(How Exact is (How Exact is Exact ?)Exact ?)

As you saw from our short As you saw from our short activity, the activity, the graduationsgraduations on the measuring on the measuring instrument used affects instrument used affects the certainty of our the certainty of our measurementmeasurement

When you used the ruler When you used the ruler with a scale of 0.1, your with a scale of 0.1, your measurement was much measurement was much more accurate than when more accurate than when you were using the ruler you were using the ruler that was scaled by 10that was scaled by 10

DefinitionDefinition Significant FiguresSignificant Figures refers to refers to those digits in a number that those digits in a number that are directly read in a are directly read in a measurement plus one measurement plus one uncertain digituncertain digit

The system of significant figures The system of significant figures was devised so we can determine was devised so we can determine how reliable a measurement was.how reliable a measurement was.

Whenever you make any measurement, Whenever you make any measurement, always include all the digits that you can always include all the digits that you can directly read PLUS one uncertain digit.directly read PLUS one uncertain digit.

REMEMBERREMEMBER If the reading is If the reading is 23.5723.57 cm, this meanscm, this means

thatthat 22,,33 andand 55 are read directly on theare read directly on the

ruler, whileruler, while 77 is estimatedis estimated. The number 23.57 has The number 23.57 has fourfour significant figures. significant figures.

One easy method of finding One easy method of finding the number of significant the number of significant

figuresfigures

in a number is through the in a number is through the

Pacific-Atlantic RulePacific-Atlantic Rule

PP stands for the PRESENCEPRESENCE of a decimal point

PP also stands for PACIFICPACIFIC

Use this rule when you see that a number has a decimal point (decimal pt. is PRESENTPRESENT)

RULE:RULE: If a number has a decimal point, count all digits (including zero) from the first nonzero digit at the LEFTLEFT

From LEFTLEFT

1) 435.06

Count from first nonzerodigit at left (4)(4) to the last digit at right (6)(6)

Answer:Answer: 5 sig. figures5 sig. figures

2) 0.005 320

Count from first nonzerodigit at left (5)(5) to the last digit at right (0)(0)

Answer:Answer: 4 sig. 4 sig. figuresfigures

1 2 3 4 5

1 2 341 2 34

Find the no. of significant figures in:(1) 57,376. 205 1 2 3 4 5 6 7 81 2 3 4 5 6 7 8 Answer: 8 sig. figs.

(2) 4,005. 900 1 2 3 4 5 6 71 2 3 4 5 6 7

Answer: 7 sig. figs.

(3) (3) 0. 000 240 1 2 31 2 3

Answer:Answer: 3 sig. figs.

(4) 0.088 000 1 2 3 4 5 1 2 3 4 5

Answer:Answer: 5 sig. figs

AA stands for the ABSENCE ABSENCE of a decimal point

AA also stands for ATLANTICATLANTIC

Use this rule when the number has no decimal point ( decimal point is ABSENTABSENT)

RULE :RULE : If a number has no decimal point, count all digits starting from the first from the first nonzerononzero digit at the digit at the RIGHTRIGHT

From RIGHTRIGHT

1) 879,000 3 2 13 2 1

Count from first nonzerodigit at right (9)(9) to the leftmost digit (8)(8)

Answer: 3 significant. 3 significant. figuresfigures

2) 762,300 4 3 2 14 3 2 1

Count from first nonzerodigit starting at right (3)(3) to the leftmost digit (7)

AnswerAnswer:: 4 significant figures4 significant figures

NUMBERS IN SCIENTIFIC NUMBERS IN SCIENTIFIC NOTATIONNOTATION

How many significant figures are How many significant figures are there inthere in

6.02 x 106.02 x 1023 23

RULE:RULE:

For numbers in scientific notation, all For numbers in scientific notation, all the numbers in the decimal part are the numbers in the decimal part are significant…significant…

ANSWER: 3 significant figures

1) 33001) 3300 2) 0.000 017 62) 0.000 017 6 3) 110.503) 110.50 4) 210.04) 210.0 5) 0.905) 0.90 6) 8.70 X 106) 8.70 X 1066

7) 45000.7) 45000. 8) 450008) 45000

How many significant How many significant figures are there in each of figures are there in each of these numbers? these numbers?

