 Hello folks today, we are going to understand how do we represent irrational numbers on a number line? Okay, so before we start we first Understand what is a number line? So in your previous grades you would have studied that if I have a line I Can I can represent all the numbers on a number line, right? Let us say this is a line and we can represent all the numbers known to us on a number line So far we have dealt with natural numbers whole numbers integers and Raster numbers is it it so we will see how do we represent these numbers? so how do you start to draw a line and Then at any point let us say this point is Point zero right and then what you need to do is you just take one unit length Let us say one centimeter or half a centimeter as a unit and then graduate this line or Divide these lines into equal parts like this like that. Okay, so Actually, it is done through a ruler and a Pencil or a compass right so hence you can draw a straight line Then you can mark the point zero and from there both on the right-hand side and on the left-hand side You can divide the lines into small small parts such that each part is of same length So hence all these points which I have drawn represent one two three Four and so on and so forth right all these points are Integers here also it is minus one minus two minus three and so on and so forth Isn't it now if I have to represent point five so where would point five be so, you know, it's very Easy point five is nothing but between zero and one exactly at the middle point So this is this might be point five and how do I? Geometrically do this so far for that what you need to do is you have to take The needle of the compass on any of these points Let us say this point and from there you roughly measure more than half the distance between These two point okay, and then from here here you draw an arc Similarly from this point you draw another arc with the same radius and you repeat the exercise on the Bottom side as well and then you Connect, you know, this is called bisecting the line segment right so wherever this dotted line meets the number line Let us say this point will be nothing, but this is zero point Five is it it now? Let us say if I have to represent One upon three one upon three. How do I represent? So let me draw the Number line once again and this time around I'll take a bigger scale. So let us say this is my Number line and this is zero and for the convenience sake, let us say this is one Okay, and I have to draw one by three So you know how to do it. So you simply draw an inclined line on zero and then you join whatever the end point is Here to here. Let us say and you have to you have to draw this O Let's say P in such a way that you can divide this line into three parts So let us say you take OP as six centimeter Why because six is a multiple of three so P is equal to six centimeter you take Then what you do is you? mark Points like this so that each division is of two units. So like that Let us say you have In you using a scale or a compass you can you can Mark two points such that this distance is to this distance is to and this distance is to as well right then what you need to do is and Once I will finish the construction, then I'll explain how and why we are doing it now Let us say this point is a now you join PA and And from all these points, let us say this is P1 and P2 you draw P1 a1 Parallel to PA and P2 a2 parallel to PA again, so if you see these are nothing but equal parts of Equal parts of OA okay divided into three parts. So hence this point will be nothing but one upon Three this is how you can find out a fraction between two points Or let's say between zero and one okay one upon three. So basically I divided this segment OA this segment OA into three equal parts using the construction So the first part is one by three. So automatically if you see this second part will be Two by three. So this part is two by three this point is two by three. Okay. This is how rational numbers are plot on a Number line. Let us now move on how to plot Irrational numbers, you know irrational numbers are nothing but numbers of the form of sorry numbers Which cannot be expressed in the form of P by Q numbers which cannot be expressed in the form of P by Q Is it it so let us say I have to represent root 2 root 2 on the number line root 2 is a irrational number. This is an Irrational number isn't it? So how do we represent root 2? on a Number line. So please note that a number line Contains all the real numbers hence it is also called a Real line we can represent all the numbers on a real line now it starts with let's say zero all integers Anyways, we know that they can be very well expressed on a real real line similarly all Rational numbers also we just saw in the previous case. So this is one two. Let us say three This is minus one minus two minus three. So if you see all the points all the point On the real number line will represent a particular number whether it is rational or irrational We just saw how to find out half one by three on the number line now in this case We are going to learn how to plot root 2 on the real line Obviously, it will be very difficult to do with a ruler and a scale even if you know the Approximate value of root 2 so hence we use geometry to solve this or to find out where exactly root 2 Will be lying on this number line I hope you know that all these numbers are nothing but all these numbers are represented by Points on the number line. So let us say root 2. So we will use Pythagoras theorem to find out root 2 on the on the Number line how so let us say I have to find out root 2. So I will take this one unit here From 0 to 1 and then I will draw a perpendicular On one such that let us say this point is o this point is a and this point is b we have to make sure that oh a is one and a b a b is One okay. Now you join ob Okay, so what do I know about ob so Basically ob square is equal to oh a square Plus a b square isn't it? Why because of Pythagoras theorem. So hence if it is one square, sorry, this is one hence if it is one If it is one then this is nothing but root of one square plus one square, isn't it? Which is nothing but root of 2 isn't it? So hence we say we say equal to one square plus one square and by geometry Ob is equal to root of 2 right now you take the needle of the compass and put it on oh and Take the radius equal to ob and try to draw an arc such that the circle the radius Oh, sorry the arc cuts cuts the number line at let us say point p Okay, so since ob is equal to root 2 and we are drawing a circle on an arc with radius ob So op also is the radius of the same arc. So this is also root 2 So hence eventually we are able to locate where root 2 will be on the number line. Okay, once again, so you draw Segment of one unit draw a perpendicular segment of one unit so that the hypotenuse becomes root 2 and hence if you draw the arc cutting the Number line then the point where it meets the arc meets the number line is the irrational number 2 Let us now say we find out we want to find out root 3 where exactly root 3 will lie So again, I have to use the Pythagoras theorem. So let us say this is 0 Okay, and you draw you draw And we know that these are the points 0 1 2 on the number line We just found out root 2 how to find out root 2 you draw one unit perpendicular on one then join these two points and Take the arc. So wherever the arc cuts is the root 2 value now This this length is root 2. Is it it now? Let us say we want to we just put another A perpendicular over root 2 point and now join these two points. Let us name them So let us say this is a this was B. This was P and this was Q so clearly if you see OP is equal to root 2 we just found out and PQ is equal to 1 and this is 90 degrees So OQ square will be OP square plus PQ square Now OQ square is equal to OP square now OP square is nothing but root 2 square plus PQ is 1 Right, so it is nothing but 3. Isn't it? So what will OQ be? OQ is root of 3 Right, so if OQ is 3 we'll repeat the process and let us say you draw this Arc like that. Okay, so this point is Root 3. So this is how you can locate any irrational number on The number line will take up some more example in the problem solving sessions