Making comparisons

...using exponents (powers of ten).

Exponents / Powers of 10

$10^5 = 10 \times 10 \times 10 \times 10 \times 10 = 100,000$
A 1 followed by 5 zeroes.

Multiplying any number by 10 can be done by just moving the decimal point to the right: $$13.8 \times 10 = 138.$$

$10^6$: Start with 1.000 000 000 ... and move the decimal point 6 times to the right: 1.000000000000000

  1. 10.00000
  2. 100.00000
  3. 1,000.0000000, 1 thousand. 1 thousand meters = 1 kilometer.
  4. 10,000.000000
  5. 100,000.0000000
  6. 1,000,000.0000, 1 million! 1 million "bytes" = 1 Megabyte.

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

    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 (positive powers) or to the left (negative powers).

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

    1 millimeter is 1/1,000 of a meter ($10^{-3}$ m).

    1 microgram is 1/1,000,000 of a gram ($10^{-6}$ gm).

    $10,000?$ 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... $1,000 \times 10 = 10,000$

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

    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.

    Practicing scientific notation handout.

    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

    I find myself doing this sort of thing every day, to check if I punched things into a calculator correctly, or just to get an approximate answer:

    $$\begineq 32,000 \times 68 &\approx 30,000 \times 70\\ &=3\times 10,000\times 7\times 10\\ &=3\times 7\times 10^4\times 10\\ &=21 \times 10^5=2.1\times 10 \times 10^5=2.1 \times 10^6\\ &\approx 2 \text{ million}\endeq $$ (The exact answer is 2,176,000: less than 10% different from my "quick and dirty" estimate.)

    Division

    $$\begineq (1.5 \times 10 ^7) / (3 \times 10^8) &= \frac{1.5\times 10^7}{3 \times 10^8}\\ &= \frac{15\times 10^6}{3 \times 10^8}\\ &=\left(\frac{15}{3}\right)\times\left(\frac{10^6}{10^8}\right)\\ &=5\times10^{6-8} =5.0\times 10^{-2}=\color{red}{0.05}\endeq $$

    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

    $\Rightarrow$ 1 minute / 60 sec : There is 1 minute per (for each) 60 seconds

    $\Rightarrow$ 60 sec / 1 minute : There are 60 seconds per (for each) 1 minute