*chalk on board writing -LEARN -*
Hi, its Craig Beals.
We're back to answer a question that I get in class all the time.
How many places do I go after the decimal?
Well, there is an answer to that and it deals with Significant Figures
...or Significant Digits
I am going to give you the rules and a few examples to help you out with it
First of all, what are significant figures?
Well in science we use instruments, we never just use numbers
So we are always dealing in objects and when we measure those objects
...we need to know how precise or how accurate our measuring instruments are.
As you read through this very loose definition
be careful because it is far more in depth than it shows here
...but what we are trying to figure out is how precise our instruments are...
So, why do significant figures matter?
Significant Figures help show the accuracy or precision of your instrument
and it shows that your measurement cannot be more accurate than your measuring instrument was...
That will make sense when we start doing a little bit of math with this.
But first let me fill up my graduated cylinder
So, I am going to take some blue liquid here
and I am going to pour it in my graduated cylinder
...like so...and it is going start to fill up until I get to here.
Now when I look at my graduated cylinder and my measuring device I can see
we are going up by increments of 50 milliliters
and I've got larger ones in between there...I've got one, two, three, four, five of those
So that must be 10 milliliters
And the little ones in between are another 5 milliliters
So lets measure the bottom of the meniscus
The bottom of the meniscus is right about...here
This is kind of a poor picture of
as it moves around there
But I'm going to say, just for our purposes right here,
that our line
goes right
to that spot
so
as I start to erase a little bit of this
we should be able to see what line we were dealing with
...this line right here...
...that line is at 190 milliliters
but our meniscus didn't quite go all the way down to that line
and it didn't go all the way up to that next line either or it would have been at 195 milliliters
So, now you can start to see the importance of precision in measurement
...and how accurate our instrument is.
We know that it is somewhere in between there.
I am going to say 193 milliliters
is our measurement
Somebody else might have said 192 milliliters
So what does that mean?
That 3 is our least significant figure
that is the one we had to estimate a little bit to get an accurate measurement
Lets look at this when you are measuring a line
um
Let me add a few numbers
We will call these centimeters
...and in between we've got 1, 2, 3, 4, 5
which means each one of those
is worth .2 centimeters
for measurement
So, if this line
Went all the way up to here
it would be 6.2 centimeters
in length
...but it doesn't...
...it doesn't quite make it up to that line.
So again, this is where we show the precision of our instrument.
It is half way, or roughly half way between there
or maybe even a little more.
which means we can't say it is 6.2
we would have to say it is about
6.1 centimeters
We know it is 6 centimeters
and now we are saying it is just a little bit more and our device only allows us to measure down to the tenth of a centimeter
and that is our estimated number
If we look at this one down here
notice the one centimeter is way over here
so each one of these is worth .1 centimeters
(.2, .3, etcetera)
and in between we have the same number of lines
so this would be .02
.04
.06
until we get to one
Now if this line went all the way up to this...
line
we would have
.52 centimeters
but it doesn't quite make it up there so...
...again we are showing the precision of our instrument
we an get that last number right at .51
or so, centimeters
So, what does that all mean?
We we need to be able to understand the rules
now that we know why we have significant figures
But this helps us show...and keep everything constant in science
...anytime I have a red box it tells me I have important information
and I should probably write it down
Um, a few examples down at the bottom
or a few numbers, lets go through this:
Number 1, all non-zero numbers are always significant
and this one here means that I've got
one, two, three, four significant figures
in that number so far
Over here, 1, 2, 3, 4
significant figures in this number
When I go to the next rule
Number 2, any trapped zeros between two significant figures are significant
There's a trapped zero
this one is not a trapped zero
because there is nothing on the other side of it.
and this one is a trapped zero
so this number
I'm sorry, this number
now becomes five
and this one over here becomes five significant figures as well
Rule number 3, any trailing zeros, any zeros at the end of the number
are significant only if there is a decimal
somewhere in the number
we've got a trailing zero over here
and we've got a decimal in this number
so this isn't five significant figures
this is six
Over here, this is a trailing zero
but there is no decimal in my number
so this remains five significant figures
This one remains six significant figures
There are a couple rules that you need to know when you are adding or subtracting
and multiplying or dividing
this keeps everything constant
so that we understand the precision of our instrument
and so we can get the same answer in a mathematical equation
if we are working on something here in a lab in the United States
or maybe over in Germany
you should end up with the same results as long as we are following the same rules
So again, this is in a red box
I am not going to go through this part
If you need to write them down, pause it and write them down.
