Significant FiguresSignificant Figures

There are 2 different types of numbersThere are 2 different types of numbers ExactExact MeasuredMeasured

Measured number = they are measured with a Measured number = they are measured with a measuring device so these numbers have measuring device so these numbers have ERROR.ERROR.

An exact number is obtained when you An exact number is obtained when you count objects or use a defined count objects or use a defined relationship.relationship.

Numbers without Significant Numbers without Significant FiguresFigures

Counting objects are always exact2 soccer balls4 pizzas

Exact relationships, predefined values, not measured

1 foot = 12 inches1 meter = 100 cm

Learning CheckLearning Check

Classify each of the following as an exact or Classify each of the following as an exact or a measured number.a measured number.

1 yard = 3 feet1 yard = 3 feet

The diameter of a red blood cell is 6 x 10The diameter of a red blood cell is 6 x 10-4 -4 cm.cm.

There are 6 hats on the shelf.There are 6 hats on the shelf.

Gold melts at 1064°C.Gold melts at 1064°C.

Learning CheckLearning Check

1 yard = 3 feet - 1 yard = 3 feet - This is a defined relationship.This is a defined relationship.

The diameter of a red blood cell is 6 x 10The diameter of a red blood cell is 6 x 10-4 -4 cm. - cm. - A measuring A measuring

tool is used to determine length.tool is used to determine length.

There are 6 hats on the shelf. -There are 6 hats on the shelf. -The number of hats is The number of hats is

obtained by counting.obtained by counting.

Gold melts at 1064°C. - Gold melts at 1064°C. - A measuring tool is required.A measuring tool is required.

Measurement and Sig FigsMeasurement and Sig Figs Every experimental Every experimental

measurement has a measurement has a degree of uncertainty.degree of uncertainty.

The volume, V, at The volume, V, at right is certain in the right is certain in the 10’s place, 10’s place, 10mL<V<20mL10mL<V<20mL

The 1’s digit is also The 1’s digit is also certain, certain, 17mL<V<18mL17mL<V<18mL

A best guess is A best guess is needed for the tenths needed for the tenths place.place.

Measurement & Sig FigsMeasurement & Sig Figs

The mass is The mass is certain to the certain to the tenths – between tenths – between 0.5 and 0.6g0.5 and 0.6g

Best guess is Best guess is needed for needed for hundredths place.hundredths place.

My guess is 0.58g My guess is 0.58g – 2 significant – 2 significant figuresfigures

What is the Length?What is the Length?

We can see the markings between 1.6-1.7cmWe can see the markings between 1.6-1.7cm We can’t see the markings between the .6-.7We can’t see the markings between the .6-.7 We must guess between .6 & .7We must guess between .6 & .7 We record 1.67 cm as our measurementWe record 1.67 cm as our measurement The last digit an 7 was our guess...stop thereThe last digit an 7 was our guess...stop there

Learning CheckLearning Check

What is the length of the wooden stick?

1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

Measured NumbersMeasured Numbers

Do you see why measured numbers have Do you see why measured numbers have error…you have to make that guess!error…you have to make that guess!

All but one of the significant figures are All but one of the significant figures are known with certainty. The last significant known with certainty. The last significant figure is only the best possible estimate.figure is only the best possible estimate.

To indicate the precision of a To indicate the precision of a measurement, the value recorded should measurement, the value recorded should use all the digits known with certainty.use all the digits known with certainty.

Significant FiguresSignificant Figures

Used to report all precisely known Used to report all precisely known numbers + one estimated digitnumbers + one estimated digit

Rule #1Rule #1

Every nonzero digit is significantEvery nonzero digit is significant

Examples:Examples:

24 = 224 = 2

3.56 = 33.56 = 3

7 = 17 = 1

Rule #2Rule #2

Zeros between non-zeros are significantZeros between non-zeros are significant

Examples:Examples:

7003 = 47003 = 4

40.9 = 3 40.9 = 3

Rule #3Rule #3

Zeros appearing in front of non-zero digits Zeros appearing in front of non-zero digits are are notnot significant significant

Act as placeholders Act as placeholders Can’t be dropped unless written in scientific Can’t be dropped unless written in scientific

notationnotation

Examples:Examples:

0.00024 = 20.00024 = 2

0.453 = 30.453 = 3

Rule #4Rule #4

Zeros at the end of a number Zeros at the end of a number andand to the right of to the right of a decimal point are significant.a decimal point are significant.

