Significant Figure Rules

One of the great things about science is that nobody's perfect (for some of
you, that may take a little getting used to!). The reason this comes up is that
science investigations rely so much on data, usually measurements in the form
of numbers. No measuring device can measure any quantity *exactly*, so
one must remember the uncertainty in every measurement when using them in calculations.
Numbers that are measurements or that come from measurements are written in
a specific way so that people reading them can know how exact the writer is
being. Digits in a measurement that are important for it are called **significant
figures** (or significant digits). Using them keeps us honest, because
we are prevented from seeming overly precise in our work.

**Gathering Data**

When gathering data, always examine the measuring tool before you begin. Most tools in our laboratory are marked using the SI System (sometimes called "the metric system"), which always uses multiples and divisions of 10. Most of our thermometers are marked every 1 degree, and our balances are marked every 0.01 grams. When recording data, you should always write down all the digits that are marked on the tool, and one more that you estimate. If you think a reading is "right on the line" then you are estimating a "0" for the last digit.

For example, using a thermometer, the fluid may be slightly above the third
mark over twenty. That's 23-and-a-little-bit. You get to estimate the "little"
bit. If it is closer to 23, you should estimate .1 or .2 or .3; if it is closer
to 24, you should estimate .7 or .8 or .9. If it is closer to the middle, you
should estimate .4 or .5 or .6. Since these are *your* estimates, the
exact digit is up to you, as long as it is in the reasonable range. You and
your partner should agree or be close.

The **significant digits** are all the digits you measured + the
one you estimated.

**Counting Significant Digits** (in someone else's measurement)

In order to present results with the proper precision, we need to know how many significat figures are present in each number we use in a calculation. There are four basic rules:

- Zeroes in the beginning of a number
**never**count. - Zeroes at the end of a number count
**only if**there is a written decimal point. - The digits 1 - 9
**always**count. - Zeroes between the digits 1 - 9
**always**count.

So how many significant digits are present in the following numbers: 5, 51,
55, 510, 501, 5010, 5010.0, 5.0, 0.00501

(the answers are 1, 2, 2, 2, 3, 3, 5, 2, 3).

**Using Significant Digits** in calculations

There are two basic rules when using significant digits in calculations. The first is that when adding or subtracting, the answer can only be as precise as the least precise number used. For example, a 250 pound person who has a hair pulled out (say, 0.001 pounds) still weighs 250 pounds. That's because the last significant digit is the 5 (in the tens place), and everything after that is not even estimated. So you have no idea how many ones of pounds the guy weighs, or how many tenths, or hundredths, or thousandths. Therefore, you have no idea what from to subtract 0.001 pounds. So he still weighs 250 pounds.

The other rule is that when multiplying or dividing, the answer has the same number of significant digits as the number used with the least. For instance, there is a quick estimation of pi (p) as 22¸7. On a calculator, that gives 3.1428571423, which is pretty close. But if you had measured a circle as 22 feet around and 7 feet across, the answer must be rounded to "3" to match the least number of significant digits.

**Other Hints**

Every measurement and calculation using measurements should take significant figures into account. Exceptions are counting numbers (like dividing by 20 students to get a class average) and exact equivalents (like 1000 mm = 1 m, which is perfect and exact by definition).

Here's another way to think of significant digits: when you write a number in scientific notation, all the zeroes that are un-necessary for precision disappear, and the remaining digits are significant. 510 become 5.1x10^2 (2 sig figs) and 510.0 becomes 5.100x10^2 (still 4 sig figs). The exponent part isn't significant because it takes over for the placeholding zeroes. For that reason, only the decimal part if pH or similar values count as significant figures; the number before the decimal is the exponent (look up pH in a chemistry text and logarithm in a math book).

One last way to think about significant digits: numbers that make a measurement more exact are significant, numbers that don't are not. Think about that for a little while.