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Significant Digits

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Measurement needs to be precise and accurate Precision:

How closely multiple measurements of the same quantity come to each other.

This will depend on the measuring device. For example, a thermometer that shows degrees in tenths is more precise than one that only shows single degrees.

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Accuracy Refers to how close a measurement

comes to the true or accepted value. This depends on both the measuring

device and the skill of the person using the measuring device.

This can be determined by comparing the measured value to the known or accepted value.

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Precision and Accuracy

Good precision

Good Accuracy

Good Precision

Poor Accuracy

Poor Precision

Poor Accuracy

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Measurement and numbers There are two kinds of numbers

Counted or definedCounted or defined - exact, not subject to estimate. Ex: number of eggs in a carton.

MeasuredMeasured - always carries some amount of uncertainty because measurement involves estimation. The size of uncertainty depends on the precision of the measuring device AND the skill of the person using the device

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Measurement and numbers

Measurements consist of two partsThe number itself (the quantity)The number itself (the quantity)The units (the nature of the The units (the nature of the

quantity measured)quantity measured)

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Estimation in measurement

When we measure, the quantity rarely falls exactly on the calibration marks of the scale we are using.

Because of this we are estimating the last digit. For instance, we could measure “A” above as about

2.3 cm. We are certain of the digit “2”, but the “.3” part is a guess - an estimate.

What is your estimate for B and C?

1 cm 2 cm 3 cm 4 cm 5 cm

A B C

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Higher Precision

A measuring device with more marks on the scale is more precise. I.e., we are estimating less, and get a more accurate reading. Here we are estimating the hundredths place instead of the tenths.

Here, we can measure A as 1.25 cm. Only the last digit is uncertain.

Usually we assume the last digit is accurate ± 1. How would you read B, C, and D?

1 cm 2 cm 3 cm

A B C D

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The ProblemArea of a rectangle = length x width

We measure:Length = 14.26 cm Width = 11.70 cm

Punch this into a calculator and we find the area as:14.26 cm x 11.70 cm = 166.842 cm2

But there is a problem here! This answer makes it seem like our measurements were more

accurate than they really were. By expressing the answer this way we imply that we estimated the thousandths position, when in fact we were less accurate than that!

A much better answer would be that the area is 166.84 cm2 because that keeps the same accuracy as our original measurements.

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Significant Figures (aka Significant Digits)

To help keep track of (and communicate to others) the precision and accuracy of our measurements, we use Significant Significant FiguresFigures

These are the digits in any measurement that are known with certainty plus one known with certainty plus one digit that is uncertaindigit that is uncertain (but usually assumed to be accurate ± 1)

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Rules for Significant Figures

1. Digits from 1-9 are always significant. 2. Zeros between two other significant

digits are always significant 3. One or more additional zeros to the right

of both the decimal place and another significant digit are significant.

4. Zeros used solely for spacing the decimal point (placeholders) are not significant.

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Examples

EXAMPLES # OF SIG. DIG.

COMMENT

453 3All non-zero digits are always significant.

50574.06

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Zeros between two significant digits are significant.

5.00106.00114.050

356

Additional zeros to the right of decimal and a significant digit are significant.

0.007 1 Placeholders are not significant

12000 2Trailing zeros in numbers with no decimal point are not significant (= placeholder)

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PracticeHow many significant digits in the following?

Number # Significant Digits

1.4682

110256.002

0.000000003

114.00000006

110

120600

59

1

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24

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Multiplication and Division with Significant DigitsThe rule for multiplying or dividing significant digits is that the answer must have only as many significant digits as the original measurement with the least number of significant digits.

Our measurements, 14.26 and 11.70 each have four significant digits. Our calculator told us the answer was 166.842, but we need to round it off. Do we round up or do we round it down?

166.842

If our original measurements had been 14.26 and 11.7, what happens?

166.842

How many significant digits would the answer to each of these have?

Problem #Sig. Digits in Result?

114.6 x 2.0004

0.0006 x 14.63

12.901 / 6.23

166.84

167

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Addition and Subtraction with Significant DigitsThe rule for adding or subtracting with significant digits is that the answer must have only as many digits past the decimal point as the measurement with the least number of digits past the decimal.

How many significant digits would the answer to each of these have?

Problem #Digits Past the Decimal?

114.6 + 2.0004

0.0006 + 14.63

12.901 - 6.23

One complication: doing math on numbers without decimal points does not follow the above rules, it just works like it normally does.

So: 1200 x 56 = 67200 not 67!

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