So, now first lecture on Numbers; we looked
at natural numbers and integers. So, now, let see what happens when we try
to divide. So, let us look at the rational numbers. So, we said that we cannot represent 19 divided
by 5 as an integer because we cannot find a number k such that 5 times k is 19. So, as we know the way we deal with this is
to represent this quantity as a fraction. So, we say that 19 divided by 5 is 3 and four-fifths. So, this number 3 and four-fifths is an example
of a rational number. So, rational number what we usually called
fractions in school, a rational number is something that can be written as p divided
by q; where, p and q are both integers. So, as you probably remember from school,
the number on the top is called the numerator. So, p divided by q; p is called the numerator
and q is called the denominator . So, just like we had the symbols n and z to
represent the natural numbers and the integers, we have a special symbol which is somewhat
unusual which is a Q . So, Q stands for the rational numbers and again, to just say it
is a special Q, we write these double lines on sides. So, this Q with these fat boundaries denotes
the rational numbers. So, one thing about the rational numbers is
that the same number can be written in many different ways. Now, this is not true of integers. Of course, we are not talking about changing
base from binary to decimal or something. But if you write a 7, there is only one way
to write 7 fix, if you are fix the notation that you are using for writing numbers. With rational numbers, this is not true because
there are many ways of writing p by q such that p by q is actually a same number. So, for instance if we take the number 3 by
5, then we all know that 3 by 5 is the same as 6 by 10 and this is the same as 30 by 50. So, when we take a rational number and multiply
it by something the same quantity on the top and the bottom, so, 3 by 2, 3 times 2 and
5 times 2, we get the same number; 6 by 10 or 3 times 10 and 5 times 10, we get the same
number 30 by 50. So, this is sometimes a nuisance, but it is
also sometimes useful. Now, there is no reasonable way to compare
two numbers like say 3 by 5 and 3 by 4 or 2 by 5 and 3 by 4. If we have two fractions which have different
denominators, there is no way to directly compare them. So, the only way to compare them is to somehow
convert them into equivalent fractions such that they have the same denominator. So, the usual way is just to find a number
such that both the denominators multiply into that number are factors of that number. Now, you can find the smallest such number
which is called the least common multiple; but you can find any number of this form. So, for instance, if you want to add 3 by
5 and 3 by 4, now you cannot do that directly; but you know that 20 is a number which divides
which which is a multiple of both 5 and 4. So, you can represent 3 by 5 as equivalently
as 12 by 20; you can represent 3 by 4 equivalent. So, this is equivalent and this is equivalent. So, you have converted these numbers into
a different fraction of the same number; but this new representation has the same denominator. And now once, the two denominator that the
same, you can add the numerators and you can get 12 plus 15 is 27 by 20. So, this kind of manipulation requires the
denominators to be the same and therefore, its actually extremely useful that we can
write the same rational number in many different ways. The same is to we want to compare two numbers. If we want to check whether 3 by 5 is bigger
or smaller than 3 by 4, there is no way to do it directly. What we have to do is again take the denominators
and make them the same and then, say that 12 by 20 is less than 15 by 20 because you
are dividing something 20 parts and you are taking 12 of them, that is less than taking
15. Now, as I said there is no reason why this
must be the smallest one. So, for instance you could take a bigger number
like 100 right. So, 5 goes into 100 and 5 goes 4 also goes
into 100. So, we could also say that 3 by 5 is the same
as 60 by 100 and 4, 3 by 4 is the same as 75 by 100 and therefore, since 60 is less
than 75; 60 by 100 is less than 75 by 100 and therefore, 3 by 5 is less than 3 by 4
. So, it is not really important that the denominator is the smallest common multiple
of the two denominators; but it must be some common multiple so that you can bring it all
to a common number that you can then compare. So, we saw that representation is not unique
for rational numbers. So, how do we find actually the best way to
represent a rational number? So, normally if you are not using it for some
arithmetic operation of some comparison, we would prefer to have it in a reduced form. So, the reduced form of a rational number
is one, where there are no common multiples between the common factors between the top
and the bottom. So, p by q is of the form, where we cannot
find any factor f such that f divides p and f divides q. So, for instance, if we take 18 by 60, then
its reduced form will be 3 by 10. Notice that 3 is of the form 3 times 1 and
10 is of the form 5 times 2 times 1. So, therefore, there is no common factor between
the top and the bottom and therefore, this is in reduced form. So, this is called the greatest common divisor
problem. So, we want to find the largest number which
divides both the top and the bottom; both the numerator and the denominator, divide
them both by this and then come to something in the reduced form. So, in this case, what we are saying is that
the gcd of 18 and 60 is actually 6 and we can do this using our prime factorization
that we talked about before. So, if we look at prime factorization for
18, then 18 is 2 times 3 times 3 right; its 2 times 3 is 6 times 3 is 18 and the prime
factorization of 60 is 2 times 2 times 3 times 5; its 4 into 3, 12 into 5 . So, now, you
can look at what are common. So, we have one 2 here and one 2 here. So, we can say that this is part of the same
