chapter 3 math toolkit. 3.1~3.2 significant figures & in arithmetic

Chapter 3Math Toolkit

3.1~3.2 Significant Figures & in Arithmetic

3-1 Significant Figures

• The number of significant figures is the minimum number of digits needed to write a given value in scientific notation without loss of accuracy.

Significant Figures

• Measurement: number + unit• Uncertainty • Ex:

0.92067 five0.092067 five9.3660105 five936600 four7.270 four

3-2 Significant Figures in Arithmetic

Addition and SubtractionAddition and Subtraction• If the numbers to be added or subtracted

have equal numbers of digits, the answer is given to the same decimal place.

P.62

• The number of significant figures in the answer may exceed or be less than that in the original data.

Significant Figures in Arithmetic

Addition & subtraction3.123 + 254.6 =?

Multiplication & division• Key number: the one with the least number of

significant figures.

(35.63 × 0.5481 × 0.05300)/1.1689 × 100 % 

= 88.54705783 % = ?

Multiplication and DivisionMultiplication and Division• In multiplication and division

Significant Figures in Arithmetic

Logarithms & antilog, see p64-65[H+]=2.010-3

pH=-log(2.010-3) = -(-3+0.30)=2.70

antilogarithm of 0.072 1.18

logarithm of 12.1 1.083

log 339 = 2.5301997… = 2.530

antilog (-3.42) = 10-3.42 = 0.0003802

= 3.8x10-4

3.3 Types of Errors

Every measurement has some uncertainty experimental error.

Maximum error v.s. time required

3.3 Types of Errors

1) Systematic error

= Determinate error = consistent error- Errors arise: instrument, method, & person

- Can be discovered & corrected

- From fixed cause, & either high (+) or low (-) every time.

- Ways to detect systematic error:examples (a) pH meter (b) buret at p. 65

One way to correct for an error of this type is by constructing an experimental calibration

Figure 3-2 Calibration curve for a 50-mL buret.

3.3 Types of Errors

2) Random error = Indeterminate erroralways present & cannot be correctedan equal chance of being (+) or (-).from (a) people reading the scale

(b) random electrical noise in an

instrument.

3) Precision & Accuracyreproducibility confidence of nearness to the truth

Precision ? Accuracy ?

3.3 Types of Errors

4) Absolute & Relative uncertaintya) Absolute : the margin of uncertainty

0.02(the measured value - the true value)

b)

0.2%0.00212.35

0.02

mL 0.0212.35 (ex)

tmeasuremen of magnitude

yuncertaint absoluteRelative

3-4 Propagation of Uncertainty• The uncertainty might be based on how well we

can read an instrument or on experience with a particular method. If possible, uncertainty is expressed as the standard deviation or as a confidence interval.

Addition and SubtractionAddition and Subtraction

3.4 Propagation of uncertainty

1) Addition & Subtraction

(ex) p.70

)e( 3.06

e 0.02)( 0.59

e 0.02)( 1.89

e 0.03)( 1.76

4

3

2

1

1%)( 3.06

0.04)( 3.06

%1.

0.04

eeee

3

1

23

22

214

3.4 Propagation of uncertainty

23

22

214 %e%e%ee

2) Multiplication & Division

use % relative uncertainties.

3.4 Propagation of uncertainty

4%)( 5.6

0.2)( 5.6

0.25.6%4.

%4.%3.%1.%1.e

%)3.( 0.59

%)1.( 1.89%)1.( 1.76

e5.640.02)( 0.59

0.02)( 1.890.03)( 1.76 (ex)

340

02

42

12

74

4

17

4

Example Example ：： Scientific Notation and Propagation of UncertaintyScientific Notation and Propagation of Uncertainty

Express the absolute uncertainty in

SOLUTION SOLUTION ：： (a) The uncertainty in the denominator is 0.04/2.11 = 1.896%. The

uncertainty in the answer is (b)

P.71

3.4 Propagation of uncertainty

3) Mixed Operations

3%)( 0.62

0.02)( 0.62

%)3.( 0.619

)0.02( 0.619

?0.6190.02)( 1.89

0.02)( 0.590.03)( 1.76

3

0

0

ExampleExample ：： Significant Figures in Laboratory Work at Significant Figures in Laboratory Work at p.73p.73

3.4 Propagation of uncertainty

4) The real rule for significant figures

The 1st uncertain figure of the answer is the last significant figure.

3.4 Propagation of uncertainty

① .

② .

③ . 0.004)`( 1.0220.002)( 0.803

0.002)( 0.821

0.0002)( 0.10660.00005)( 0.02500

0.000003)( 0.002664

0.0002)( 0.09460.00005)( 0.02500

0.000003)( 0.002364