Finding Square Root Using Repeated Subtraction - Squares & Square Roots

okay now in this video let us look at
how do we find out the square roots of a
number so we already saw a couple of
methods of finding squares now we will
look at how to find square roots and of
the different approaches of a label to
find a square root we will be looking at
three different methods the first method
is that of repeated subtraction and I
will go into the details of this method
in this video in the next two videos we
are going to cover the other two methods
that is the prime factorization which we
already learned in our earlier classes
the method of prime factorization but
will apply the concept of prime
factorization to finding square roots
and then we will also look at the
division method so now let us look at
finding square root using repeated
subtraction
now let us say that we are given a
number 25 and I know it's
straightforward you can just do it 5
times 5 is 25 and you could do it
verbally but I would like to explain the
method and therefore I am taking a
simple number so how do we use this
repeated subtraction method what
essentially we are going to do is to use
the property of squares being
represented as the sum of an odd number
so let's say that you have a number n
okay and you want to find out the square
all you do is you add the first n odd
natural numbers to get the square ok it
actually gives you square of n that's
what so we are going to use this
property which means that if we start
subtracting or if we subtract the first
n odd natural numbers then we should be
able to arrive at the square of this 25
so let me explain that now what are the
first n odd number so let's start from
you know we start subtracting the odd
numbers one by one so let's say 25 minus
the first odd number is 1 so this gives
us 24 now this is our first subtraction
now the second subtraction would be with
this next odd number that is 24 minus 3
what does it give it gives us 21 now let
us subtract with the third odd number
that is 21 so this result needs to be
subtracted with the next odd number that
is 5 or 21 minus 5 would give you 16 ok
now let us continue this further and we
I would say that I would like to
subtract 16 minus the next odd number is
7 so in this case it is going to be 9
right and then the next odd number is
actually 9
9 and you get zero so as you can see how
many times we have subtracted so we have
subtracted five times and the odd
numbers that we have is like one three
five seven and nine so one plus three
plus five plus seven plus nine equals
twenty five and you can see that this is
the sum of first n odd natural numbers
that is five right
hence the square root of 25 is 5 and you
can do this for other numbers just take
up a number let us say 81 pause the
video take this number 81 and try to
find the square root of this number
using this repeated subtraction so all
you do is you start subtracting this
number by one first then you subtract
the result by the next odd number which
is three then by five by seven by nine
and you keep on doing the subtraction
till you get a zero that's what you need
to do now what you think is a problem
with this method though it is simple the
problem with this method is imagine what
happens if you are given let us say a
number which is big for example let's
take 625 now this is a huge number and I
can recollect from my memory that it's
the square of 25 the square root of 625
will be 25 however if you were to do 25
repeated subtractions from 625 so you
start with 625 minus 1 then you get 624
then you say 624 minus 3 and so on and
on and you have to do this 25 times so
the fundamental problem here is that as
you get a big square number applying
this repeated subtraction method becomes
a big pain and therefore we should be
looking at another method ok so in this
video we learnt about what the repeated
subtraction method is and what are the
challenges that we face with the
repeated subtraction method in the next
video with the
look at the prime factorization method