BASIC RULES FOR COUNTING SIGNIFICANT FIGURES NONZERO …

BASIC RULES FOR COUNTING SIGNIFICANT FIGURES

NONZERO INTEGERS

ZEROS

Nonzero integers ALWAYS count as significant figures.

A. Leading Zeros

Leading zeros NEVER count as significant figures.

B. Captive Zeros Captive zeros ALWAYS count as significant figures.

C. Trailing Zeros Trailing zeros are significant ONLY if the number contains a decimal point.

EXACT NUMBERS Numbers that do not come from measurement, but from definition.

Exact numbers NEVER limit the number of significant figures in a calculation.

BASIC USE OF SIGNIFICANT FIGURES IN MATHEMATICAL OPERATIONS For Addition or Subtraction

When adding or subtracting measurements, round off (the result) to the decimal place of the least precise measurement. This is the measurement having the fewest decimal places, not the one with the fewest sig figs.

55.555 + 44.44 + 33.3 = 133.295

133.3 1 deci. place 4 sig figs

1011164.3 + 21.604852 + 7610.2964 = 1018796.201

1018796.2

1 deci. place

8 sig figs

For Multiplication or Division

When multiplying or dividing measurements, report the result of the

calculation to the same number of sig figs as the measurement having the

fewest sig figs.

(2.00)(3.026) = 1.239909855

4.881

1.24 3 sig figs

Mixed Operations

(5.0831 + 0.03) = 2.11285124

2.42

03.4 + 10.2) = 9.991532599

2.362

2.11 3 sig figs 9.99 3 sigfigs

RULES FOR ROUNDING In a series of calculations. carry the extra digits through to the final result. then round.

Look at all digits beyond the last place desired a) if more than "halfway" to the next digit, the last place is increased by one (round up) b) if less than "halfway" to the next digit, the last place stays the same c) if exactly halfway, then round to the nearest even digit

ABSOLUTEANDRELATIVE UNCERTAINTY(ERROR) The resultsof an analysisare 36.97g, comparedwiththe acceptedvalueof 37.06 g. What is the relativeerrorin partsper thousand?

PROPAGATION OF ERROR From the knowledge of the uncertainties in each number, it is possible to estimate the actual uncertainty in the answer. The errors in the individual numbers will propagate throughout a series of calculations, in either a relative or absolute fashion.

= For Addition and Subtraction e4 vier + e~ + e~ (use absolute uncertainties)

e4 = uncertainty in final answer

e., ez, e3 = uncertainty in individual tetms

= (65.06 :t 0.07) + (16.13 :t 0.01) - (22.68 :t 0.02) 58.51 (:I: ?)

For Multiplication and Division

%e4 = V %ei+ %e~+ %~

= (13.67:t 0.02)(120.4 :t 0.2)

356.0 (:I: ?)

4.623 :t 0.006

(use percentrelative uncertainties)

SIGNIFICANT FIGURES AND PROPAGATION OF ERROR The Real Rule: Thefirst uncertain figure of the answer is the last significantfigure.

= (73.1 :t 0.2)(2.245 :t 0.008) 164.1 :I: 0.7

= = (73.1 :t 0.9)(2.245 :t 0.008) 164.1 :t 2.1 164 ::t: 2

Provide the answers to the following calculations to the proper number of significant figures:

(38.68 :t 0.07) - (6.16 :t 0.09) =

= (12.18 :t 0.08) + (23.04 :t 0.07)

3.247 :t 0.006

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