so welcome to my course on calculus of one
variables those who are taken my course on
basic calculus they have learned both one
variables and two variables but too much thing
was compressed into little time so i decided
to make the things much more detailed much
more relaxed today we are going to start speaking
about numbers you might think that why you
need to speak about numbers numbers also fundamental
we use it every day to buy our groceries to
pay for a tickets to to look at the cricket
score everything is done through numbers so
human society thrice because of the existence
of numbers in fact a very famous german mathematician
whose name is leopold kronecker he once made
a very famous statement the statement is that
god gave us the natural numbers that is one
two three four five six seven eight nine and
then everything is mans hand evolved once
you have that you can do everything
but dont ever think that numbers is something
which you didnt really not bothered about
it so common of course you know about this
basic numbers so it so obvious that i dont
need to explain to you why and how work this
now the importance of the numbers is a following
these numbers introduce into our study what
we call abstraction so what does abstraction
mean abstraction means using a single symbol
to represent different things for example
if i talk about say two oranges and i say
about two elephants so this same two is representing
that quantity two the same symbol so it doesnt
matter whether it is orange or whether it
is an whether it is elephants so this is something
we have to keep in mind that it brings in
the notion or abstraction however let us look
at from modern point of view from modern point
of view the mathematical language is the language
of set theory set theory you must have learnt
at school those these are somethings which
contains objects dont try to define sets as
well defined collection of objects and all
those things because that will get into logical
problems so you know and i even know what
a set means so you always had some say number
of one set is set of your friends or set of
your classmates set of all say cards in india
or set of all potholes in the cities city
roads
so ah set theory or the theory of sets was
first introduced and studied a somebody called
george cantor and in his language the set
of natural numbers is put into a set n symbol
as n so this set consist of all the natural
numbers that we have so this is called the
set of natural numbers this called the set
of natural numbers now if you observe very
carefully this one two three four five six
seven eight all this keeps on going it doesnt
end and this is the first time you are possibly
you see in your very basic thing of life one
two three four five six seven eight nine ten
dot dot dot dot dot you are coming face to
face with the infinite thus this is i think
your first example that you get as you taught
in mathematics is of example of an infinite
set the set whose elements are infinite not
finite so this is not the kind of set that
you have studied really in school and George
cantor really bothered about sets of this
kind his teacher leopold kronecker who leopold
kronecker supervise a thesis of george cantor
but unfortunately his studies of taking having
a headlong crash with infinite with the infinite
was not really ah appreciated by his teacher
for his teacher infinity was just a notional
thing where you take limits and all sorts
of things which we will soon learned but to
him to george cantor infinity was a real thing
and we will see something in the next classes
also
so this is the set of natural number some
people try to define what is called set of
whole numbers by introducing zero in the set
of natural numbers so i will use this symbols
of union intersection all those things assuming
that you know this basic facts of set theory
from your school days so is this enough for
us the fact is that it is not enough for us
see given an object here i can we can define
something called addition of these two numbers
so if you have two apples in one box three
apples in another box i can bring them together
make them five apples but there is a issue
of subtraction so if i have three apples in
a box can i take away five apples from there
if i ask very stupid possibly a question like
that then is there any answer to this that
gives us idea of negative numbers i fact blaise
pascal one of the very big mathematician so
the seventeen century he seventeenth i guess
ah yeah seventeenth century he made a statement
that how can you think of taking away something
from nothing so but negative numbers are really
your bank balances go in negative many of
many of us have made the bank balances so
in that case negative numbers are the real
so this introduces us to the set of all integers
so the set of al integers contains zero contains
one plus one plus two plus three which is
nothing but one two three just to separate
them from minus one minus two minus three
this is also an infinite set this z this symbol
comes from the word zahl zahl actually means
number in german i am not mistaken possibly
richard dudgeon the german professor had first
introduced this symbol now at your school
level you are also taught about fractions
that if i have single cake and i have to divide
them between four people then i have divide
