Significant Figures…

Bluefield High School

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What is a significant digit?Significant digits is a set of internationally

accepted rules for measurement. The rule of thumb is to record all digits that

are certain, plus one that is estimated.The rules of significant digits apply to all

reporting and calculating unless you are dealing with defined values (those by definition. For example …60 minutes = 1 hour).

1. Every nonzero digit is significant. 24.7 has three significant digits

2. Zeroes between nonzero digits are significant. 2.07m and 109m each have three significant digits.

3. Zeroes in front of all nonzero digits are not significant. They are merely place holders. The measurements of 0.00037m and 0.46m each have two significant digits.

4. Zeroes at the end of the number and to the right of the decimal point are significant. The measurement 43.00m, 1.010m and 9.500m all have four significant digits.

5. Zeroes at the end of a measurement and to the left of the decimal point are unclear. You may consider all such zeroes to be not significant. To avoid confusion, express the number in scientific notation.

The Rules of Significant Digits…

When to use Significant figuresWhen a measurement is recorded only those digits that are dependable are written down.

When to use Significant digits…If you measured the width of a paper with your ruler you might record 21.7cm.

To a mathematician 21.70, or 21.700 is the same.

But, to a scientist 21.7cm and 21.70cm is NOT the same

21.700 cm to a scientist means the measurement is accurate to within one thousandth of a cm.

significant figures video

How many sig digits?7400.50.000037 x 105

7,000,000

111111

How many sig digits here?1.2210056.764.000.07927,083,000,000

224334

How many sig digits here?340121002100.05.000.004128,000,050,000

425336

Counting Significant Figures

Number of Significant Figures

38.15 cm 45.6 ft 265.6 lb ___122.55 m ___

All non-zero digits in a measured number are (significant or not significant).

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Leading ZerosNumber of Significant Figures

0.008 mm 1

0.0156 oz 3

0.0042 lb ____

0.000262 mL ____

Leading zeros in decimal numbers are (significant or not significant).

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Sandwiched ZerosNumber of Significant

Figures

50.8 mm 3

2001 min 4

0.702 lb ____

0.00405 m ____

Zeros between nonzero numbers are (significant or not significant).

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Trailing Zeros Number of Significant

Figures 25,000 in. 2

200 yr 1

48,600 gal 3

25,005,000 g ____Are trailing zeros, serving as place holders in numbers without decimals, significant or not significant?

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Learning Check A. Which answer contains 3 significant

figures?1) 0.4760 2) 0.00476 3) 4760

B. All the zeros are significant in

1) 0.00307 2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

1) 535 2) 535,000 3) 5.35 x 105

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Solution

A. Which answers contain 3 significant figures?

2) 0.00476 3) 4760

B. All the zeros are significant in

2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

2) 535,000 3) 5.35 x 105

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Learning CheckIn which set(s) do both numbers contain the same number of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000015 and 150,000

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Solution …

3) 0.000015 and 150,000

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Learning CheckState the number of significant figures in each of the following:A. 0.030 m 1 2 3

B. 4.050 L 2 3 4

C. 0.0008 g 1 2 4

D. 3.00 m 1 2 3

E. 2,080,000 bees 3 5 7

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Solution

A. 0.030 m 2

B. 4.050 L 4

C. 0.0008 g 1

D. 3.00 m 3

E. 2,080,000 bees 3

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Significant Numbers in CalculationsA calculated answer cannot be more

precise than the measuring tool. A calculated answer must match the

least precise measurement.Significant figures are needed for

final answers from 1) adding or subtracting

2) multiplying or dividing

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Adding and SubtractingThe answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2 one decimal place

+ 1.34 two decimal places

26.54answer 26.5 one decimal place

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Learning Check In each calculation, round the answer to the correct number of significant figures.A. 235.05 + 19.6 + 2.1 =

1) 256.75 2) 256.8 3) 257

B. 58.925 - 18.2 =1) 40.725 2) 40.73 3) 40.7

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Solution A. 235.05 + 19.6 + 2.1 =

2) 256.8

B. 58.925 - 18.2 =3) 40.7

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Multiplying and Dividing

Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

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Learning Check A. 2.19 X 4.2 =

1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 = 1) 61.58 2) 62 3) 60

C. 2.54 X 0.0028 = 0.0105 X 0.060 1) 11.3 2) 11 3) 0.041

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Solution A. 2.19 x 4.2 = 2) 9.2

B. 4.311 ÷ 0.07 = 3) 60

C. 2.54 x 0.0028 = 2) 11 0.0105 x 0.060

Continuous calculator operation = 2.54 x 0.0028 0.0105

0.060

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Significant Figures

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Multiplication or Division

The number of significant figures in the result is set by the original number that has the smallest number of significant figures

4.51 x 3.6666 = 16.536366= 16.5

3 sig figs round to3 sig figs

6.8 ÷ 112.04 = 0.0606926

2 sig figs round to2 sig figs

= 0.061

Accuracy – how close a measurement is to the true value

Precision – how close a set of measurements are to each other

accurate&precise

precisebutnot accurate

not accurate&not precise

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A short-hand way of writinglarge numbers without writing all of the zeros.

The Distance From the Sun to the Earth

93,000,000

Step 1Move decimal leftLeave only one number in front of decimal

Step 2Write number without zeros

Step 3Count how many places you moved

decimalMake that your power of ten

The power often is 7 becausethe decimalmoved 7 places.

93,000,000 --- Standard Form

9.3 x 107 --- Scientific Notation

Practice Problem

1) 98,500,000 = 9.85 x 10?

2) 64,100,000,000 = 6.41 x 10?

3) 279,000,000 = 2.79 x 10?

4) 4,200,000 = 4.2 x 10?

Write in scientific notation. Decide the power of ten.

9.85 x 107

6.41 x 1010

2.79 x 108

4.2 x 106

More Practice Problems

1) 734,000,000 = ______ x 108

2) 870,000,000,000 = ______x 1011

3) 90,000,000,000 = _____ x 1010

On these, decide where the decimal will be moved.

1) 7.34 x 108 2) 8.7 x 1011 3) 9 x 1010

Complete Practice Problems

1) 50,0002) 7,200,0003) 802,000,000,000

Write in scientific notation.

1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011

Move the decimal to the right

3.4 x 105 in scientific notation

340,000 in standard form

3.40000 --- move the decimal

Write in Standard Form

6.27 x 106

9.01 x 104

tutorial

6,270,00090,100

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Dimensional Analysis Method of Solving Problems

1. Determine which unit conversion factor(s) are needed

2. Carry units through calculation

3. If all units cancel except for the desired unit(s), then the problem was solved correctly.

1 L = 1000 mL

How many mL are in 1.63 L?

1L1000 mL

1.63 L x = 1630 mL

1L1000 mL

1.63 L x = 0.001630L2

mL

The speed of sound in air is about 343 m/s. What is this speed in miles per hour?

1 mi = 1609 m 1 min = 60 s 1 hour = 60 min

343ms x

1 mi1609 m

60 s 1 minx

60 min 1 hourx = 767

mihour

meters to miles

seconds to hours

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Please do the following questions…Page 349 in your text…

#’s 2,3,4,5,6,8 and 9

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