1. Chapter 3 Math Toolkit

2. 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. P.64

3. Figure 3-1 Figure 3-1 Scale of a Bausch and Lomb Spectronic 20 spectrophotometer. P.62

4. Significant Figures Measurement: number + unit Uncertainty Ex: 0.92067  five 0.092067  five 9.3660  10 5  five 936600  four 7.270  four

5. 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

6. The number of significant figures in the answer may exceed or be less than that in the original data. P.62

7. Multiplication and Division In multiplication and division P.63

8. Example ： Significant Figures in Molecular Mass Find the molecular mass of C 14 H 10 with the correct number of significant digits. SOLUTION: 14×12.010 7=168.149 8 ←6 significant figures because 12.010 7 has 6 digits 10×1.007 94=10.079 4←6 significant figures because 1.007 94 has 6 digits 178.229 2 P.64

9. Logarithms and Antilogarithms The base 10 logarithm of n is the number a, whose value is such that n=10 a : The number n is said to the antilogarithm of a. P.64

10. A logarithm is composed of a characteristic and a mantissa. The number 339 can be written 3.39×10 2. The number of digits in the mantissa of log 339 should equal the number of significant figures in 339. P.64

11. In converting a logarithm to its antilogarithm, the number of significant figures in the antilogarithm should equal the number of digits in the mantissa. P.65

12. Significant Figures and in Arithmetic Logarithms & antilog, see p64-65 [H + ]=2.0  10 -3 pH=-log(2.0  10 -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

13. 3.3 Types of Errors Every measurement has some uncertainty  experimental error. Experimental error is classified as either systematic or random. Maximum error v.s. time required

14. 3.3 Types of Errors 1)Systematic error = Determinate error = consistent error -Errors arise: instrument, method, & person -Can be discovered & corrected -Is from fixed cause, & is either high (+) or low (-) every time. -Ways to detect systematic error: examples (a) pH meter (b) buret

15. 3.3 Types of Errors 2)Random error = Indeterminate error Is always present & cannot be corrected Has an equal chance of being (+) or (-). (a) people reading the scale (b) random electrical noise in an instrument. (c) pH of blood (actual variation: time, or part) 3)Precision & Accuracy reproducibility confidence of nearness to the truth

16. Precision ? Accuracy ? P.69

17. 3.3 Types of Errors 4)Absolute & Relative uncertainty a) Absolute : the margin of uncertainty  0.02(the measured value - the true value) b).

18. 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 P.69

19. 3.4 Propagation of uncertainty 1)Addition & Subtraction (ex) p.69

20. 3.4 Propagation of uncertainty 2)Multiplication & Division use % relative uncertainties.

21. 3.4 Propagation of uncertainty P.71

22. Example ： Scientific Notation and Propagation of Uncertainty Express the absolute uncertainty in SOLUTION ： (a) The uncertainty in the denominator is 0.04/2.11 = 1. 896 %. The uncertainty in the answer is (b) P.71

23. 3.4 Propagation of uncertainty 3)Mixed Operations Example ： Significant Figures in Laboratory Work at p.73

24. 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.

25. 3.4 Propagation of uncertainty .‚.ƒ..‚.ƒ. P.72