Recognizing and Counting Significant Figures
The first and easiest rule for counting significant figures is that any non-zero number is counted as significant. Thus the number 356 has three significant figures (3 sig figs), 24.996 has five sig figs and .33 has two.
Zeroes are more complicated. They are sometimes counted and sometimes not depending on their placement in the number. Zeros come in three flavors in a number:
- zeros between non-zeros
- zeros at the beginning (the left end) of a number
- zeros at the end (the right end) of a number
Zeros between non-zeros
Zeroes that are between two non-zero numbers are always counted. Thus 307 has three sig figs, and 1.0042 has five sig figs. This is because estimates rarely, if ever include zeros anywhere but at the end. As an example, the news might tell you that 250,000 people attended a parade (an estimate with the zeros at the end). On the other hand if you attend a baseball game they might announce that 4,093 people attended the game. The zero here is reliable, because nobody would go to the trouble of counting the last 93 people if they were unsure whether it was 4000 or 4100 people before that last 93.
Zeros at the beginning of a number
Zeros at the beginning of a number (to the left of all non-zeroes) are never counted. This is true whether or not there is a decimal point in the number and no matter where that decimal point may be. The number 0000234 (perhaps from the odometer of a car) has three sig figs, the number 0.45 has two, and the number 0.00000000000005 has only 1 sig fig. Again, this does NOT imply that the zeroes don’t matter, they are very important place holders, but that is their only purpose. The zeroes here are not part of the measurement.
Another way to look at this is to imagine you are given a piece of a broom handle and are told that it is one meter long. You are then then asked to measure the distance across the classroom. Laying the broom stick one the ground over and over again lets you "measure" the width of the room to be 12 m. If you converted that distance to kilometers, you would find that the width of your room was 0.012 km. The second number (0.012km) cannot be more reliable or precise than the first number -- since you only measured once. Therefore the extra zeroes serve only to mark the place of the 12. They are not “significant.”
Zeros at the end of a number
Lastly, zeroes to the right of all non-zero numbers count if, and only if, there is a decimal point in the number. The zeroes do not need to be after the decimal point or even near it. So the number 20.00 has four sig figs, the number 300. has three (notice the decimal point after the second zero), and the number 2.990 has four. This idea is how a scientist would distinguish between the counted fruit flies and the ducks mentioned in the discussion of reliability. “Like a million” would be written 1,000,000 (no decimal point and therefore only one sig fig) and the counted million would be written 1,000,000. (with a decimal point and therefore with seven sig figs).
Just to be clear... the number 0.0003060 has 4 significant figures. The 3 and the 6 count because they are not zeros. The zeros at the left end do NOT count, because zeros at the beginning never do. The zero between the 3 and the 6 counts because zeros between non-zeros always count. Lastly, the zero after the 6 counts because zeros at the end count if there is a decimal point somewhere in the number.