into four parts and every body gets one fourth
of the whole right if i want to divide them
equally
so fractions are also part of our daily life
which we cannot ignore when we use pure integers
to denote fractions we introduce something
called a rational number very loosely we can
tell that we are talking about fraction but
fraction actually mean something more bigger
its not just ah what we are trying to say
so here we are going to talk about rational
numbers so we essentially trying to talk about
the fractions that we study at school so what
are rational numbers rational numbers are
any number say r which is expressed as the
ratio of two integers p and q where q is not
equal to zero of course you know very well
that division with zero is not permissible
and here p and q both are in the set of integers
z so now we have got our natural numbers or
negative numbers we have got zero we have
got fractions the question is do we need anything
more other than this rational numbers see
this thing that there can be something other
than rational numbers was first came from
geometry it did not come from basic algebraic
manipulations of you take a right angle triangle
and this is the most famous description away
whose both sides are say one centimeter then
applying the pythagoras theorem the hypotenuse
would have a length root two centimeter and
it was one of the pythagoreans you know the
great pythagoras had a clan a secret clan
who had access to knowledge about the world
and they wanted to keep it secret they believe
that everything was natural numbers and fractions
they didnt believe there was something else
so when this came out and one of his students
call who actually proved that root two this
number root two cannot be expressed in the
form p by q so we will show all this things
in the third lecture what ah we are just going
to mention that this root two are shown by
one of the pythagoreans to be not rational
and the pythagoreans were all hast was thrown
in the sea and killed because he made such
a statement that n there is something other
than rational numbers
so what we have obtained this far using the
language of set theory is the following that
we have the natural number set which when
added with zero means the whole number set
then when negative numbers coming you have
the integer set and every integer can be written
as that integer divided by one so all the
integers come in the set q the set q is called
a set of rational numbers which i have not
introduced earlier so there are certain numbers
which are not expressible as p by q for example
root two root two is thus called an irrational
number in fact we will show in the next lecture
that irrational numbers are more in abundance
than rational number so here so any number
which is not rational that is that cannot
be expressed in the form p by q is called
irrational
now is there anything beyond irrational of
course you can imagine that there cannot be
anything beyond irrational so we have the
most obvious things this fractions and standard
numbers negative all all that you can do with
numbers so you would be perfectly happy just
of rational numbers but unfortunately geometry
ah which is also very real thing gives us
something very ah different story because
here you have taken sides whose lengths are
rational numbers one one centimeter so irrational
number is a reality with which mathematicians
have to put up and they play of pretty major
role in mathematics so then if i denote where
c denotes the complement of a set this is
a complement of the set of all rational numbers
is a set of irrational numbers this is what
we have and now we introduce the super number
set which is call the set of real number these
are the numbers which we really need we real
in the sense that this number seem to exist
actually in reality in the sense that ok we
can actually find them even irrationals root
two so r the set of real numbers is nothing
but the union of q with q complement of course
you know q intersection q complement is empty
ah there can may not be a number which is
both rational and irrational
so hence union of these two forms what is
call the rational number set now we will start
talking of properties of rational numbers
and in that we will start first talking about
the natural number set its properties that
thats quite important actually you should
have some knowledge about the set of natural
numbers so we are now going to going to talk
about the properties of natural numbers so
we are started dealing with something call
the infinite set from the very beginning we
have started dealing with the infinite and
the infinite is very much or the reality in
calculus we cant really get out of that so
these natural number sets are are classified
into two different sets call odd sets set
of odd numbers and even numbers which you
know form very basic school mathematics which
i just repeat for your convenience which need
not have bother also with now apart from these
two classification there is a interesting
class of numbers one of the see there are
four arithmetical operations that you carry
on on these numbers one is addition subtraction
multiplication multiplication tables of course
have to do with natural numbers and and you
also carry out division
so when you carry out division which is inverse
of multiplication you come to some interesting