1) 2) 2 2) 32) 3 3) 53) 5 4) 44) 4 5) 25) 2 6) 36) 3 7) 57) 5 8) 28) 2

Now, wasn’t that easy!!!Now, wasn’t that easy!!!

Let’s Review !Let’s Review !

Don’t lose it…

The number of significant figuresThe number of significant figures in a measurement indicates the in a measurement indicates the reliability of that measurereliability of that measure

When you do a mathematical operation (add, When you do a mathematical operation (add, subtract, multiply….) on that number be sure to subtract, multiply….) on that number be sure to ROUND OFF properly so the answer reflects the ROUND OFF properly so the answer reflects the reliability of the original measurementreliability of the original measurement

Say what?Say what?

ROUND OFF !!!ROUND OFF !!!

One skill you learned in grade One skill you learned in grade school is how to round off a school is how to round off a numbernumber

Can you still remember how you Can you still remember how you solved such problems…solved such problems…

RULE: If the number following the underlined decimal RULE: If the number following the underlined decimal place is 5 or greater than 5 , add 1 to the underlined place is 5 or greater than 5 , add 1 to the underlined digit. Drop all the numbers after it.digit. Drop all the numbers after it.

SOLUTION:SOLUTION: o You must round up to 3You must round up to 3o The digit following 3 is 8The digit following 3 is 8o 8 is greater than 5, so add 1 to 3 and8 is greater than 5, so add 1 to 3 and

the underlined digit becomes 4the underlined digit becomes 4o Drop all numbers after 4Drop all numbers after 4

ANSWER: 64.04ANSWER: 64.04

PROBLEM: Round off the number 64.0384 to the underlined place

PROBLEM: Round off the number 8.3722 to the underlined placeRULE : If the number following the underlined RULE : If the number following the underlined

decimal place is less than 5, keep the decimal place is less than 5, keep the underlined digit. Drop all the numbers after it.underlined digit. Drop all the numbers after it.

SOLUTION:SOLUTION:o You must round up to 7You must round up to 7o The digit following 7 is 2The digit following 7 is 2o 2 is less than 5 so keep the underlined 2 is less than 5 so keep the underlined

digit (7)digit (7) Drop all the numbers after Drop all the numbers after

ANSWER: 8.37ANSWER: 8.37

Now it’s YOUR turn!Now it’s YOUR turn! Round each number to the underlined Round each number to the underlined

digit:digit:

(1) 18.(1) 18.4421 21 Think …must round up to 4Think …must round up to 4

… … 4 is followed by 24 is followed by 2

Since 2 is less than 5, Since 2 is less than 5,

… …must keep 4must keep 4

Drop numbers after 4Drop numbers after 4

Answer: 18.4Answer: 18.4

Next number Next number

(2) 7.(2) 7.22619 619 Think …must round up to 2Think …must round up to 2

… … 2 is followed by 62 is followed by 6

Since 6 is greater than 5, Since 6 is greater than 5,

… …must add 1 to 2must add 1 to 2

So 2 becomes 3So 2 becomes 3

Drop all numbers after 3Drop all numbers after 3

Answer: 7.3Answer: 7.3

And one more… And one more… (3) 8(3) 82234.67 34.67 Think …must round up to 2Think …must round up to 2

… … 2 is followed by 32 is followed by 3

Since 3 is less than Since 3 is less than 5, 5,

… …must keep 2must keep 2

Drop all numbers after Drop all numbers after 2,2,

but to indicate place but to indicate place value value

keep two zeroskeep two zeros

Answer: 8200Answer: 8200

Don’t forget to connect ROUNDING OFF with SIGNIFICANT FIGURESProblem 1: Round off 56.18754 to three significant figuresA decimal point is Present, Present, so follow so follow Pacific Pacific

rulerule

(count from the first nonzero digit at the (count from the first nonzero digit at the leftleft) )

56.18754 56.18754

1 2 3 4 56 71 2 3 4 56 7

You must round up to 1 ( 56.You must round up to 1 ( 56.118754)8754)

ANSWER: 56.2ANSWER: 56.2

Problem 2: Round off Problem 2: Round off 0.006392 0.006392 to to two two significant significant figuresfigures

A decimal point is Present,Present, so follow so follow PacificPacific rulerule

(count from the first nonzero digit at the (count from the first nonzero digit at the leftleft))

0.06392 0.06392 1 2 3 4 1 2 3 4

You must round up to 3 ( 0.06You must round up to 3 ( 0.063392)92)

ANSWER: 0.064ANSWER: 0.064

Practice Problems:

Round these Round these measurements to measurements to three significant three significant figures:figures:

(1)(1) 85.779 kg.85.779 kg.

