Making comparisons

Decision theory

If the time between the lottery and the show is very large or very small, then you don't need to think a lot to decide whether you can make it or not.

A very rough estimate of "how long it takes to get to Chicago" is enough.

Only if the time gets *close* to your rough estimate do you actually need to start thinking (and calculating) in detail...

Unit conversions

When making comparisons we are often interested in questions like is one thing smaller or larger than another thing, and don't need to know exactly how much bigger/smaller.

Still, we have to use the same units to make comparisons!

1 inch = 2.54 cm
100 cm = 1 meter
1 m = 39 inch ~ 36 inches = 1 yard = 3 ft
2.2 lb = 1 kg
1 mile = 1.6 km
1000 m = 1 kilometer

renewable energy


How are we doing compared to other countries in solar energy production?

References

Exponents / Powers of 10

$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000$

$10^{-3} = \frac{1}{10 \times 10 \times 10} = 0.001$

Think of the 'power of ten' as the number of times you move the decimal point from its initial position in $1.0$ to the right or left.

$10^{-3}?$: Take 3 steps towards smaller numbers... $1.0\to 0.1 \to 0.01 \to 0.001=10^{-3}.$

$10,000?$ How many times do you have to move the decimal point to reach 1.0?
$10,000\to 1,000\to 100 \to 10 \to 1.0$: 4 steps so 10,000$=10^4$.

So, move the decimal point in 1.0 zero steps? That means... $10^0 = 1.0$!

Multiplying powers of ten

$10^ 3 \times 10^1 = 10^{3+1} = 10^4$

This is the same problem as... $1000 \times 10 = 10000$

$10^3 \times 10 ^{-1} = 10^{3+(-1)}=10^2$

Multiplication problems turn into addition problems!

Division:

$10^3 / 10 ^5 = 10^ {3-5} = 10^{-2} = 0.01$

This is the same problem as 1000 / 100,000 = 0.01.

Division problems turn into subtraction problems!

Metric prefixes

These prefixes correspond to powers of ten:

milli- = $10^{-3}$; micro- = $10^{-6}$; nano- = $10^{-9}$; pico- = $10^{-12}$, also centi- = $10^{-2}$

kilo- = $10^3$; mega- = $10^6$; giga- = $10^{9}$; tera- = $10^{12}$

A kilometer is $10^3$ m = 1000 m.

Now we're in a better position to examine solar production again

Scientific notation

$32,000=3,200 \times 10^1=320 \times 10^2 = 32 \times 10^3 = 3.2 \times 10^4=0.32 \times 10^5$

Multiplying two numbers

$32,000 * 68 = 3.2 \times 10^4 * 6.8 \times 10^1 = (3.2 * 6.8) \times (10^4 * 10^1) \approx 21 \times 10^{4+1} = 21 \times 10^5 = 2.1 \times 10^6 = 2,100,000$

Division

$(1.5 \times 10 ^7) / (3 \times 10^8) = (1.5/3) \times (10^7/10^8) = .5 \times 10^{7-8} = .5 \times 10^{-1} = .5\times 0.1=0.05$

Metric system

Even if you have a problem purely in English units, it is often easier to convert to metric units first, because then you can start using powers of 10 to convert to bigger / smaller units.

1 inch = 2.54 cm
1 m = 39 inch ~ 36 inches = 1 yard = 3 ft
2.2 lb = 1 kg
1 mile = 1.6 km

Welcome to the 21st century

Nowadays, the much easier way to convert units is to use Google or Wolfram Alpha to do it for you.

For example

160 lbs in kg

oil consumption in USA

Units

Often, you don't need formulas to figure out an answer. You just need to know the units that are required.

How long does it take for light, traveling at $3 \times 10^8$ m/s to reach Earth from the Sun--a distance of $1.5 \times 10^{11}$ m?

The answer to "How long" must have units of time. ("Seconds" rather than minutes or hours in the numbers we have so far.):

time (s) = $\frac{1.5 \times 10^{11}\rm{ m}}{ 3 \times 10^8 \rm{ m/s}} = 0.5 \times 10^3 \rm{s} = 5 \times 10^2 \rm{s} = 500 \rm{s}$

600 s$*\frac{1 \rm{ minute}}{60\ \rm{s}}$is 10 minutes, so, 500 s sounds like about 8 minutes.

Unit conversions

What does per mean? in each

Seconds per year?

1 year * 365 days / yr * 24 hours / day * 60 minutes / hour * 60 seconds / minute $\approx 400 * 20 * 4000 sec = 32 \times 10^6 sec$.

60 sec = 1 minute

1 minute / 60 sec