How big?

Measurements and Significant Digits

How small?

How accurate?

Agenda

• MORE PRACTICE ON SIGNIFICANT DIGITS• HW: complete scientific notation, rounding,

sig. digits worksheets

Using Scientific Measurements

Precision and Accuracy1. Precision – the closeness of a set of measurements

of the same quantities made in the same way (how well repeated measurements of a value agree with one another).

2. Accuracy – is determined by the agreement between the measured quantity and the correct value.

Ex: Throwing Darts ACCURATE = CORRECT

PRECISE = CONSISTENT

Accuracy vs. Precision

Random errors: reduce precision

Good accuracyGood precision

Poor accuracyGood precision

Poor accuracyPoor precision

Systematic errors: reduce accuracy

(person)(instrument)

                             

                                                                                                                     

    

Precision Accuracy

reproducibility

check by repeating measurements

poor precision results from poor technique

correctness

check by using a different method

poor accuracy results from procedural or equipment flaws.

Percent Error

is calculated by subtracting the experimental value from the accepted value, then dividing the difference by the accepted value. Multiply this number by 100. Accuracy can be compared quantitatively with the accepted value using percent error.

Measurement

• Exact number - results from counting items that cannot be

subdivided - has an infinite number of significant digits.• Approximate number - results from measuring- does not express absolute accuracy- has a defined number of significant digits that

depends on the accuracy of the measuring device

What time is it?

• Someone might say “1:30” or “1:28” or “1:27:55”• Each is appropriate for a different situation• In science we describe a value as having a certain number

of “significant digits”• The # of significant digits in a value includes all digits that

are certain and one that is uncertain• “1:30” likely has 2, 1:28 has 3, 1:27:55 has 5

Reporting Measurements

• Using significant figures

• Report what is known with certainty

• Add ONE digit of uncertainty (estimation)

Counting Significant Figures

• When you report a measured value it is assumed that all the numbers are certain except for the last one, where there is an uncertainty of ±1.

• Example of nail: the nail is 6.36cm long. The 6.3 are certain values and the final 6 is uncertain! There are 3 significant figures in the value 6.36cm (2 certain and 1 uncertain). All measured values will have one (and one only) uncertain number (the last one) and all others will be certain. The reader can see that the 6.3 are certain values because they appear on the ruler, but the reader has to estimate the final 6.

Significant Figures

• Indicate precision of a measurement.

• Recording Significant Figures (SF)– Sig figs in a measurement include the known digits

plus a final estimated digit

2.35 cm

Practice Measuring

4.5 cm

4.54 cm

3.0 cmcm0 1 2 3 4 5

cm0 1 2 3 4 5

cm0 1 2 3 4 5

20

10

15 mL ?

15.0 mL?

1.50 x 101 mL

There are rules that dictate the number of significant digits in a value. 1. Read the handout up to A.2. Try A3. Bored? There are more:

a. 38.4703 mL b. 0.00052 g c. 0.05700 s d. 500 g

a. 6 b. 2 c. 4 d. 1

The rules for counting the number of significant figures in a value are:

1. All numbers other then zero will always be counted as significant figures.

2. Captive zeros always count. All zeros between two non-zero numbers are significant.

3. Leading zeros do not count. Zeros before a non-zero number after a decimal point are not significant.

4. Trailing zeros count only if there is a decimal. - All zeros after a non-zero number, after a decimal point

are significant.- Zeros after a non zero number with no decimal point

are not significant.

Answers to question A

1. 2.832. 36.773. 14.04. 0.00335. 0.026. 0.24107. 2.350 x 10 – 2

8. 1.000099. 310. 0.0056040

3 4 3 21446

infinite5

Rounding Rounding using the statistical approach: When a number ends in 5 and only 5 when you need to round: • If the preceding number is even –leave it, don’t round upEx. The number 21.45 rounded off to 3 significant figures becomes • If the preceding number is odd – round upEx. The number 21.350 rounded off to 3 significant figures becomes BUTIf any nonzero digits follow the 5, raise the preceding digit by 1.Ex. The number 21.4501 rounded off to 3 significant figures becomes

21.4

21.4

21.5

Scientific notation• All significant digits must be maintained• Only one number is written before the decimal point and

express the decimal points as a power of ten.

9.07 x 10 –

2 m0.0907m

5.06 x 10 –

4 cg0.000506cg

2.3 x 1012 m2 300 000 000 000m

1.27 x 102 g127g

Scientific notationDecimal notation

Scientific notation

• If your value is expressed in proper scientific notation, all of the figures in the pre-exponential value are significant, with the last digit being the least significant figure. “7.143 x 10-3 grams” contains 4 significant figures

• If that value is expressed as 0.007143, it still has 4 significant figures. Zeros, in this case, are placeholders. If you are ever in doubt about the number of significant figures in a value, write it in scientific notation.

