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All right, let's look at scientific notation.
Because many physics, chemistry, biology,
when you do experiments, sometimes
you end up with very small numbers or very large numbers,
when you're doing astronomy or something.
Very small numbers come when you're
looking at something microscopic.
So in order to write a scientific notation, what
we use is the largest place value
with a non-zero digit as a reference.
And then write the number as decimal number between 1 and 10
and then times 10 to the power that the place value is from.
So, for example, in 2012, the US debt
is approximately $16 trillion dollars.
What does that mean?
So take a look here.
So you have these three 0's, thousand.
This is going to be million, billion, trillion.
So to write that in scientific notation,
1 is our largest place value.
So we would write the number as 1.6.
And then the place value, you're going to have 13 places.
So that's 10 to the 13th, right?
6, 7, 8, 9, 10, 11, 12, 13.
You can also say, 16 times 10 to the 12th.
The diameter of an E. coli bacteria is about 0.0000012
meters.
So in scientific notation, we would
write that as 1.2 times 10 to the negative 6 meters.
Or in the metric system, we also can call that as 1.2 microns.
That's one thing nice about the metric system.
You can go up and down between units
by just multiply or dividing by powers of 10.
So here's some exercises that we would like you to try.
So pause this video right now.
Do these exercises.
And then come back.
All right, assuming you have come back
from doing the problems.
So let's start with this.
1, 2, 3, 4, 5, so five decimal places.
So that would become 3.45 times 10 to the fifth.
What do you think is the next one?
So our decimal point is here.
So we're going to go 1, 2, 3, 4, right?
And then this time, because you're
working with a small number, you're
going to have negative 4 power.
Good.
So how do you decide whether it's
positive power or negative power of 10?
Look at what number you're representing.
If you multiply by a positive power of 10,
this digit, 3.45 times 10 to the fifth,
is going to become a large number.
Whereas 4.9 times 10 to the negative 4
is going to be a tiny number.
So whether you have tiny number or big number
would decide the positive or negative exponent.
That's the best way to think of that.
What if you were given scientific notation?
How do you put a decimal?
So, again, remember what I said.
Multiplying by powers of 10, and we
saw this in previous lecture, means move the decimal six
places.
So 1, 2, 3, 4, 5, 6.
So the number is going to be what?
Anyone?
Remember, this is thousand.
This is million.
So I will let you think about how
you're going to write this in words.
This will be a good exercise for you
to recall what each of these place values represent.
All right, so here we're going to move the decimal places
12 places.
So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.
So that's where the decimal place will go.
If you're not sure, just rewind and watch again to make sure
you can go from decimal numbers to scientific notation and vice
versa.
All right, rounding decimals.
When you're rounding, we want to know to what degree of accuracy
you are giving us the number, so nearest 1/10 of a mile
or nearest 1/4 of an inch or nearest 100.
That helps us communicate with each other.
Visualizing a number on a number line
helps in rounding a decimal.
You can select the number on the tick mark
nearest to the unrounded number.
For the numbers exactly halfway between tick marks,
depending on the context, you can
use two different conventions are there.
One is to round up to the right of the number line.
Or one is to round to the nearest even digit.
So if I asked you to round this to the nearest
10th, nearest 10th, the 10th place is 0.75.
And 0.75, the number closest to that is this one.
This is equally far apart.
But convention is to go to the next digit
closest on the right side.
All right, let's round this to the nearest thousand.
So here is the number.
3,435 is going to be a number somewhere here.
Which thousands is closer?
Remember, 3,500 is closer to 3,435 than 3,000 and 4,000.
However, since you want thousand,
is 3,000 closer to 3,435 or 4,000?
So you can see by where it's placed
that 3,000 is closest to it.
So that's why we round to the nearest thousands,
it's going to be 3,000.
So this digit, 4, is less than 5.
So you round to the 3.
If this was 3,500 something, then
you would round that to the 4,000.
All right, what if I ask you round to the nearest hundred.
Then, which hundred is closer to 3,435, 3,400 or 3,500?
And you can see that it's going to be 3,400.
Rounding is used in many natural science experiments
to indicate the accuracy of measurement.
And it depends on what equipment you are using to measure.
So, for example, when we say weight of a person
is 160.0, when it's written like that, it's not 160 pounds.
What that means is that the actual weight is measured
to the nearest 10th of a pound.
And so the actual weight is less than 160.05 pounds
and greater than or equal to 159.95 pounds.
This is important to remember.
So if you read a label, a nutrition label,
and it says 0 grams of fat or 0.0 grams of fat,
think about what that really means.
Often we use English units to measure.
And we can round to the nearest 1/16 or 1/8 of an inch.
So take a look.
Here's a sewing tape, measuring tape,
and here's a measuring tape, one of those hard measuring tapes
that you use to measure height of a wall and stuff like that.
Look at the difference.
Look at the number of tick marks here.
1, 2, 3, 4, 5, 6, 7, 8, so this is eighth of an inch
tick marks.
Whereas here you have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16.
So this is 16th of an inch.
So when you measure things, it's not always 10 tick marks.
So if you pick things, anything that you measure in your life,
you will see what kind of scale you're using to measure.
Here's some measuring cups where you can use ounces.
Or you can use liters.
Or you can use cup measures.
So here I have water.
And I'm saying that the closest I can measure is 8 ounces.
Whereas in this case, I can say it's about 225 milliliters.
So it's important to know how to round
so you can measure things.
So measuring is something practical.
You have measuring in many jobs, construction, roads, bridges,
cooking.
Pretty much anything you do, you're
probably measuring something out all the time in cups
or in ounces or tablespoons or meters and so on.
I know that I showed you my face at the beginning
of the lectures.
But you haven't gotten a chance to see
Dr. Paul Martin at Marathon.
So we thought how we would add this picture here.
This is Dr. Paul Martin.
And he and his family built this bridge on their own
on one of their lands.
So this is a bridge construction.
He had to use a lot of measuring.
So it's perfect for the end of this lecture here.
And so in the old English old system here,
we use pounds, yards, feet.
And it's cumbersome to go from yards to feet to inches.
But in the metric system, you just
go like decimal system moving in powers of 10.
Like 1 kilometer is 1,000 meters.
1 meter is 100 centimeters.
It makes it easy to convert from kilometers to meters
to centimeters.
And we'll see this in a little bit later.
Here's your homework.