In this segment, we will talk about an example. What we want to be able to see that if we
know what the absolute relative approximate error is, can we figure out how many significant
digits are at least in my answer? So, here is the example, which is given to us, which
we are asked to solve. There are six terms of the Maclaurin series for e to power X we
are calculating e to the power of 0.7 and we want to be able to figure out how many
significant digits can we trust in the solution. Keep in mind that even your finding the answer
you are not privy to the exact value of e to the power zero point seven. So, if you
look at the Maclaurin series for e to the power x, it is given by this particular polynomial.
And, if we want to use the first six terms of the series, so let’s suppose, six terms
of the series, we can approximately calculate each power zero point seven as follows: by
using the first six terms of the series. And you can see that we have six times one, two,
three, four, five, and six. And the value which I get from here is two point zero one
three six. Now, if I want to figure out how many terms how many significant digits can
I trust in my solution here, then I'll have to calculate the value of the previous approximation,
which will be hey, what is the value of each part zero point seven is five terms of the
series? So, here we have the first five terms of the series. One, two, three, four, and
five. And this number here turns out to be two point zero one two two. Now what I can
do is this is my current approximation, and this is my previous approximation. So, I can
take my current approximation and a previous approximation and calculate my absolute relative
approximate error, which is the current approximation minus the previous approximation, divided
by the current approximation, and I'm going to calculate it as a percentage. And this
turns out to be zero point zero six nine five two seven percent. And by looking at this
absolute relative approximate error, I can basically say that two significant digits
are at least correct in my answer. How do I come to that conclusion because we have
this theorem that if the absolute relative approximate error is less than or equal to
0.5 times 10 to the power of 2 minus m percent, then at least m significant digits are correct
in my answer? So, we, I basically applied that because when M is equal to 1, epsilon
A has to be less than equal to 5%, when M is equal to two, Epsilon A, absolute relative
approximate error, has to be less than or equal to 0.5 percent, and when M is equal
to 3, absolute relative approximate error has to be less than 0.05 percent. And now
I can see that if I look at this number right here it is less than 5, it is less than 0.5
percent, but it is not less than 0.05 percent. Hence, the number of significant digits which
will be correct, and the answer is 2. So, M is equal to two is my answer. Now, sometimes
what happens is that when people try to apply this theorem, they might get a little bit
concerned about that hey I have to calculate it for one significant digit, then two then
three, then four and then see where it falls. You can also say that hey we are trying to
figure out hey when is Epsilon A less than equal to 0.5, 10 to the power of two minus
M. What is the largest value M for which this absolute relative approximate satisfies this
particular inequality? So, you get zero point zero six nine five two seven, which is my
absolute relative approximate error, when is it less than or equal to 0.5 times 10 to
the power of two minus M? So, if I do that, I get this: taking zero point five to the
left-hand side. Now, here in order to be able to find M, one of the ways I can do it is
I can take the log to the base 10 of both sides, that's the easiest thing to do, you
can take natural log as well, it will give you the same answer. So, here I will get minus
zero point eight five six seven, this will be equal to two minus M. That means M is equal
to two plus zero point eight five six seven, and that's two point eight five six seven.
So, the number of significant should be at least correct will be two. So, by doing this
way also, you are able to find out how many significant digits can you trust in your solution
of e to power zero point seven being approximately to two point zero one three six. That means
you can trust this two, and you can trust this zero, but it's quite possible that more
than two significant are at least correct in this answer, but it's only telling you
that hey it's quite possible that more than two significant digits that at least are correct
in your answer. But we are from the theorem only able to say how many are at least correct
in this case is 2. So, this is one, and this is two. And that's the end of this segment.