What are significant figures? (aka sig figs) Significant figures are all the digits in a measurement that are known with certainty plus a last digit that must be estimated. With experimental values your answer can have too few or too many sig figs, depending on how you round.

How Rounding Influences Sig Figs 1.024 x 1.2 = 1.2288 Too many numerals (sig figs) Too precise 1.024 x 1.2 = 1 Too few numerals (sig figs) Not precise enough

Why This Concept is Important We will be adding, subtracting, multiplying and dividing numbers throughout this course. You MUST learn how many sig figs to report each answer in or the answer is meaningless. Lab Reports: Correct number of sig figs needed or lose 1 point. Tests/Quizzes: Correct number of sig figs +/- 1 needed or lose 1 point.

How Do We Find the Correct Number of Sig Figs In an Answer? First, we will learn to count number of sig figs in a number. You must learn 4 rules and how to apply them. Second, we will learn the process for rounding when we add/subtract or multiply/divide. We will then apply this process in calculations.

Rules for Counting Sig Figs Rule #1: Read the number from left to right and count all digits, starting with the first digit that is not zero. Do NOT count final zero’s unless there is a decimal point in the number! 3 sig figs 4 sig figs 5 sig figs 23.4 234 0.234 2340 203 345.6 3.456 0.03456 34560 3405 678.90 6789.0 0.0067890 67008 60708

Rules for Counting Sig Figs Rule #2: A final zero or zero’s can be designated as significant if a decimal point is added after the final zero. 3 sig figs 4 sig figs 5 sig figs 2340 23400 234000 2340000 2340. 2000. 20000. 23400.

Rules for Counting Sig Figs Rule #3: If a number is expressed in standard scientific (exponential) notation, assume all the digits in the scientific notation are significant. 2 sig figs 3 sig figs 4 sig figs 2.3 x 102 2.0 x 103 2.30 x 102 2.00 x 103 2.300 x 102 2.000 x 103

Rules for Counting Sig Figs Rule #4: Any number which represents a numerical count or is an exact definition has an infinite number of sig figs and is NOT counted in the calculations. Examples: 12 inches = 1 foot (exact definition) 1000 mm = 1 m (exact definition) 24 students = 1 class (count)

Practice Counting Sig Figs How many sig figs in each of the following? 1.2304 mm 1.23400 cm 1.200 x 105 mL 0.0230 m 0.02 cm 8 ounces = 1 cup 30 cars in the parking lot

Answers to Practice How many sig figs in each of the following? 1.2304 mm (5) 1.23400 cm (6) 1.200 x 105 mL (4) 0.0230 m (3) 0.02 cm (1) 8 ounces = 1 cup (infinite, exact def.) 30 cars in the parking lot (infinite, count)

General Rounding Rule When a number is rounded off, the last digit to be retained is increased by one only if the following digit is 5 or greater. EXAMPLE: 5.3546 rounds to 5 (ones place) 5.35 (hundredths place) 5.355 (thousandths place) 5.4 (tenths place) You will lose points for rounding incorrectly!

Process for Addition/Subtraction Step #1: Determine the number of decimal places in each number to be added/subtracted. Step #2: Calculate the answer, and then round the final number to the least number of decimal places from Step #1.

Addition/Subtraction Examples Round to tenths place. Example #2: Round to hundredths place. Example #3: Round to ones place. 23.456 + 1.2 + 0.05 -------------- 24.706 Rounds to: 24.7 3.56 - 0.14 - 1.3501 --------------- 2.0699 2.07 14 + 0.735 + 12.0 26.735 27

Process for Multiplication/Division Step #1: Determine the number of sig figs in each number to be multiplied/divided. Step #2: Calculate the answer, and then round the final number to the least number of sig figs from Step #1.

Multiplication/Division Examples Round to 1 sig fig. Example #2: 2 sig figs. Example #3: 3 sig figs. 23.456 x 1.2 x 0.05 -------------- 1.40736 Rounds to: 1 3.56 x 0.14 x 1.3501 --------------- 0.67288984 0.67 14.0/ 11.73 1.193520887 1.19

Practice Example #1: Example #2: Example #3: . 23.456 x 4.20 x 0.010 -------------- Rounds to: ? 0.001 + 1.1 + 0.350 --------------- 17/ 22.73

Answers to Practice Example #1: Example #2: Example #3: . 23.456 -------------- 0.985152 Rounds to: 0.99 (2 sf) 0.001 + 1.1 + 0.350 --------------- 1.451 1.5 (tenths) 17/ 22.73 0.747910251 0.75 (2 sf)

Other Rules If you are using constants which are not exact, try to select one which has at least one or more sig figs that the smallest number of sig figs in your original data. That way, the constant will not impact the number of sig figs in your final answer. Example: pi = 3.14 (3 sig figs) = 3.142 (4 sig figs) = 3.1459 (5 sig figs)

Important Rounding Rule When you are doing several calculations, carry out all the calculations to at LEAST one more sig fig than you need (I carry all digits in my calculator memory) and only round off in the FINAL result.