We had seen that numbers which are not rational
are called irrationals and further
irrational numbers
after the of decimal the digits keep on
repeating
they do not terminate so they are non
terminating decimals,
and also non recurring, that is the digits
after the decimal
do not repeat or recur in a particular
sequence.
So this is irrational numbers which are 
non terminating and non recurring .
Now how do I plot these irrational numbers
on a number line
that is the question. We had seen
integers on a number line
between any two integers we had seen fractions
that is the rational numbers which are
integers plus fractions.
Now how do I plot these irrational numbers?
Where do they lie on the number line?
So let's start with irrational number root two
we want to plot under root two
that is we know that is an irrational number.
Now let's take a unit distance here and a
unit distance vertical.
So one ,this is one, a unit distance and
this is also one.
Now if I join these two lines
I get a triangle, not just that
I get a right angled triangle
horizontal vertical they are at right
angles and I get a triangle. So I know I have
a right angled triangle
Now as soon as I have a right angled triangle 
what strikes me is the Pythagoras theorem.
So I can apply pythagoras theorem here,
let's say that this is 'l'.
the length of this purple line here is 'l'.
So applying pythagoras theorem ,
I can say that this is the hypotenuse
opposite to the right angle, then 'l' square
would be equal to the sum
of squares of the other two sides, this
is one,
this is also one,so it would be
one square plus one square
so 'l' square turns out to be one square
plus one ,which is two
taking positive square root
I get 'l' as under root two.Why positive,
because this
length, length has to be positive 
we can ignore the negative
square root, so I get that 
this length this purple
side is of length under root two.
so no how do I plot under root two on
the number line
I know this is under root two distance,
this point from this point, but
I need to find under root two distance on
the number line.
this point does not lie on the number
line so what do I do
I take this as my centre,
I take under root two as my radius
and I draw  a circle around it. With this as
the centre, that is zero as the centre
and root two as radius.
So if I draw a circle I get this.
Now I know that this line which is under
root two, which is nothing but the radius
of the circle
if I take any other point on this
Lets say I take this point on the circle,
now I know that the radius
has to be the same, so if I look at
this point and see its distance from
zero
this should also under root two ,because
this is nothing but the radius of the
circle
this is this centre, a point on the circle
so that means any point on this circle
would give me under root two length from zero
that is the radius. So now let's look at
this point
I wanted a point which lies on the
number line,
let's look at this point and similarly
this point
from zero
in front of zero ,this is  behind zero
so again this point would be what
this point which I get would be under root two
this point
Why?  because it's again radius
under root two, so this under root two
is actually the length
which is from 0 to this point,so this is
that radius of this circle,
so this is under root two.
Similarly you can say that this point would
be minus
under root two.
negative of under root two
so which is this length, distance is the
same, length is the same
what changes is the ,just the direction
magnitude is the same
direction change is a negative number,
you get a negative number. So this is how you
can plot under root two on the number line.
So I get under root two on the number line
which lies between one and
two. I get under root two.
now can we find under root three?
so I know this is under root two
again I take one vertically,distance one
this is under root 2, now completing the triangle again,
I get that this is a right
angled triangle.
This is a right angled triangle, this is
the hypotenuse, I want to find this
distance
again applying Pythagoras theorem,
I can find the length of this purple
line, which is 'l',
so 'l' would be what, 'l' square would be
hypotenuse square is sum of the other
two sides
sum of squares of other two sides
so this would be
Under root two square
plus, this is under root two, this is one
so,what I get is under root two square
is number two
plus 1. This is 3
'l' square is three,taking positive square root,
I get 'l' as under root 3.
'l' cannot  be negative, so I ignore the negative
square root, I get 'l' as under root 3.
So this is what I get, now again I need
to find this number on the number line,
So what I do is I take zero as the centre
under root 3, this purple line as the radius
and I draw a circle around it.
now again any point on this circle
would be
under root 3 distance away from zero
so I mark this point.
So this would be under root 3,
this point would under root 3,that
is the distance
from zero till this point would be
under root 3
and that's how I can mark under root 3
on the number line.
So that how I get under root 3 also on the number line
similarly what you
can do is
if you take this same distance on the
opposite side
that is this point,
you can also find negative
of under root 3, same distance but in the
opposite direction
so that so how can find
under root 3.Now can you locate under root 5
on the number line? under root 2 we found,
under root 3 we know,under root
4 is nothing but number two.Now what
would
under root 5 be, where would that
be lying on the number line?
So let's see
let's take the unit distance two now this
two unit distances
this is two, this is one, one unit
distance vertically
now again, if I complete this triangle
this would be a right angled triangle, where this line
is nothing but the
hypotenuse
opposite to the right angle and I can find the
hypotenuse applying the
Pythagoras theorem in a right angled triangle.
So by applying pythagoras theorem, if this is 'l',
I can say that 'l' square, hypotenuse square
would be sum of squares of the other
two sides
this side is of length two. one plus one
so 2 square plus, this side is
one
one square
so I get 4+1,
which is nothing but 5,so 'l' square turns out to be 5,
what would 'l' be?
since 'l' is the length , 
I will take the positive square root,
length cannot be negative so 'l' would be
positive square root 5,
 so I get 'l' as
under root 5.
So this length is root 5,that is the
distance of this point from zero
is root 5.Now I need to find this
point on the number line ,where would it
lie on the number line?
so again taking zero as the centre and this
length root 5 as my radius, 
I draw a circle,
you see we can draw a circle, now the point
where it touches the number line
this gives me under root 5
and in the other direction I get
negative of root 5, so you can see it 
touches the circle at two points,
this circle touches this line at
two points
which gives me under root 5 and
negative of under root
five.
So this point is nothing but under root five
now you can see
drawing this  circle takes a whole lot of
space so instead of this
drawing this circle every time, all you can
do is
with this as centre and this as radius
just draw a part of this circle, an arc
which cuts the line,
the number line at a point. So what
I do is
instead of drawing the whole circle,
with this
as centre and this as radius, I draw a small arc
that cuts the number line.
Now I know where the point, where this
arc meets the number line gives me the
point root 5
that is the radius, the length from zero
to this point is root 5, which gives
me the position of under root 5
on the number line.
So we've seen how to plot under root 2
under root 3 and under root 5
on a number line. You can plot other
irrationals too following the
same method.
