Knowing how to properly round numbers is extremely important in math and in
Chemistry. In fact if you round incorrectly this
can cause you to lose points on your exam and we both don't want that to
happen. So let's go over the rules for rounding
when rounding a number always look at the number to the right
of where you want to stop to determine whether to round up or down.
For example, if we wanted to round 3.2 to the nearest whole number and
note a whole number has no decimals, we would start with looking
at the number to the right of three which is two
and if this number is less than five we will round down.
Our answer in this case is three. If we wanted to round
three point five to the nearest whole number we would look at the number to
the right of 3 so 5 and if a number is 5 or greater
then we will round up our answer in this case is 4. In a
math class you would be asked to round to the nearest whole number or
a specific decimal place like the tenths, hundredths, thousandths place and so on
but in chemistry you are asked to round to the appropriate amount of significant
figures or sig figs. What I'm about to teach you is
going to keep popping up this entire semester
because your final answer for any chemistry problem
has to have the correct number of significant figures.
That being said let's go over the rules for significant figures.
Rule one any number that is not zero is significant. For example, every single
one of these numbers are non-zero numbers
so they all count as sig figs there are five significant figures.
Rule two zeros that are in the middle of non-zero numbers
are significant. This example has two zeros in the middle of non-zero numbers
the five and the three so all of these numbers count as sig figs there are four
significant figures. Rule three any zeros after the decimal are
significant in this case there are three zeros after
the decimal so these count as significant
all these numbers count as sig figs there are four significant figures.
Rule 4 any coefficients in scientific notation
are significant. Remember a coefficient refers to the number in front
we do not look at this portion of our scientific notation
so there are only two significant figures for this example.
Rule 5 any leading or beginning zeros are not significant these leading zeros
do not count as sig figs only these non-zero numbers count
there are three significant figures. Rule six any zeros within a number that has
no decimal are not significant for example if we had seven thousand
only the seven would count as a sig fig and the zeros would not count there is
only one significant figure. Okay go ahead and try
this question out round this number to two sig figs go for
it.
Okay if you said 34 you just made one of the most common mistakes
and I want you to think of this in terms of money.
If you were promised 341 dollars but given 34.
would you think these numbers were the same yeah, probably not.
So make sure not to completely change the value of the original number
the correct answer here is 340. We found this by seeing that we need to round it
to sig figs but we will look at the number to the right
of those two sig figs and round down because one is less than five we
get 340 and remember this zero is not significant because there's
no decimal shown. We could have also written the number in
scientific notation and rounded to two sig figs
so 3.4 times 10 to the second would also be correct.
If you didn't get that one right don't worry here's another chance to redeem
yourself go ahead and try this question out on
your own.
For this question we know that two sig figs stops here at the five
but we need to look at the number to the right of this and since it is greater
than five we will round up so around the five up to six to get
seven thousand six hundred our zeros again do not count as
significant only the seven and the six do so our answer is seven thousand six
hundred we could have also written the number in
scientific notation and rounded to two sig figs
so 7.6 times 10 to the third would have also been correct.
Now let's go over the significant figure rules for the following operations
starting with adding and subtracting. When adding or
subtracting numbers we must always round to the least
number of decimal places. Let's do an example.
Always start with identifying the number of decimal places
each number has this first number has one, two, three decimal places the second
number doesn't even have a decimal shown so
there are zero decimal places and the third number has one, two decimal
places. Next determine what the least number of
decimal places are, in this case the second number has the
least number of decimal places which means our final answer will have
zero decimal places. Now we'll add everything together
and round up to 77 and this once again has no decimal places based on the least
number of decimal places we found in the first step. When multiplying or
dividing numbers we must always round to the least number of
significant figures. For this example we will start with
identifying how many significant figures each number has. The first number has
three sig figs and the second number has four sig figs
based off of rule one from our significant figure rules.
Next we'll multiply these numbers together to get this
and we will round the least number of significant figures which was 3
so round up to 17.6. There are some questions that combine
all of these operations together but before we go over how to figure
out the sig figs for these types of questions it's important to review our
basic order of operations or PEMDAS. Where anything in parentheses
is first, followed by exponents, multiplication
division, addition, and subtraction is last.
I want you to try this question out right now based on what you remember
from order of operations.
Alright, let's go over this together we'll use PEMDAS to help us out.
Start with parentheses first so 16 plus 2 is 18
and this is the part that may have tricked a few people.
According to PEMDAS the next step is to divide
not add. Eighteen divided by three is six and last step is to add the three our
answer is nine. For those who said seven you added first
instead of dividing so make sure to remember to use PEMDAS so you don't get
the answer wrong. Let's do a more challenging question
together. We will start with parentheses so 3 times 9 is 27.
Now the question is do we multiply or divide next?
You would actually divide first and yes I know multiplication is before division
but whenever you have both operations you will solve these going from left to
right. So applying this we will go from left to
right so divide first then multiply after. 21 divided by three is seven
next we will multiply negative twenty seven times nine to get negative two
hundred and forty-three. Now the question is do we add or
subtract next? Same thing applies here we will perform
the operations going from left to right so we will subtract first then add.
7 minus 243 is negative 236 and now we will add so negative 236 plus
5 is negative 231
Overall the rules for order of operations is to start with parentheses
first, then any exponents, next is multiplying
and dividing going from left to right and last is adding and subtracting again
going from left to right. Now that you have order of operations
down let's see how this applies to chemistry specifically
significant figures you'll need to remember order of operations and your
sig fig rules for adding and subtracting and multiplying and dividing.
For this example start with order of operations
where parentheses are first and then we will subtract
but instead of just doing this we need to account for
our sig fig rules and we will round at the very
end. So if we are multiplying we want this to have the least number of sig
figs. This first number has four sig figs and
the second number has two sig figs since the least number of sig figs is
two we would have rounded to two sig figs after multiplying
but you should always say rounding to the very end
so i will underline where this number should have stopped at
then bring down the rest of the problem we no longer are multiplying we are
subtracting. Our sig fig rules have just changed to
rounding to the least number of decimal places.
Start with identifying how many decimal places each number has.
For this first number, we are focusing on the underlined portion
we said that this number would have stopped at two sig figs from our
previous multiplication rule so there is only one decimal place and
we would ignore the rest of this. The second number only has two decimal
places. Subtract the entire number for each then
round to the least number of decimal places which is one. We'll round up to 0.3. Let's do another example.
Start with parentheses first we'll add these two numbers together to get this.
Identify how many decimal places each number has. The first number has two decimal places and the second number has three decimal
places. Since we are rounding we will need to
underline where this number should have rounded to
which is the least number of decimal places. That would be two decimal places
so I will underline this four. Bring down the rest of this problem.
Order of operations tells us to multiply and divide going
from left to right, so we will multiply first. The sig fig
rule for multiplication is to round to the least number of sig figs.
This number has five sig figs and this number has four sig figs.
Multiply the numbers together and underline where we would have rounded to
the least number of sig figs which is four. Last part of this is to divide. Our
sig fig rule for division is to round to the least number of sig figs
remember to focus on where we underlined each number and identify the least
amount of sig figs. The first number has four sig figs based
off of where we underlined and the second number also has four sig
figs based off of where we underlined. Divide both numbers entirely and then
round up to four sig figs, our answer is 1.631. Ready for those
practice problems? Well using the link in the description
you'll find the practice problems for this
entire playlist with step-by-step video explanations.
So try those out on your own then come on back for the next video.
Alright, let's put in some work, together of course!