classes of numbers so they are call the prime
numbers and they they are very very important
in the understanding about in if you want
to understand about natural numbers so what
are prime numbers these also taught at school
level any number that is divisible by itself
and one is called a prime number so prime
numbers means divisible by itself and one
so you can say ok i can start with one one
is divisible by itself and one but one is
not taken to be a prime number we will soon
tell you why it is not taken to be a prime
number so our prime number starts with two
then two is the only even prime number and
then it goes to three five seven and so and
so eleven and important fact about natural
numbers is something called unique prime factorization
which i am writing on the side there important
fact is called unique prime factorization
so what does it tell me it tells me that if
you give me any number n you can try out i
can always express it as a product of the
powers of prime factors right means a factor
is a number which divides numbers among the
factors there is some number which have prime
numbers and you can always express it in this
way for example so m can be written as say
if you take p one to a prime number p one
to the power q one say p r to the power q
r you can form of for example sixty sixty
is fifteen into four fifteen is three into
five and four is two square thats it so all
the numbers are prime so and this representation
is unique now suppose i have considered one
to be a prime number then sixty could have
this representation unique also means this
powers are also unique every number has that
so once you do a take one of the prime number
this is what can happen you dont have a unique
prime factorization that is reason why one
is not taken by convention to a prime number
so this is something important and has to
be kept in mind this is a very very powerful
result actually this idea would be used to
show that every every know natural number
has a prime factor would be used to show that
there infinite primes so that that is a very
key result about numbers which has to be understood
and one of the important properties of natural
numbers is that it allows something called
the method of induction so what is the method
of induction method of induction is a following
that if you have a a mathematical statement
involving the symbol m so you have to decide
whether the statement given statement is true
or false
so the method of induction says fist check
is p one is true and then if p n plus one
is true when p n is true then p n holds for
all n holds for all natural numbers this is
a method of induction for example if you take
this very simple statement so my p n is the
following this is what i want to check is
this true so obviously p one is one here and
you can put n equal to one and see the left
side is also one so p one of course satisfies
this so now you take assume that p n is true
hm so what is p n plus one in this case the
statement p n plus one is of course one plus
two plus three plus n plus n plus one this
simply tells me simple fact because i know
that this is true i can use this that up to
n it is true so this simply gives me n plus
two into n plus one by two which is same as
n plus one into n plus one plus one by two
and hence this holds for n plus one also showing
that this result actually holds for all n
so combining these two features or induction
as well as the fact that ah you have a unique
prime factorization that hence we will now
show that we have an infinity of prime so
this result is a very ancient result which
which also present in euclids element a famous
one most famous books in mathematics are all
times now we will not do euclids proof we
will do a very modern proof taken from a book
call the proofs from the book actually a the
great mathematician paul erdos used to think
that god whom we called also supreme fascist
that you have he has a book in which all the
best and most elegant proofs of mathematical
results are kept so we have to figure out
that whether we can do better than the book
or not so it doesnt mean so mathematical research
doesnt always consist of the fact that it
you would do always something new you can
have look look at old things from very new
angle and get new new or insights so here
is our first theorem so because we are doing
mathematics we have to come into the structure
of the subject and our first theorem is that
primes are there are there are an infinity
of primes to do this proof we will take the
help of something call fermats number or fermat
numbers a fermat number is a number so we
are doing a proof so fermat number is a number
of this form for any n fermat number is two
two to the power n plus one you can ask that
who conjure of this strange things that you
would proof like this of course it need guess
and test it has come through many many guessing
and testing the original; proof of euclid
might look much most straight forward compare
to this but this is say something interesting
and so i just want to show you this that what
happens at if you realize that this number
is always an odd number this two to the power
two to the power n any power of two is an
even number so this is an odd number now i
will take two fermat numbers and show that
they are relatively prime that is their greatest
