463 in. Like before, 0.029463 in becomes 0.03 inches and 3.395833 in becomes 3.40 inches. Notice that we now have 3 sig figs instead of two. To wrap up, as measurements get closer together, the last digits become more likely to become significant (determined by the standard deviation).

The formal rules are as follows: non-zero digits to the left of the (implicit) decimal place are significant (12400 has 3 sig figs), non-zero and zero digits to the left of the explicit decimal are significant and exact (12400. has ∞ sig figs), zero digits as a placeholder for the explicit decimal are not significant (0.02 has 1 sig fig), zero digits following non-zero digits to the right of an explicit decimal are significant but not exact (10.0000 has 6 sig figs). Multiplication and division force the answer to have the lesser sig figs of the two (or more) values (9.00000/3.0 = 3.0, 14/7.000 = 2.0, or 14./7.000 = 2.000). Addition and subtraction fore the answer to have the lesser of the digits to the right of the decimal (9.000 – 3.3 = 5.7, 7. + 8.88 = 15.88, 0.33 – 0.31 = 0.02). Notice that the number of significant figures can be less than the original with addition or subtraction. It will be common practice to report an answer to a problem with the same sig figs (least) that were given in the problem, even if addition and/or subtraction occur. IMPORTANT: leave more sig figs than allowed during calculation and round your final answer at the end of each part of the problem, carrying the extra digits for subsequent parts.

I hope to have left an impression of the importance of significant figures and UNITS. Good luck.

-Last updated on 02/20/2017 by TLH