factor, we have one 3 here and another 3 there. The second 2 is not present in the first term. So, we have a 2 and 3 and 18 which are factors. We have a 2 and 3 in 60 which are factors
and this gives us the fact that 6 is a common factor. There is no bigger common factor because we
want to assemble a bigger common factor, we have to pull out one more prime from each
side; but there is no prime left which is present on both sides. 3 is there in 18; 2 and 5 are there on 60,
but we do not have a matching one of the other side right. So, this way, the common prime factors are
one 2 and one 3 and so, 2 times 3 equal to 6 is the gcd. Now, this is not the best way to find the
gcd, there are more efficient ways to find the gcd. But this intuitively tells us what the gcd
is. You take the prime factorization of both the
numbers and you collect together all the primes that occur in both the numbers, the same number
of times. So, here is another interesting property about
rational numbers. Now, for each integer, we know intuitively
that there is something which is the next integer and the previous integer. If I tell you 22 and ask you what is the next
integer? Then, you will know it is 23. What is the previous one? It will be 21. So, for every integer m, the next one is m
plus 1 and the previous one is m minus 1 and it does not matter, if this is positive or
negative. So, for instance if I am at 17, then the next
integer is 18, the previous one is 16 right. If I am at minus 1, then the next integer
is 0 and the previous integer is minus 2. So, I can always take the integer that I am
at, add 1 and get the next integer, subtract 1 you will get the previous integer. So, the property of this next and previous
is that there is nothing in between right. So, there is no integer between m and m plus
1, there is no integer between m and m minus 1. So, that is what next means, it is not some
bigger integer or some smaller integer. It is the immediate neighbour in the integer
of the in this number line. Now, what about rationals? Is it possible to talk about the next and
the previous rational number? Now, it turns out that this is not possible
for a very simple reason . So, between any two rationals, we can always
find another one because we can always take the average of 2 numbers. So, remember that if you take the average
of any 2 numbers, then it must be between those 2 numbers right because it is the sum
of the numbers divided by 2. So, the average cannot be smaller than both
or cannot be bigger than both. So, if the 2 numbers are not the same, then
it must lie strictly between them. If the numbers are the same, then the average
is the same. So, if somebody has 37 marks and 37 marks,
then their average marks is 37. But if they have 37 marks and 52 marks, even
without calculating the average, you know that their average is bigger than 37, but
smaller than 52 right. So, in the same way, if I give you 2 fractions
m by n and p by q and I tell you that m by n is smaller than p by q. Remember that in order to do this, we would
have to normally get the denominators to be the same and so on. But supposing I know that m by n is smaller
than p by q. So, I know that say m by n is here and I know
that say p by q is here and supposing you claim that m by n and p by q are adjacent,
that is p by q is the next rational after m by n. Well, I will say no; let me take these 2 numbers
and find its average right. So, this average now is also a rational number
because you can also represent it as another p; p or a by b right. If you just workout this m by n plus p by
q divided by 2, you can simplify this whole expression and you will get a new number which
is also of the form a by b. So, this is also a rational number and this
rational number as we argued must be between the 2 numbers and therefore, between any 2
rational numbers by just taking the average of the mean of the 2 numbers, I can find another
one . So, in other words, the rational numbers are
dense right. So, dense in the usual sense, so dense just
means that they are closely packed together. So, so, basically you cannot find any gaps
in the rational numbers because any between any 2 rational numbers, you will find another
rational number and this is not true of the integers because we saw that in the number
line, there is a gap between m and m plus 1, there is no integer there right. So, we say that the rational numbers are dense
and conversely, we say that the integers and the natural numbers are discrete. So, a discrete set has this kind of next property
and a dense set has no next property between any 2 numbers, will find another number right. To summarize, we use this funny symbol Q to
denote the rational numbers and a rational number is just the ratio. So, that is where it comes from actually;
so, ratio. So, rational number comes from the word ratio
and so, it is a ratio of 2 integers p divided by q. Now, there is no unique representation of
a rational number because we can multiply both the numerator and the denominator by
the same quantity and get a new rational number which is exactly the same in terms of the
quantity that it represents. And we use this fact for things like arithmetic
and comparisons, but if we really want to talk about rational numbers in a canonical
way, in a unique way; then, we get this reduced form, where we cancel out the common factors
using prime factorization. So, that we get a number whose gcd of the
numerator and the denominator is 1. And finally, we saw that we cannot talk about
the next or the previous rational number because between any 2 rational numbers, there is another
rational number. In particular, if you take the average of
the 2 numbers, you will find a number that is in between. So, unlike the integers and the natural numbers
which are discrete for which next and previous makes sense; for the rational numbers, there
is no such quantity.