The document discusses significant figures and types of errors in measurement, explaining how to determine the number of significant figures in calculations using addition, subtraction, multiplication, division, and logarithms. It also covers absolute and relative uncertainty, propagation of uncertainty through calculations, and the rule for determining the number of significant figures in the answer of a calculation involving measurements with uncertainty.

1. Chapter 3
Math Toolkit

2. 3.1~3.2 Significant Figures
and in Arithmetic

3. Significant Figures
• Measurement: number + unit
• Uncertainty
• Ex:
0.92067  five
0.092067  five
9.3660105  five
936600  four
7.270  four

4. Significant Figures
and in Arithmetic
Addition & subtraction
3.123 + 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 % = ?

5. Significant Figures
and in Arithmetic
Logarithms & antilog, see p54-55
[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

6. 3.3 Types of Errors
Every measurement has some
uncertainty ⇒ experimental error.
Maximum error v.s. time required

7. 3.3 Types of Errors
• 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

8. 3.3 Types of Errors
• Random error = Indeterminate error
Is always present & cannot be corrected
Has an equal chance of being (+) or (-).
From (a) people reading the scale
(b) random electrical noise in an
instrument.
5) Precision & Accuracy
reproducibility
confidence of nearness to the truth

9. Precision ? Accuracy ?

10. 3.3 Types of Errors
1) Absolute & Relative uncertainty
a) Absolute : the margin of uncertainty
± 0.02(the measured value - the true value)
b). Relative = absolute uncertainty
magnitude of measurement
(ex) 12.35 ± 0.02 mL
0.02
⇒ = 0.002 = 0.2%
12.35

11. 3.4 Propagation of uncertainty
Uncertainty (random error) expressed
standard deviation
confidence interval

12. 3.4 Propagation of uncertainty
• Addition & Subtraction
1.76 ( ±0.03) ← e1 e 4 = e1 + e 2 + e 3
2 2
2
+ 1.89 ( ±0.02) ← e2
= 0.04 1
− 0.59 ( ±0.02) ← e3
⇒ 1.3 %
3.06 ( ±e 4 )
3.06 ( ±0.04)
⇒
(ex) p.58 3.06 ( ±1%)

13. 3.4 Propagation of uncertainty
• Multiplication & Division
use % relative uncertainties.
e4 = ( %e1 ) 2
+ ( %e 2 ) + ( %e 3 )
2 2

14. 3.4 Propagation of uncertainty
1.76 ( ±0.03) × 1.89 ( ±0.02)
(ex) = 5.64 ± e 4
0.59 ( ±0.02)
1.76 ( ±1.7 %) × 1.89 ( ±1.1%)
⇒
0.59 ( ±3. 4 %)
e4 = (1.7%) 2
+ (1.1%) + ( 3. 4 %) = 4.0 %
2 2
4.0 % × 5.6 4 = 0.2 3
5.6 ( ±0.2)
⇒ 
5.6 ( ±4%)

15. 3.4 Propagation of uncertainty
1) Mixed Operations
1.76 ( ±0.03) − 0.59 ( ±0.02)
= 0.619 0 ± ?
1.89 ( ±0.02)
0.619 ( ±0.02 0 ) 0.62 ( ±0.02)
 ⇒
0.619 ( ±3.3 %) 0.62 ( ±3%)

16. 3.4 Propagation of uncertainty
1) The real rule for significant figures
The 1st uncertain figure of the answer is
the last significant figure.

17. 3.4 Propagation of uncertainty
a. . 0.002364 (±0.000003)
= 0.0946 (±0.0002)
0.02500 (±0.00005)
0.002664 (±0.000003)
d. . = 0.1066 (±0.0002)
0.02500 (±0.00005)
0.821 (±0.002)
g. . = 1.022 (±0.004)
0.803 (±0.002)