hey it's professor dave, let's talk about scientific notation
and dimensional analysis
humans like to count things. in the first civilizations
we had to count in order to trade goods with one another. because we have 10
fingers
we based our counting on a system of 10. if you count apples we can get to nine
but if we add one more we put a 1 in the tens place
and then start building up the units place again. there's a hundred place
thousands and so forth. each time we reach the next factor of 10
we add a digit. but what if we wanna talk about a really big number
like the number of molecules in this glass of water, or the number of stars in
the universe
it's going to get tedious very quickly to have to write numbers with dozens of
digits
we need a way to abbreviate very large and very small numbers
that way is called scientific notation
a number written in scientific notation will always consist of a single digit
followed by a decimal point and then some other numbers
depending on how many sig figs are in the value. that number is multiplied by
10
which is raised to an exponent. what this represents
is the magnitude of the number. 10 squared means ten times ten
which is a hundred so 1.42 times ten to the second
is 142. we can get that by moving the decimal to the right twice
ten to the sixth or one with six zeros
is a million so five times ten to the sixth is 5 million
we can get this by moving the decimal over six times adding zeros if we need to
we can also discuss very tiny numbers this way. ten to the negative second
means one over 10 squared or one 100th so 3.5 times ten to the -2
is .035. we can get this by moving the decimal two places to the left
and eight times 10 to the -9 will be this number once we move the decimal
nine times. this is useful because now we don't need inches and feet and miles
we can just use one unit and express values in scientific notation
for length we use meters for time we use seconds
and so forth. now we can discuss the diameter of an atom
and the distance to the nearest star in the same unit: meters
just with scientific notation. we can also use
unit modifiers to symbolize different magnitudes. kilo
means tend to the third which is a thousand so a kilogram
is a thousand grams. nano is 10 to the -9
which is one 1 billionth so a nanometer is this many meters
so we can express values with a word to indicate magnitude instead
a specific factor of 10. here are the different modifiers we use
we can apply them to any unit and they will do the same thing
to convert between units we need to do dimensional analysis
let's say we have a value in meters that we want to express
in nanometers. we take the value we have and we multiply by a fraction
that is equal to 1. we do this because multiplying by one
doesn't change the value but we multiply by a version of one
that changes the way the value is expressed. we do this by putting things
on the top and bottom
that are equal to each other. in this case ten to the nine nanometers on the
top
and one meter on the bottom. these are equal because if a nanometer is a
billionth of a meter
then there are one billion of them in one meter
so since these are equal this fraction equals 1
but the meters cancel just like an algebraic variable
and now we have a value in nanometers. it's the same value we had
but expressed differently. we can do this to convert between
any units, just multiply by fractions that equal one
until you have the value expressed in the units you want.
for example how many seconds are in a year?
lets use some conversion factors we know: 365 days in a year
24 hours in a day, 60 minutes in an hour
sixty seconds in a minute. all the units cancel
except seconds and we just multiply the numbers to get the answer
these are exactly the same length of time just expressed
differently. let's check comprehension
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