In this segment, we'll talk about what the relationship between the absolute relative
approximate error and the number of significant digits you find to be correct is. So, for example,
you are a numerical method going to be calculating absolute relative approximate errors,
which is basically calculating your current approximation and subtracting it from your
previous approximation, divided by the current approximation. And assuming you're calculating
it as a percentage, we multiply by 100. So, when do we know that hey we need to
no longer find the current approximation, because, for example, we are going to be
finding the sum of a series we will continue to add one term at a time and those become our
current approximation but when do we know when we have to stop the current approximation. So,
in this case, we may have something called a pre-specified tolerance. What that simply
implies is that you have pre-decided that hey you want the absolute relative approximate to be less
than some number Epsilon S. And that's when you want to stop the numerical method which you are
using to calculate an approximate answer. So, in that case, what we have basically is that when
Epsilon A is less than equal to Epsilon S that is when you want to stop your numerical method,
whether it is to find the sum of a series, or to find the root of an equation, whether it is to
integrate a function, and so on and so forth. Now, sometimes what we will find out is that
the pre-specified tolerance might be given in terms of number of significant digits you want
correct in your answer. Let's suppose somebody says hey I want two significant digits to be
correct in my answer, while you are conducting this numerical method, what does that mean? So,
there's a nice theorem which relates the number of significant digits which are at least correct to
the absolute relative approximate error you're getting. What is that theorem? It says that,
if you want at least M significant digits to be correct in your answer, then you will need to
have an absolute relative approximate which is less than this particular quantity: 0.5 times
10 to power 2 minus M percent. So, for example, we were saying hey we want two significant digits to
be correct in my answer, what that simply is telling you is that you want absolute
relative approximate to be less than or equal to 0.5 times 10 to the power of two minus two,
which is equal to 0.5 percent. So, this becomes your pre-specified tolerance. And
it also relates that hey if you are absolutely relative approximate turns out to be less than
0.5 percent, then two significant digits are at least correcting your answer. So, let's suppose,
if I want one significant digit to be correct in my answer, then my absolute relative approximate
needs to be less than 0.5 times 10 to the power of two minus M, which is 5 percent. So, one of the
things which is important to note here is that, if it goes both ways, that if somebody says hey
you want two significant needs to be correct in your answer, then you will say hey the absolute
relative approximate is less than 0.5 percent. However, if your absolute relative approximate
turns out to be less than or equal to 0.5 percent, then you are guaranteed that at least two
significant digits to be at least correct in your answer. So, you should be able to go both ways and
interpret it correctly, so far as this theorem is concerned. And that is the end of this segment.