I am going to give you some examples
So, lets look at the first one here
Notice, when adding or subtracting...
in this one we are obviously adding
so we need to count up the number of significant figures after the decimal
we've got one, two, oh and here's a trailing one
but we count the trailing ones whenever there is a decimal
so we're going to count that
this one over here?
we've got (get rid of this)
get rid of this
we've got three significant figures after the decimal
This one is easy
there are no zeros to try to deal with
all of those are significant which means
our answer can have three numbers
or three significant figures after the decimal
When I run the math on this
my number should be 533.536 grams
and this already has 3 significant figures in it so that one is done
the next one
Again, we are subtracting so we are going to count up after the decimal
there's 1, 2, 3 and a trailing zero and there is a decimal
but these one's don't count as significant figures
because they're in the front, they never do
which means we've got 1, 2, 3, 4 significant figures
on this number
after the decimal
and this one we've got 1, 2
so that's going to be our limiting
and our answer can have two significant figures after our decimal
When I plug this into the calculator
if you need one,
you get -.1182 meters
And when we change that into the correct number of significant figures
we have to round that to .12
now on this answer up here I may have left grams off that
I shouldn't have, you should always have them labeled
for this one I need to make sure to put the meters down there.
I am going to clear this and go to the next one.
And down here at the bottom
this one, lets start here
has 2 significant figures after the decimal
it has 3 in the whole thing
but we are only concerned about the ones after the decimal
And this one doesn't have a decimal
so what do we do?
Well, we know that the precision of this instrument was not that good
and we can't be more precise than the instrument we used
so this has 0 significant figures
or, I'm sorry, zero significant figures after the decimal
so our answer has to have 0 significant figures after the decimal
we get 108.2
I'm sorry 108.02
for our answer
and that's milliliters but we can only have 0 significant figures after the decimal
I need to round this to 109 milliliters
What do we do when we get into multiplying and dividing?
So, operation rule #2
The first one we are looking at grams per milliliter
a density problem
here we've got two significant figures
over here we've got 1, 2, and we've got a trapped zero
so total we've got three
two in here, three in here. Again, we don't have to worry about after the decimal
we're looking at the whole number
and two is our lowest number
so when we get our answer
for the density of this object we get 3.36633 and it goes on and on
and we obviously need to round that
to 2 significant figures
so we are going to end up with
3.4 grams per milliliter
Get that down if you need it, its about to go away.
And the next one. We've got 1, 2, and this trailing zero and we have a decimal
so we've got 3 total significant figures there.
This one, we've got 1, 2, 3 and a trapped zero
that tells us we are dealing with four significant figures
so our answer can have three significant figures
This is 12.132
and again, lets simplify that down
to the number of significant figures
and we have to go 12.1 centimeters squared
don't forget what you've got going on here
And down at the bottom
I wanted to make sure to include one in scientific notation
because in chemistry or in physics these are the ones you are really going to end up with.
and it is really important to understand what scientific notation is
and how to do the math with it.
so
When we look at this it has 1, 2, 3
significant figures
um...I'll show you how
because, when we write this out
2.61
and I take that to times 10 to the
that means I move it 1, 2, 3, 4, 5, 6 places
I fill these in with zeros
and if I clean that up I get 2,610,000
thats two million six hundred and ten thousand
Significant figures wise I've got 1, 2, 3
significant figures in this
So, just like I told you down here I've got 3 significant figures
So I'm going to switch screens here to show you how to plug this into the calculator
because it is important to be able to do that
and do the math with these large numbers
when we go to put this in
we are going to put 2.61
instead of trying to do times 10, use the little carrot thing and put the 6 up there
Scientific calculators have this magic number
called exponential notation number
Usually is is an EE or maybe one large E
some calculators have it as EXP
you'll just need to find it on your calculator
but when I press that on here I can see an "E" that means exponential notation
so when I press the 6
now my calculator knows I've got
I've got 2.61, this E means times ten to the 6
so this is 2.61 times 10 to the 6th
I don't have to worry about parenthesis, the little carrot,
the hat, or whatever you want to call it
its good just like it is right here
so I take that and I divide it
by my other number which was .0034 seconds
I hit enter and I get this huge number
and I am going to transfer that number over to our slide
here's our number on the slide
we know that in our answer we can only have two significant figures
because in division we have to count up the smallest number of significant figures
This has way to many.
So I am going to turn it into exponential notation
I am going to take this decimal and move it 1, 2, 3, 4, 5, 6, 7, 8 places
that direction
which means now I will have
7.676, but I need to round it up because I can only have two so I am going to have
Which means I am going to have 7.7 times 10 to the 1, 2, 3, 4, 5, 6, 7, 8
and this is Joules per second
or watts
So, this is how you do significant figures or significant digits
And hopefully now you will never have to ask that question again...
"How many places am I supposed to go after the decimal?"
If you follow the three simple rules and the two operational rules
and you will get this right every time
so, GOOD LUCK!