In this video we�re going to talk just for a minute about something called scientific
notation now this is a very, very helpful math tool that helps out in a lot of different
scientific field like astronomy, in physics, and chemistry and what it does it helps us
very easily express really, really big number and really, really small numbers. So just
too kind of motivate this a little bit let�s say we�re doing an astronomy problem and
where talking about light years that�s the distance that light travel in an entire year
which is obviously really long distance well that distance would be nine quadrillion four
hundred and sixty trillion seven hundred and thirty billion four hundred and seventy two
million five hundred and eighty thousand eight and eight hundred meters, really long distant
and if we wanted to work with that number and do some calculation with that number such
as the number of light years and or the distance of five light years are something like that
it would just be too bulky and cumbersome to multiply this guy times another number
or add it to another number or subtract it you�d it just too big to handle so it will
be better if we had this something that�s a little bit more of a condense notation and
that is exactly what scientific notation is. So scientific notation a number written in
scientific notation looks like this is A times 10 to the B where there is some restriction
on A and B. A has to be greater than or equal to one but strictly less than 10 and then
B has to be an element of the integer, so B has to be an integer. Now what�s the deal
with A being greater than one but less than 10 what we�ll want to make sure is that
A only has a ones place like 9 point 7 would be great for A because it just has a ones
place. The number we would want to stay away from for A would be like 13 point 221 because
having thirteen that has a tenth places as well which is larger then what we want for
A. So it has to be under ten to make sure you�d at most 9 point something to have
a ones place only but greater than or equal to one so that you�re not just a decimal
that doesn�t even have a ones place and then the reason why we want B to be an integer
is because it�s super easy to multiply by multiplies of ten every time you multiply
by ten you�re going to move a decimal place and so if you had times ten to the 5th then
you�ll only moving the decimal place 5 times and so on and so forth. So basically you�re
keeping A and then moving the decimal place left to right depending on weather your B
is positive or negative. So let�s quiz ourselves here real quick and see if we can tell which
of these number are in scientific notation. The first one is 2 point 71 times 10 to the
5th that looks great to me the A is has a ones place only it�s 2 point 71 it�s not
12 point 71 or a hundred and twelve point 71 it just has a leading 2 and 5 is an integer.
Now 0 point 399 times 10 to the negative 8 I don�t think that going to be in scientific
notation the negative 8 is an integer that�s fine but the 0 point 399 is not between one
and ten, 2 point 71 is between one and ten but point 399 is not it doesn�t even have
a ones place. Lastly 1 point 291 times 10 to the negative 1 point 5 that�s not in
scientific notation the 1 point 291 is between one and ten but negative 1 point 5 is not
an integer so make sure you�re very confident in what a number in scientific notation look
like. Now probably one of the more common question that you�ll see is that someone
will ask you to convert back and forth from scientific notation into what we call standard
notation. Now that will go from scientific notation back to standard notation you�re
basically going to be moving decimal of A left or right and whether its left or right
depends on your exponent B you�re going to move your decimal right if B is positive
cause your making the number bigger you have the multiple of ten and then you going to
move the decimal left if the exponent is negative. So simple example let say that we had 2 point
71 times ten to the 4th well since this is 10 times 10 times 10 times 10 four factors
of 10 this number is going to get larger and we�re going to move this decimal place right
four units since the base or the B not the base but the B was greater than 0 so this
would be 2 7 1 0 0 move it ones twice three times four time. So 2 point 71 times 10 to
the 4th is the same as 27 thousand 1 hundred and this makes perfectly good sense if you
wrote 10 to the 4th out long ways 2 point 71 times a 1 with four zeros so this would
actually be the same as 2 point 71 times ten thousand which would make it 27 thousand one
hundred. Now if the exponent B had been negative we would have move this decimal place left
a certain number of unit and made the number much, much smaller. Alright and lastly what
if we go in the other direction what if we had a number in standard notation and we want
to put it in scientific notation well the most important thing is to make sure this
leading A only has a ones place so what we�ll do is move the decimal left or right or however
many units we need to move it so that it�s just passed the first nonzero digit so we�ll
have one digit and then the decimal so then we�ll just be filling up a ones unit not
a ones and a tenth place. And then lastly depending on how many places you moved your
decimal left or right we need to choose an appropriate value for B either positive or
negative. So let�s try an example let�s say we had 0 point 00325 well this is in standard
notation let see if we could write it in scientific notation. So we would need to have as our
A 3 point 25 that the same number but I moved this decimal place 123 units to the right
so we�d have 3 point 25 times 10 to the something now if we start with 3 point 25
and we want to get back to point 00325 this decimal place would need to the left to get
there which means we�ll need to move negative 3 units or negative 3 places or 3 places sorry
3 places to the left since it�s negative of course. So anyway yeah we�re trying to
make this number smaller so it goes 123 places to the left and we designate that with a negative
exponent for the ten. Alright let�s try one more in closing let�s say we had 4 million
9 hundred and 90 thousand and that�s good 4 million and 9 hundred and 90 thousand so
we call this 4 point 99 but now 4 point 99 we would need to make larger to get to 4 million
to 9 hundred and 90 thousand so how many decimal places will we need to move this guy to the
right well if the decimal place was here we need to move it 123456 places to the right
so it be 4 point 99 times 10 to the 6th power. So anyway this is how we go from standard
notation back to scientific notation. In the upcoming video we�ll work a little bit more
with scientific notation such as how to do scientific notation on a calculator and how
to multiply and divide with scientific notation as well.