Significant Figures In Measurements

Significant Figures At the conclusion of our time together, you should be able to: 1. Explain what significant figures are in a measurement 2. Determine the number of significant figures in any measurement

Significant Figures The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated. The numbers reported in a measurement are limited by the measuring tool.

How many sig figs are there in a given measurement?

Measurement and Significant Figures Every experimental measurement has a degree of uncertainty. Every experimental measurement has a degree of uncertainty. The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The volume, V, at right is certain in the 10’s place, 10mL<V<20mL The 1’s digit is also certain, 17mL<V<18mL The 1’s digit is also certain, 17mL<V<18mL A best guess is needed for the tenths place. A best guess is needed for the tenths place.

To indicate the precision of a measurement, the value recorded should use all the digits known with certainty, plus one additional estimated digit that usually is considered uncertain by plus or minus 1. To indicate the precision of a measurement, the value recorded should use all the digits known with certainty, plus one additional estimated digit that usually is considered uncertain by plus or minus 1. No further insignificant digits should be recorded. No further insignificant digits should be recorded. The total number of digits used to express such a measurement is called the number of significant figures. The total number of digits used to express such a measurement is called the number of significant figures. All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate.

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

Reading a Meterstick. l I.... I 3....I.... I 4.. cm First digit (known)= 2 2.?? cm Second digit (known)= ? cm Third digit (estimated) between Length reported=2.77 cm or2.76 cm or2.76 cm or2.78 cm

Known + Estimated Digits In 2.77 cm… Known digitsandare 100% certain Known digits 2 and 7 are 100% certain The third digit 7 is estimated (uncertain) The third digit 7 is estimated (uncertain) In the reported length, all three digits (2.77 cm) are significant including the estimated one In the reported length, all three digits (2.77 cm) are significant including the estimated one

Learning Check. l I.... I I.... I 10.. cm What is the length of the line? 1) 9.6 cm 2) 9.62 cm 3) 9.63 cm How does your answer compare with your neighbor’s answer? Why or why not?

Zero as a Measured Number. l I.... I I.... I 5.. cm What is the length of the line? First digit 5.?? cm Second digit 5.0? cm Last (estimated) digit is 5.00 cm

Always estimate ONE place past the smallest mark! 11.5 mL

So how many sig figs are there in a given measurement? 52.8 mL

How to Determine Significant Figures in a Problem Use the following rules: Use the following rules:

Rule #1 Every nonzero digit is significant Every nonzero digit is significantExamples: 24 = = 3 7 = 1

Rule #2 – Sandwiched 0’s Zeros between non-zeros are significant Zeros between non-zeros are significantExamples: 7003 = = 3

Rule #3 – Leading 0’s Zeros appearing in front of non-zero digits are not significant Zeros appearing in front of non-zero digits are not significant Act as placeholders Act as placeholders Can’t be dropped, show magnitude Can’t be dropped, show magnitude Examples: Examples: = = 3

Rule #4 – Trailing 0’s with DP Zeros Zeros at the end end of a number and and to the right right of a decimal point are significant. Examples: = = = 3

Rule #5 – Trailing 0’s without DP Zeros at the end of a number and to the left of a decimal point aren’t significant Zeros at the end of a number and to the left of a decimal point aren’t significantExamples: 300 = 1 27,300 = 3

Easier Way to do Sig Figs!! Pacific/Atlantic Pacific/Atlantic PA If a decimal point is present, start on the Pacific (P) side and draw an arrow through the number until you hit a non-zero digit. Count all numbers without an arrow through them. If a decimal is absent, start on the Atlantic (A) side and draw an arrow through the number until you hit a non-zero digit.

Examples: grams decimal present, start on “P” side, draw arrow, count digits without an arrow through it. Answer = 6 10,100 centimeters Decimal absent, start on “A” side, draw an arrow, count digits without an arrow through it. Answer = 3

Learning Check A. Which answer(s) contain 3 significant figures? 1) ) ) 4760 B. All the zeros are significant in 1) ) ) x 10 3 C. 534,675 rounded to 3 significant figures is 1) 535 2) 535,000 3) 5.35 x ) 535 2) 535,000 3) 5.35 x 10 5

Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and ) 22.0 and ) and 40 3) and 150,000

Significant Figures and Numbers Some numbers are exact: There are 60 seconds in 1 minute 25 cents in 1 quarter 12 eggs in one dozen There is no uncertainty in any of these numbers. In other words there are eggs in 1 dozen (add as many zeros as you like)

Counting Numbers Counting numbers have infinite sig figs. Counting numbers have infinite sig figs. Ex: 3 apples Ex: 3 apples

Significant Figures Lets’ see if you can: 1. Explain what significant figures are in a measurement 2. Determine the number of significant figures in any measurement

State the number of significant figures in each of the following: A m B L C g D m E. 2,080,000 bees Learning Check