common divisor or hcf if you talk about a
school language is one right so there are
no other prime numbers which are a common
dividers to this and every prime number is
bigger than one so there cannot be no two
no prime number which is a common divisor
to two different fermats number so that is
why they are called relatively prime or co
prime
so if i have a number is that two two numbers
so given of a prime number if it is a divisor
of say a then it cannot be a divisor of b
if a and b are co prime now you can understand
as i keep on changing n because the set of
natural numbers is infinite yeah i have infinite
number of fermats number so this fermats number
is infinite infinite numbers there be infinite
number of fermats numbers while i will show
that give me any pair of fermats number they
are co prime so they are cannot be a given
common prime which is a factor to both of
them so for each of the fermats number there
should be different primes see fermats number
infinite so are the number of primes that
is the whole idea ok in order to do this we
really use this very important recursive relation
we will show it by induction actually and
n of course is greater than equal to one now
assume that so fk so here k is of course strictly
less than n this k is one zero to n minus
one so let q be a divisor of fk and fn so
you are talking about common divisors now
if it is so if if it divides all the fks and
fn so this is the common divisor right so
we have a so so for some k q has a divisor
when a q is a divisor of fk and also of fn
so what would happen here i have a product
of numbers
so if o divide this side it is completely
divisible because q is dividing but then this
side should also be completely divisible then
the in equality make sense which means f because
fn is also divisible by q which means two
also has to be divisible by q which means
q this would imply let q is equal to one or
q is equal to two right now q cannot be two
because if q is two if q but an q divides
fk then fk must be even two only divides even
number two doesnt divide odd number so q cannot
be two this is a conclusion since f k is odd
which means what q is equal to one so only
common divisor possible between any fk and
fn here is one so for any pair you take k
less than n for any pair fk fn that you chose
or any fk one k two where my for any pair
fk fn that you chose doesnt matter now you
can do the same thing if you chose any k less
than n you can do the same sort of induction
same sort of recursiveness same structure
with some thing lesser so whatever k you chose
from here the only common factor is one so
then which means we have proved that for any
two pairs of fermat numbers they are relatively
prime and hence it shows as we have discussed
that prime numbers are infinite you see this
is very beautiful and simple proof and now
by induction we can show that this is actually
true
so here instead of n minus one i have to add
one here so i come to this simple fact k equal
to zero to n minus one so once you do this
i would request you as this all given in the
notes i am not going to do the proof that
put in the put in fn what is what is the structure
of fn you know that this fn is given as this
one so use that and finally prove that this
is nothing but f of n plus one minus two that
is what you can prove so now we are coming
more into rational number things how do we
represent rational numbers there is an interesting
way of representing rational numbers call
decimal representation so decimal representation
means i am looking at multiples of ten right
so or division through multiples of tens so
that when when we give representation through
multiples of ten that gives raise to decimal
representation of numbers
now so any rational number or any fraction
if you want to be this more lose fraction
is actually more broader term than rational
number because i will tell you later why once
we introduce functions you see if you take
a rational number say p by q then if you multiply
them by any non zero number or then p by q
is same as r p by rq this is something you
have to understand now decimal representation
means that you can represent there is something
called a decimal point and which is a i am
sorry there is something called decimal point
and on the right hand side of the decimal
point there will be some chain of numbers
so decimal representation is important because
it shows if the number is between zero and
one the left hand side of the decimal point
where you have the whole numbers actually
that part would be zero so you have already
done some decimal ah points called trying
to learn division in school but we will talk
about something so in more detail so rational
numbers are two types so you take the set
q there two types one which is represented
by terminating decimals another is representing
by infinite decimal that is their own terminate
and decimal points are as you keep on dividing
of bigger number ah smaller number by a bigger
number youre the there on the right hand side
you will keep on getting your division will
continue your division will continue
so these decimal things arise through the
division process right for example if you
say half is point five but luckily your the
number one can be expressed as a non terminating
decimal you might be wondering how come this
would be equal to this actually it means as
