[Music] so we started with the natural numbers and the integers and then we moved on to the rational numbers which are defined as P by Q where P and Q are both integers so we decided that the rational numbers are dense right and that means that on this number line between any two rationals you can find a rational so if I want to now talk about this number line then I know that if I take any two positions then I'll find the ration between them and everyone rational between there and so on so it makes sense to ask this question which is that if I take any two points in the rational between them any two points then is this entire number line composed only of rational numbers of course some of those rational numbers are integers so an integer is a rational number because I can write 7 for instance as 7 by 1 right so this is of the form P by Q so any rational number which in reduced form as denominator 1 is an integer so an integer is a special case of a rational number so do all the rational numbers fill up this number line that is the question so it turns out this is not the case so remember that a square of a number is the number multiplied by self so if I take a number m and multiply it by itself I get M Squared which is M times M and if I take this operation and turn it around then the square root of a number is that number R such that R times R is equal to M right so I want to find out which number I have to square in order to get M and that's called the square root so if we take the so-called perfect squares like 1 4 9 16 25 and so on their square roots are integers so 1 squared is 1 so the square root of 1 is 1 2 squared is 4 so the square root of 4 is 2 5 squared is 25 so square root of 5 is 25 is 5 16 squared is 256 so square root of 256 is 16 and so on so some integers are clearly squares of other integers and so you can get the square root and find any teacher now what happens if something is not a square right so supposing I take a number which is not a square like 10 and I take its square root I know that the square root is not an integer it's somewhere between 3 & 4 because 3 squared is 9 and 4 squared is 16 question is is it a rational number or not so what happens to the square roots of integers that are not perfect squares so the smallest such number which is not a perfect square because one remember is a perfect square one times one is one the smallest such number that is not a perfect square is actually two and it is one of the very old results that the square root of two cannot be written as P by Q this was certainly known to the ancient Greeks in fact to Pythagoras and one way to do this is to see that you can actually draw a line of so this is not an unreal number in that sense right so you can actually draw a line of this length because if you take a square whose sides are 1 right so this is 1 then if you remember your Pythagoras's theorem then the hypotenuse of this triangle is going to be square root of 1 + 1 1 squared plus 1 squared technically which is square root of 2 so I can actually physically draw a line whose length is square root of 2 so this is a very real quantity on the other hand for reasons that we will not describe here but there will be a separate lecture explaining this for if you are interested square root of 2 cannot be written as a rational number P by Q so here is a number which is a very measurable quantity I can actually draw this quantity as a length at the same time it does not fit into this number line of rational numbers which seems to cover all the rational numbers all the numbers because they are dense so square root of 2 since it is not a rational number and yet it exists is called an irrational number and these numbers which constitute all the rational numbers and the real irrational numbers together are called the real numbers so the real numbers are denoted by this double line R so we had n for the natural numbers Z for the integers Q for the rational numbers and now we have the real numbers R so the real numbers extend the rational numbers by these so-called irrational numbers which are very much on the number line but which cannot be written in the form P by Q now it's not difficult to argue that like the rationals the real numbers are dense for the very same reason because if you have two real numbers R and R prime such that R is smaller than R Prime then you can just take their average R plus R prime divided by two this must be a number which is bigger than R and it is smaller than R Prime and therefore it must lie between them so between any two real numbers you will find another real number so the real numbers are also dense so there are some irrational numbers which we use a lot in mathematics and which you have probably come across one of them is this famous number pi which comes when we are talking about circles because it is the ratio of the circumference to the diameter okay and this is an invariant pi is always the diameter the circumference divided by the diameter for any circle is pi okay so pi is an irrational number we cannot write it in the form P by Q and it has this if we write it in this decimal form it has this infinite decimal expansion another number which is very popular as an irrational number is this number E which is used for natural logarithms so it is two point seven one eight two eight one eight and so on right so there are a lot of rational numbers so square root of two as we have seen as a rational number it will turn out that square root of anything square root of three is also an irrational number square root of six is also an irrational number anything which is not a perfect square its square root is actually in irrational number but many of these numbers are not very useful to us but PI and E are certainly very useful irrational numbers so now we have seen that we can find more numbers on the night line than just the rationals and these are the real numbers so do we stop here well let's look at the square root operation which we use in order to claim that there are irrational numbers so what happens if we now take the square root of a negative number like minus 1 so remember that we had a sign rule for multiplication the sign rule for multiplication said that if I multiply any two numbers then if the two signs are the same that is there two negative signs or two positive signs I will get a positive sign and the answer only if the two signs are different if I have one minus sign in one plus sign will I get a negative answer so if I want to multiply two numbers and get a minus one one of them must be negative and one must not be negative but by definition a square root is a number which is multiplied by itself the same number has to be multiplied by itself so it will have the same sign so any square root which multiplies by itself must give me a positive number so if I take a negative number there is no way to find a square root for it so if we want to find square roots for negative numbers we have to create yet another class of numbers called complex numbers so complex numbers extend the real numbers just like real numbers extend the rational numbers and rational numbers extend the integers and so on but the good news for you is that we don't have to look at complex numbers for this course so to summarize real numbers extend the rational numbers by adding the so-called irrational numbers which cannot be represented of the form P by Q and a typical example of an irrational number is the square root of an integer that is not a perfect square so square root of 2 for example is not a rational number and this is also the case was square root of 3 square root of 5 square root of 6 and so on so except for the perfect squares none of the square roots are actually rational numbers now just like we said that the rational numbers are dense because the average of any two rational numbers is a rational number similarly the real numbers are dense because the average of any two real numbers is a real number so we have a progression in terms of numbers so every natural number that we started with is also an integer because the integers extend the natural numbers with negative quantities now every integer is also a rational number because we can think of every integer as a ratio P by Q where the denominator is 1 and finally every rational number is a real number because we said that the real numbers include all the rational numbers plus all the irrational numbers and finally we said that there are even things beyond rational numbers like complex numbers but we won't discuss them