Give the number of significant figures in the following values:

a. 6.19 x 101 years b. 7 400 000 yearsc. 3.80 x 10-19 J

• Helpful Hint :Convert to scientific notation if you are not certain as to the proper number of significant figures.

• When solving multiple step problems DO NOT ROUND OFF THE ANSWER UNTIL THE VERY END OF THE PROBLEM.

ANS: a. 3 b. 3 c. 3

Significant Digits• It is better to represent 100 as 1.00 x 102

• Alternatively you can underline the position of the last significant digit. E.g. 100.

• This is especially useful when doing a long calculation or for recording experimental results

• Don’t round your answer until the last step in a calculation.

• Note that a line overtop of a number indicates that it repeats indefinitely. E.g. 9.6 = 9.6666…

• Similarly, 6.54 = 6.545454…

Fill in the table Ordinary Notation ( g) Scientific Notation (g) # of Significant Figures

0.00120.00102 0.001201.20012.0012001200

1.2 x 10 -3 21.02 x 10 -3

1.20 x 10 -3

1.200 x 10 0

1.200 x 10 1

1.2 x 10 3

33442

31.20 x 10 3

Fill in the table considering the number of significant figures.

Previous Number (mL) Number ( mL) Following Number (mL) 120120.0120

110

119.9119

130 120.1

121

Significant Figures in Calculations 1. In addition and subtraction, your answer should have

the same number of decimal places as the measurement with the least number of decimal places.

Example: 12.734mL - 3.0mL = __________

Solution: 12.734mL has 3 figures past the decimal point. 3.0mL has only 1 figure past the decimal point. Therefore your final answer should be rounded off to one figure past the decimal point.

12.734mL- 3.0mL

9.734 -------- 9.7mL

Adding with Significant Digits

• How far is it from Warsaw to room C40? To B12?• Adding a value that is much smaller than the last sig.

digit of another value is irrelevant• E.g. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36

13.640.075

67.

80.71581

267.89.36

258.44 • Try question B on the handout

–++

B) Answers

83.14

i)83.25

0.1075–

4.02

4.020.001+

ii)

1.82

0.29831.52+

iii)

Multiplying with Significant Digits 2. In multiplication and division, your answer should have

the same number of significant figures as the least precise measurement (or the measurement with the fewest number of SF).

Examples: a. 61cm x 0.00745cm = 0.45445 = =

2SF 3SF 2SF

b. 608.3m x 3.45m = 2098.635 = 4SF 3SF 3SF

• 4.8 g 392g = 0.012245 = 2SF 3SF 2SF

• Try question C and D on the handout (recall: for long questions, don’t round until the end)

0.45cm2

2.10 x 103 m2

0.012 or 1.2 x 10 – 2

4.5 x 10-1 cm2

C), D) Answersi) 7.255 81.334 = 0.08920ii) 1.142 x 0.002 = 0.002iii) 31.22 x 9.8 = 3.1 x 102 (or 310 or 305.956)

i) 6.12 x 3.734 + 16.1 2.3 22.85208 + 7.0 = 29.9

ii) 0.0030 + 0.02 = 0.02

135700 =1.36 x105

1700134000+

iii)

iv) 33.4112.7+

0.032+146.132 6.487 = 22.5268

= 22.53

Note: 146.1 6.487 = 22.522 = 22.52

Calculations & Significant Digits In multiple step problems if addition or subtraction AND multiplication or division is used the rules for rounding are based off of multiplication and division (it “trumps” the addition and subtraction rules).There is no uncertainty in a conversion factor; therefore they do not affect the degree of certainty of your answer. The answer should have the same number of SF as the initial value.

a. Convert 25 meters to millimeters.

b. Convert 0.12L to mL.

?mm→ 25 m X 1000 mm = 25 000 mm 1 1 m 2SF

?mL→ 0.12L X 1000 m = 120 mL 1 1 L 2SF

Unit conversions & Significant Digits• Sometimes it is more convenient to express a value in

different units.• When units change, basically the number of significant

digits does not.E.g. 1.23 m = 123 cm = 1230 mm = 0.00123 km• Notice that these all have 3 significant digits• This should make sense mathematically since you are

multiplying or dividing by a term that has an infinite number of significant digits.

conversion factors= infinite # of sig. digits E.g. 123 cm x 10 mm / cm = 1230 mm• Try question E on the handout

E) Answers

• A shocking number of patients die every year in United States hospitals as the result of medication errors, and many more are harmed. One widely cited estimate (Institute of Medicine, 2000) places the toll at 44,000 to 98,000 deaths, making death by medication "misadventure" greater than all highway accidents, breast cancer, or AIDS. If this estimate is in the ballpark, then nurses (and patients) beware: Medication errors are the forth to sixth leading cause of death in America.

i) 1.0 cm = 0.010 m

ii) 0.0390 kg = 39.0 g

iii) 1.7 m = 1700 mm or 1.7 x 103 mm

Real World Connections :

• Information from the website “Medication Math for the Nursing Student” at http://www.alysion.org/dimensional/analysis.htm#problems