(2)(2) 0.007 231 56 m0.007 231 56 m

(3)(3) 457.4932 cm457.4932 cm

(4)(4) 256,769 mg256,769 mg

(5)(5) 5.9256 m5.9256 m

(6)(6) 405,670,000 mL405,670,000 mL

Did you get these answers?

(1)(1) 85.8 kg.85.8 kg.

(2)(2) 0.007 23 m0.007 23 m

(3)(3) 457 cm457 cm

(4)(4) 257,000 mg257,000 mg

(5)(5) 5.93 m5.93 m

(6)(6) 406,000,000 mL406,000,000 mL

CALCULATIONS CALCULATIONS INVOLVING SIGNIFICANT INVOLVING SIGNIFICANT

FIGURESFIGURES

PARTPART 44

RULE:RULE: Round off the answer so that it Round off the answer so that it

has the same number of has the same number of significant figures as the significant figures as the measurement with the fewest measurement with the fewest significant figures significant figures

5.3 x 1.54 x 3.21 = 26.20002 5.3 x 1.54 x 3.21 = 26.20002 (2 sf) (3 sf) (3 sf)(2 sf) (3 sf) (3 sf)

You must round off 26.20002 so that it has 2 sig. figuresit has 2 sig. figures . ANSWER:ANSWER: 2626

MULTIPLICATION and MULTIPLICATION and DIVISIONDIVISION

Now give these questions a try:Now give these questions a try:

(1)(1) 8.2 cm x 6.35 cm =8.2 cm x 6.35 cm =

(2)(2) 4,170 m4,170 m2 2 ÷ 5.0 m =÷ 5.0 m =

(3)(3) 9.3 mm x 0.025 mm x 200.3 9.3 mm x 0.025 mm x 200.3 mm =mm =

(1) 52.07 becomes 521) 52.07 becomes 52

Answer: 52 cmAnswer: 52 cm2 2

(2) 834 becomes 830(2) 834 becomes 830

Answer: 830 mAnswer: 830 m

(3) 46.56975 becomes 47(3) 46.56975 becomes 47

Answer: 47 mmAnswer: 47 mm33

RULE:RULE: Round off the answer so that it Round off the answer so that it

has the same number of has the same number of decimal decimal placesplaces as the measurement with as the measurement with the least number ofthe least number of

decimal places.decimal places. 23.5 (one decimal place) + + 22. 1 . 1 (one decimal (one decimal place)place) 77.26.26 (two decimal (two decimal places)places) 3232 .86 .86 Round off the answer to one Round off the answer to one decimal place decimal place ANSWER:ANSWER: 32.932.9

ADDITION AND ADDITION AND SUBTRACTIONSUBTRACTION

How about these How about these problems?problems?

Problem 1:Problem 1: Find the sum of 6.730 cm 1.1 cm 3.45 cm

Problem 2:Problem 2: Get the difference of 68.731 m - 14.22 m

So how well did you do? Problem 1: 6.730 cm (3 decimal places) 1.1 1 cm (1 decimal place)cm (1 decimal place)

3. 3. 45 45 cm (2 decimal places) cm (2 decimal places)

11.280 cm 11.280 cm

Answer must be rounded off to 1 decimal Answer must be rounded off to 1 decimal placeplace

Answer:Answer: 11.3 cm 11.3 cm

Problem 2:Problem 2:

68.731 m (3 decimal places) - 14.22 m (2 decimal place) 54.511 m

Answer must be rounded off to 2 decimal places Answer:Answer: 54.51 m

So good luck on sig figsSo good luck on sig figs and let’s keep and let’s keep trying !!!trying !!!