In this video, we will learn about significant figures.
We will discuss how to identify significant figures in a number and look at how to round
numbers to significant figures.
Let's see five rules that will help us decide whether a digit in a number is significant
or not.
Here they are.
All nonzero digits are significant, zeros sandwiched between nonzero numbers are significant,
zeros at the end of a number are significant if they are after the decimal point, zeros
in the beginning of a number are not significant, and if there is no decimal point, zeros at
the end of a number are not significant.
Let's apply these rules in a few examples.
Since all nonzero digits are significant, the 2, the 5 and the 6 in 256 are all significant.
Zeros sandwiched between nonzero numbers are significant, so all four numbers in 4007
are significant.
Zeros at the end of a number are significant if they are after the decimal point, so the
zero at 23.810 is significant, along with all other numbers.
However, zeros in the beginning of a number are not significant, so the three zeros in
0.00248 are not significant, because of the first rule, 2, 4 and 8 are significant.
If there is no decimal point, zeros at the end of a number, in other words, trailing
zeros are not significant, so in 58,000, 5 and 8 are significant, however, the three
zeros at the end are not.
There is a way to show that these trailing zeros are significant, which you see in 3200,
as it has a decimal point at the end, even though there are no numbers after the decimal
point.
So here, 3 and 2 are of course significant, and because of that decimal point, the two
zeros are also significant.
This notation is rarely used, so there is no need to worry about it too much.
Let's see some examples about rounding numbers to significant figures.
SF here stands for significant figures.
In the first number, we have to round to two significant figures, so let's find the first
two significant figures in this number.
These are 9 and 8.
So we have to round the 8 based on the next number, which is 6.
This means that the 8 must be rounded up, so we get 9, 9, and the common mistake here
is to leave this number as 99, assuming that the 6 and the 7 become nonsignificant numbers,
so they can simply be omitted.
However, it clearly doesn't make sense to round the number that is close to 10,000 down
to 99.
So instead of the 6 and the 7 disappearing, they simply become zeros at the end of the
number, because these zeros are not significant.
In the next number, the first three significant figures are 4, 7, and 0, so we will round
based on the 8.
We need to round the 0 between the 7 and the 8 up, so we get 4, 7, 1, and the last three
digits become zeros, so we end up with three significant figures in this number.
The first three significant figures in the next number are 5, 3, and 0.
We have to round based on the other 0, so we will leave the 0 after the 3 as it is,
and get 5.30.
Remember that a 0 after the decimal point at the end of the number is a significant figure,
so this number now has three significant figures.
The first significant figure in the next number is 2, so we have to round based on the 8,
so round up, which gives us 0.03.
Since here adding more zeros at the end of the number would add more significant figures,
in this case we just leave the zeros off.
In the next number, the first four significant figures are 2, 6, 7, and 7, so we will round
based on the 8, which means rounding up, and we get 2, 6, 7, 8, and the remaining five
numbers are all zeros.
For our last number, the first two significant figures are 4 and 7, so we have to round based
on the 5, which gives us 4, 8, and for the remaining two numbers we will add two zeros.
Note that originally this number had four significant figures because of the decimal
point at the end of the number, but since night has to have two significant figures,
so only the 4 and the 8 are significant, we will not put a decimal point at the end to
show that the two zeros are not significant.
Let's see what we have learned about identifying and rounding to significant figures.
All non-zero digits are significant, zeros that are sandwiched between non-zero numbers
are significant, if zeros are at the end of a number and after a decimal point, they
are significant, zeros in the beginning of a number are not significant, and if there
is no decimal point, zeros at the end of a number are not significant.
This wraps up our discussion about significant figures.
In the next video, we will learn about orders of magnitude.