In this segment we’ll talk about round off errors.
So, in order to talk about round off errors uh what we need to do is we need to talk about
where the errors come from in computation. So, we talk about the errors which are inherent in
numerical methods; so, the two types of errors which are inherent in numerical
methods one is the round off error and another one is the truncation error.
And it's important that we understand the distinction between the two uh because sometimes
they get conflated with each other: so round off error comes from, caused by, it’s caused by
approximating numbers. Because if we cannot take a number and represent it exactly it will have to be
represented by an approximate number; that's where the round off error comes from. Truncation error,
which is a separate lesson, comes from approximating mathematical processes.
So, it's important for you to understand that hey we have round off error here which
is caused by simply by approximating numbers. So, round off error is defined as actual number
and from that you subtract the representation
of number. So, a good example of that is one-third; let's suppose one-third we know that
if you write in decimal format, it will be written like this. Now if we say hey, we want to represent
one-third with only six significant digits let's suppose then this will be the approximation of
one third so if that's how it is represented then the round of error will be one-third minus
this number right here. And that number will turn out to be six zeroes right here
and then we'll have three, three, three recurring from there. So that's how the round-off error is
taking place. Now round of error also takes place in irrational numbers just pi, square root of two,
and so on and so forth as well. That's to be thought about. Also, what you got to think
about is that since numbers are not represented in base 10 in the computer, they are represented
in base 2. So, let’s suppose I have 0.3 in base 10; this number here will have infinite number of
zeros and ones zero all ones after the radix point right here. So, what that means is since
the computer represents in base 2 and in base 2 it will require infinite number of
digits after the radix point you can very well see then since the computer cannot have
infinite number of bits to represent the number there is going to be a difference between how
0.3 is represented actually in the computer so there's going to be a difference and that will
be the round off error also. Also, for round off error you got to think about two things:
are you chopping a number or are you rounding, uh, the number. So that's
something which you will see in later lessons as well. Chopping is where, for example, if you
have a number like 0.3378 and somebody says hey go ahead and write it up to three significant digits,
you will write 0.333, that is chopping. If somebody says hey go ahead and use rounding,
you will say 0.334 because this digit here is 5 or more. So, that's why you will there's a difference
between these two things the chopping rounding but the error which is caused by chopping or rounding
is so called the round off error and that's the end of this segment