The Science of Numbers Sig-figs Originally created by S. Koziol 1 Nov. 2015

Introduction – (DO NOT COPY THIS SLIDE) Significant figures are critical when reporting scientific data because they give the reader an idea of how well you could actually measure/report your data. Before looking at a few examples, let's summarize the rules for significant figures.

The Rules (copy them, know them, Live them!) 1) ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.

The Rules (continued) 2) ALL zeroes between non-zero numbers are ALWAYS significant.

The Rules (continued) 3) ALL zeroes which are SIMULTANEOUSLY to the right of the decimal point AND at the end of the number are ALWAYS significant.

The Rules (continued) 4) ALL zeroes which are to the left of a written decimal point and are in a number >= 10 are ALWAYS significant. A helpful way to check rules 3 and 4 is to write the number in scientific notation. If you can/must get rid of the zeroes, then they are NOT significant.

Mathematical Operations (DO NOT COPY THIS SLIDE) To find out how to use this information when performing mathematical operations (addition, subtraction, multiplication, division), see "Significant Figures: Mathematical Operations.“

ADDITION AND SUBTRACTION (+, -) The answer cannot CONTAIN MORE PLACES AFTER THE DECIMAL POINT THAN THE SMALLEST NUMBER OF DECIMAL PLACES in the numbers being added or subtracted. Example: 23.112233 (6 places after the decimal point) 1.3324 (4 places after the decimal point) + 0.25 (2 places after the decimal point) -------------- 24.694633 (on calculator) 24.69 (rounded to 2 places in the answer) Note: There are 4 significant figures in the answer.

MULTIPLICATION AND DIVISION (x, ÷) The answer cannot CONTAIN MORE SIGNIFICANT FIGURES THAN THE NUMBER BEING MULTIPLIED OR DIVIDED with the LEAST NUMBER OF SIGNIFICANT FIGURES. Example: 23.123123 (8 significant figures) x 1.3344 (5 significant figures) -------------- 30.855495 (on calculator) 30.855 (rounded to 5 significant figures) Note: There are 5 significant figures in the answer.

Sig Fig - Summary Ignore leading zeros. Ignore trailing zeros, unless they come after a decimal point. Everything else is significant. Or The simplest rule: "Convert the number into scientific notation. Any leading or trailing zeros the decimal point bumps past in the conversion will vanish. Everything else is significant."

Atlantic-Pacific Rule "If a decimal point is Present, ignore zeros on the Pacific (left) side. If the decimal point is Absent, ignore zeros on the Atlantic (right) side. Everything else is significant." Number Atlantic-Pacific rule 0.001010 decimal point Present: ignore zeros on the Pacific side. 4 sig. digits. 0.30000 decimal point Present: ignore zeros on the Pacific side. 5 sig. digits. 100.0000 decimal point Present: ignore zeros on the Pacific side (none!) 7 sig. digits. 12303000 decimal point Absent: ignore zeros on the Atlantic side. 5 sig. digits.

Exact Values All exact values or conversion factors have an infinite (never ending) number of significant figures. You might count 8 people or 9 people but it is not possible to count 8.5 people. Examples of exact values: 12 complete waves ; 17 people ; 28 nails There are exactly: 60 seconds in one minute 1000 meters in one kilometer [this is the definition of kilo (k)] 12 eggs in one dozen 7 days in one week 2.54 cm in one inch

Constants - Simplified Fundamental constants are numbers without units of any kind that come strictly from mathematics; they are not ``measured'' like most quantities in science. When you have one of these numbers, you should never let it determine how many significant figures you have. = 3.14159265359...

References H. M. Stone, "Atlantic-Pacific sig figs (INS)", J. Chem. Educ., 66, 829 (1989). "Significant Figures - Rules." Significant Figures - Rules. N.p., n.d. Web. 01 Nov. 2015. "Significant Figures Rules." Significant Figures Rules. N.p., n.d. Web. 01 Nov. 2015.