Chapter 3 Math 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: five five 10 5 five four four
3-2 Significant Figures in Arithmetic Addition 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 & subtraction =? Multiplication & division Key number: the one with the least number of significant figures. (35.63 × × )/ × 100 % = % = ?
Multiplication and Division In multiplication and division
Significant Figures in Arithmetic Logarithms & antilog, see p64-65 [H + ]=2.0 pH=-log(2.0 ) = -( )=2.70 antilogarithm of 1.18 logarithm of 12.1 log 339 = … = antilog (-3.42) = = = 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 error always present & cannot be corrected an equal chance of being (+) or (-). from (a) people reading the scale (b) random electrical noise in an instrument. 3) Precision & Accuracy reproducibility confidence of nearness to the truth
Precision ? Accuracy ?
3.3 Types of Errors 4) Absolute & Relative uncertainty a) Absolute : the margin of uncertainty 0.02(the measured value - the true value) b)
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 Subtraction Addition and Subtraction
3.4 Propagation of uncertainty 1) Addition & Subtraction (ex) p.70
3.4 Propagation of uncertainty 2) Multiplication & Division use % relative uncertainties.
3.4 Propagation of uncertainty
Example ： Scientific Notation and Propagation of Uncertainty Express the absolute uncertainty in SOLUTION ： (a) The uncertainty in the denominator is 0.04/2.11 = %. The uncertainty in the answer is (b) P.71
3.4 Propagation of uncertainty 3) Mixed Operations Example ： Significant Figures in Laboratory Work at p.73
3.4 Propagation of uncertainty 4) The real rule for significant figures The 1 st uncertain figure of the answer is the last significant figure.
3.4 Propagation of uncertainty ......