in this video we're going to quickly cover how to write numbers in scientific notation and
then we're going to cover how to convert numbers and scientific notation back into standard form
and as we can see at the top here the main idea with scientific notation is that any number can be
written as a number between 1 and 10 multiplied by a power of 10. when i say between 1 and 10 i'm not
including 10 itself so what we're talking about here is a single digit before the decimal point
and then multiplied by 10 to some power and the basic idea behind this is that for example the
number 1000 can be written as a power of 10 as 10 to the third so that's our basic
key to writing large numbers as powers of 10 and then a number like 1000th .001 or one thousandth
well i realize a thousand in the denominator is a ten to the third in the denominator
but that simplifies to ten to the negative three and so large numbers will have positive
exponents of ten and small numbers will have negative exponents of 10. so let's apply this
to the first example i'm given the number of 43 500 i'm told to assume that 5 is the last
major digit here and i'll explain why that matters in a moment and i'm trying to write this in
scientific notation and i immediately realized oh this thing is just four point three five
times ten thousand but ten thousand can be written as a power of ten
and i've got my answer four point three five times ten to the fourth now why does it matter that we
were told that five was the last major digit well the standard notation is ambiguous when i see 43
500 i can't tell whether or not that number was rounded to the hundreds place
or whether it was rounded to the tenths place and it just happened to be a zero in that place or
even if it was rounded to the ones place so when we're told that five is the last major digit we're
being told that we have three significant digits here and the scientific notation should have three
digits in it if we were told that the tens place was the last measured place value we could simply
write scientific notation as 4.350 times 10 to the fourth and this is a major advantage of scientific
notation over the standard notation the trailing zeros in the standard notation are ambiguous
but in scientific notation however many digits you have in front of that power of 10 that's
how many sig figs you have let's take a look at part b this time we have a small number .0052
and i could write that as 5.2 times point zero zero one or one thousandth and then i remember one
thousandth is ten to the negative three and we're done with that one now i do wanna point out that
in practice you're probably going to find the power of 10 by counting decimal places so for
part a i would start with my decimal point after the ones place and then count how many places does
it take to get to the point where there's only one digit left in front of the decimal one two
three four spaces and that's why i have a ten to the fourth for the small number in part b one two
three spaces to get the decimal after the first non-zero digit and because it's a small number
that's a ten to the negative 3. now let's go the other way around in part two in part a i'm
told the speed of light is 2.998 times 10 to the 8th meters per second and because i have a 10 to
the eighth it means that that decimal bouncing idea had to happen eight times to arrive at the
scientific notation and we're just going to reverse the process by bouncing the other way
eight spaces so i got one two three and then i'm out of digits so i need to start putting
in place holders so here's a four five six seven and eight and then i should have commas in here
and i'll rewrite the answer just to clean it up and this is 299 million eight hundred thousand
don't forget the units meters per second in part b we take a very small number in scientific
notation so the diameter of the hydrogen atom is 1.06 times 10 to the negative 10
meters and again i'll just think of this in terms of bouncing the decimal place and this time i'm
going the opposite direction in order to arrive at a very small number so i bounce one decimal place
and then two and i need a placeholder three and so on and so i arrive at this very small
decimal with nine leading zeros before i get to the 106 and i'll clean it up and put units on it
and we're done with that one if you enjoyed this video or at least found it useful
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