Chapter 1R: Review
How do you quantify time? Depends on the application.
How do you quantify a length?
Imperial System of Measurement
Metric System
Some countries still use the imperial system. Science has adopted the metric system, a system of measurement that uses units with ratios that are multiples of 10.
Source: Wikimedia
Science uses SI Units (International System of Units) to describe quantities. The base SI units are as follows:
Property | Unit | Abbreviation |
---|---|---|
Mass |
kilogram |
kg |
Length |
meter |
m |
Time |
second |
s |
Temperature |
kelvin |
K |
Amount of substance |
mole |
mol |
Electric current |
ampere |
A |
Luminous intensity |
candela |
cd |
Used to modify a basic unit
Prefix | Abbeviation | Meaning | Example |
---|---|---|---|
Tera- |
T |
1012 (trillion) |
1 terameter (Gm) = 1×1012 m |
Giga- |
G |
109 (billion) |
1 gigameter (Gm) = 1×109 m |
Mega- |
M |
106 (million) |
1 megameter (Mm) = 1×106 m |
Kilo- |
k |
103 (thousand) |
1 kilometer (km) = 1×103m |
Deci- |
d |
10–1 (one tenth) |
1 decimeter (dm) = 1×10–1 m |
Centi- |
c |
10–2 (one hundredth) |
1 centimeter (cm) = 1×10–2 m |
Milli- |
m |
10–3 (one thousandth) |
1 millimeter (mm) = 1×10–3 m |
Micro- |
μ |
10–6 (one millionth) |
1 micrometer (μm) = 1×10–6 m |
Nano- |
n |
10–9 (one billionth) |
1 nanometer (nm) = 1×10–9 m |
Pico- |
p |
10–12 (one trillionth) |
1 picometer (pm) = 1×10–12 m |
Femto- |
f |
10–15 |
1 femtometer (fm) = 1×10–16 m |
Which quantity would be most appropriate for describing the distance between Athens, GA and Akron, OH (ca. 665 mi)?
Source: Openstax
°F → °C
\[t/^{\circ}\mathrm{C} = \dfrac{\left ( t/^{\circ}\mathrm{F} - 32 \right )}{1.8}\]
°C → K
\[t/\mathrm{K} = t/^{\circ}\mathrm{C} + 273.15\]
°C → °F
\[t/^{\circ}\mathrm{F} = \left ( t/^{\circ}\mathrm{C} \times 1.8 \right ) + 32\]
K → °C
\[t/^{\circ}\mathrm{C} = t/\mathrm{K} - 273.15\]
What is 93.0 °F in °C and K?
33.9 °C
What is absolute zero (0 K) in °C and °F?
–273.15 °C
\[b^n = \underbrace{b \times b \times \cdots \times b}_{n~\mathrm{times}}\]
\[10^1 = 10 \] \[10^2 = 10 \times 10 = 100\] \[10^3 = 10 \times 10 \times 10 = 1000\]
\[b^{-n} = \underbrace{\dfrac{1}{b} \times \dfrac{1}{b} \times \cdots \times \dfrac{1}{b}}_{n~\mathrm{times}}\]
\[10^{-1} = \dfrac{1}{10} = 0.1\] \[10^{-2} = \dfrac{1}{10} \times \dfrac{1}{10} = 0.01\]
\[10^{-3} = \dfrac{1}{10} \times \dfrac{1}{10} \times \dfrac{1}{10} = 0.001\]
Used for small and big numbers
\[ 1,234 = 1.234\times 10^1 \times 10^1 \times 10^1 = 1.234\times 10^3 \]
\[ 0.01234 = 1.234 \times 10^{-1} \times 10^{-1} = 1.234\times 10^{-2} \]
\[ 0.1234 = \dfrac{1.234}{10^1\times 10^1} = 1.234\times 10^{-2} \]
Write the following numbers into scientific notation.
\[502\]
\[1,535,234\]
\[0.00008753\]
\[5.02\times 10^2\]
\[1.535234 \times 10^{6}\]
\[8.753\times 10^{-5}\]
\[3.4\times 10^{4} + 9.7\times 10^{5}\]
Transform each number into the same power of 10.
\[0.34\times 10^{5} + 9.7\times 10^{5}\]
Add or subtract the numbers.
\[10.04\]
Rewrite in scientific notation.
\[10.04 \times 10^{5} = 1.004 \times 10^6\]
Powers of 10 get added.
\[200 \times 100 = 20,000 = 2.00\times 10^{4}\]
\[\begin{align*} \left (2.00\times 10^{2} \right) \times \left (1.00 \times 10^{2}\right) &= \\[2ex] \left (2.00\times 1.00 \times 10^{2} \times 10^{2} \right) &= 2.00\times 10^4 \end{align*}\]
Powers of 10 get subtracted.
\[\begin{align*} \dfrac{200}{100} = 2 &= 2\times 10^{0} \\[4ex] \dfrac{2.00\times 10^{2}}{1.00 \times 10^{2}} &= 2\times 10^0 = 2 \end{align*}\]
Not all digits are significant. Here, the graduated cylinder is a “measuring stick” for volume. How accurate are we able to make a reading?
We know by inspection that the water is clearly greater than 21 mL but less than 22 mL. We estimate that the water is perhaps 21.6 mL (at the meniscus).
The “2” and the “1” are certain but the 0.6 is uncertain. All certain/exact digits as well as the first uncertain/inexact digit are considered significant. Our measurement has three significant figures.
Source: Openstax
The numbers you use in this class are treated as exact (infinite sig. figs.) or inexact numbers.
Exact
Obtained through counting
Inexact
Obtained through measurement
Imperial-to-metric conversions
See table for some official SI Units and Conversions
For this class, treat all imperial-to-metric conversions as being exact.
See Conversions notes.
Precision - indicates how well several measurements agree
Accuracy - the agreement of a measurement with the accepted value
Source: Openstax
Measurements have error.
Experimental Error
“Error in measurement” can be an average value of n measurements.
\[\mathrm{error~in~measurement} = \mathrm{measured~value} - \mathrm{accepted~value}\]
Percent Error
\[ \%~\mathrm{error} = \dfrac{\mathrm{error~in~measurement}}{\mathrm{accepted~value}}\times 100\% \]
A coin has a diameter of 28.054 mm. Two people make a series of measurements (reported in mm). What is the average diameter and percent error obtained in each case? Which person was more accurate on average?
Person 1 | Person 2 |
---|---|
28.246 |
27.9 |
28.244 |
28.0 |
28.246 |
27.8 |
28.248 |
28.1 |
Person 1
Average: 28.246 mm
\[\begin{align*} \mathrm{\%~error} &= \dfrac{28.246~\mathrm{mm} - \mathrm{28.054~\mathrm{mm}}}{28.054~\mathrm{mm}}\\[1.5ex] &~~~~\times 100\% \\[1.5ex] &= 0.684\% \end{align*}\]
Person 2
Average: 27.95 mm
\[\begin{align*} \mathrm{\%~error} &= \dfrac{27.\bar{9}5~\mathrm{mm} - \mathrm{28.054~\mathrm{mm}}}{28.054~\mathrm{mm}}\\[1.5ex] &~~~~\times 100\% \\[1.5ex] &= -0.4\% \end{align*}\]
Press q
for answer.
Determinate
identifiable; can be avoided
Indeterminate
random; arises from uncertainties in measurements
The std. dev. of a series of measurements is equal to the
\[\mathrm{std.~dev.} = \sqrt{\dfrac{\sum{\left(x_i - \bar{x}\right)^2}}{N-1}}\]