They show how precise the measurement wasThey show how precise the measurement was

Examples:Examples:

43.00 = 443.00 = 4

1.010 = 41.010 = 4

1.50 = 31.50 = 3

Rule #5Rule #5

Zeros at the end of a number Zeros at the end of a number when no decimal point decimal point is present aren’t significant.

You can’t tell if the number was rounded

Examples:Examples:

300 = 1300 = 1

27,300 = 327,300 = 3

Rule # 6Rule # 6

All numbers use in scientific notation are All numbers use in scientific notation are significant.significant.

They show precision of measurement.They show precision of measurement.

Examples:Examples:4.00 x 10 = 34.00 x 10 = 35.0977 x 10 = 55.0977 x 10 = 51.5000 x 10 = 51.5000 x 10 = 5

2

4

4

Practice with ZeroesPractice with Zeroes

45.8736 .000239 .00023900 48000. 48000 3.982106 1.00040

•6 - All digits count

•3 - Leading 0’s don’t

•5 - Trailing 0’s do

•5 - 0’s count in decimal form

•2 - 0’s don’t count w/o decimal

•4 - All digits count

•6 - 0’s between digits count as well as trailing in decimal form

Sig. Fig. Math RulesSig. Fig. Math Rules

Multiplication / Division:Multiplication / Division: Your answer can’t have more sig. figs. Your answer can’t have more sig. figs.

than than the number in the problem with the least the number in the problem with the least amt. amt. of sig. figs.of sig. figs.

Your are only as good as your least accurate Your are only as good as your least accurate measurementmeasurement

Example = 60.56227892 x 35.25Example = 60.56227892 x 35.25Calculator says – 2134.890832 (wrong)Calculator says – 2134.890832 (wrong)Answer - 2135 Answer - 2135

Mult/Div Significant FiguresMult/Div Significant Figures

Sig. Fig. Math RulesSig. Fig. Math Rules

Addition / Subtraction:Addition / Subtraction:Answers can’t have more numbers to Answers can’t have more numbers to

the right of the decimal point than the the right of the decimal point than the number in the problem with the least amt. number in the problem with the least amt. of numbers to the right of the decimal of numbers to the right of the decimal point.point.

Example = 24.1 + 3.35 + 2.23Example = 24.1 + 3.35 + 2.23Calculator says – 29.68 (wrong)Calculator says – 29.68 (wrong)Answer – 29.7Answer – 29.7

Add/Sub Significant FiguresAdd/Sub Significant Figures

Line up decimal just like you always do Line up decimal just like you always do when adding or subtracting with decimalswhen adding or subtracting with decimals

Round your answer to have the same Round your answer to have the same number of sig figs as the measurement number of sig figs as the measurement with the least.with the least.

Add/Sub ExamplesAdd/Sub Examples

56005600

+ 997.5+ 997.5

6597.56597.5

66006600

Can only have 2 sig fig so Can only have 2 sig fig so you must round answer.you must round answer.

RoundingRounding Once you decide how many digits to retain, the rules for Once you decide how many digits to retain, the rules for

rounding off numbers are straightforward:rounding off numbers are straightforward: RULE 1. RULE 1. If the first digit you remove is 4 or less, drop it If the first digit you remove is 4 or less, drop it

and all following digits. 2.4271 becomes 2.4 when and all following digits. 2.4271 becomes 2.4 when rounded off to two significant figures because the first rounded off to two significant figures because the first dropped digit (a 2) is 4 or less.dropped digit (a 2) is 4 or less.

RULE 2. RULE 2. If the first digit removed is 5 or greater, round If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 is 4.6 when up by adding 1 to the last digit kept. 4.5832 is 4.6 when rounded off to 2 significant figures since the first rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater.dropped digit (an 8) is 5 or greater.

If a calculation has several steps, it is best to round off If a calculation has several steps, it is best to round off at the end.at the end.

Learning CheckLearning Check

32.27 1.54 = 49.6958

3.68 .07925 = 46.4353312

1.750 .0342000 = 0.05985

3.2650106 4.858 = 1.586137 107

6.0221023 1.66110-24 = 1.000000

49.7

46.4

.05985

1.586 107

1.000

Learning CheckLearning Check

.56 + .153 = .713

82000 + 5.32 = 82005.32

10.0 - 9.8742 = .12580

11 – 9.8742 = 1.12580

.71

82000

.1

1

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