you keep on increasing the nine the distance
between one and that number would go to zero
here also is hidden the stamp of the infinite
which which we will come in more detail later
on for example one third would be zero point
three three three three three so these so
this is the terminating decimal this is the
example of a termination decimal these are
examples of non terminating decimals ok of
take this for example this nice ah explanation
given by niven that when does a of course
you would liked our terminating decimal or
the non terminating decimal terminating decimal
looks much more nicer so you ask the question
when would p by q give me a terminating decimal
thats a question when will it give me when
will it give me a terminating decimal
so consider this example this is nothing but
eight six two five that is you have learnt
in school divided by ten thousand as a four
digit so of one after one you put four zeros
and this is same as sixty nine by eighty you
observe that the greatest common divisor between
eight six two five and ten thousand is one
twenty five a greatest common divisor or hcf
whatever you want to call gcd between eight
six two five and ten to the power four is
one twenty five so if we divide we have basically
divided both of both the sides by the greatest
common divisor and if you look at eighty so
this is the terminating decimal actually now
we have into a rational number form now this
look at this eighty and look at its prime
factorization its prime factorization gives
me what is what is this it is you can write
it as twenty to four so five into four into
four five into two square into two square
so its five into two to the power four six
into five thats it so this is this is is this
prime factorization so what is happening that
when you have a terminating decimal it looks
like as if the lower when you put the thing
in the see what is happening that the lower
thing the lower because you are ultimately
dividing by the gcd to get the p by q in the
lowest form you cannot do any more your cross
cuttings that you are learnt in school any
more divisions
so what in this case what happens what is
eighty what is this this is nothing but a
factor of or rather this number is a multiple
of this number because you have divided by
the gcd so gcd into eighty is this which is
multiple of this number so you have observe
that because ten is equal to two into five
the only prime factors of this number is two
into five and so only prime factor of this
number will also be two and five so at decimal
can be a terminating decimal if you have a
rational number and if qs only prime factors
are two and five we will always there are
terminating decimal and that is the correct
and this can be as we have proved which we
will not statement which you can look into
it later on now but among these ah so this
is our terminating decimal what is what about
non terminating decimal so we will discuss
about non terminating decimals and finish
do they have some pattern non terminating
decimals have some interesting pattern see
if you look at this number say five by eleven
so i am giving this explanations from very
beautiful small book called numbers rational
and irrational by ivan niven very well written
very beautifully explained book so if you
look you see four five four five four five
get so whats happening so for example you
take three zero nine seven by nine nine zero
zero you have zero point three one two eight
two eight two eight so initially two things
are not repeated then again its repeated so
the shortened of writing this is zero point
four five bar on the shortened of writing
this is zero point three one two eight bar
the shortened so it means that these things
are repeated periodically so every non terminating
decimal has some periodic structure built
in do it that one particular patterns after
some time one particular pattern will keep
repeating itself so why this happens this
just because of division is simple division
for example you take the number two by seven
let us see what sort of a seven is a prime
number so it cannot have two and five that
is prime factors right the only factor are
seven and one so this means it cannot be a
non it cannot be a terminating decimal it
must be a non terminating decimal let us see
if i divide two by seven what would happen
so here is a seven two can be written as two
point zero zero zero zero zero zero zero so
first is of course zero zero then you put
a point any of twenty so you to seven twos
fourteen six it at sixty and i am just writing
it down i am writing down answer which you
can verify because this is ah very simple
to do i need not do this thing repeatedly
you can do this so what is happening you have
started with do these divisions and finally
you come back to two twenty so again you start
over from this two and once you start from
this twenty see seven if it is seven what
could be the reminders reminders will less
than seven it has to be six five four three
two one so once you have record if any one
of the things have record here then you go
back the division process again goes back
so this number again gets repeated
so two by seven is nothing but zero point
two eight five seven one four so it is not
that the very first number would always to
get repeated may be the second or third number
reminder suddenly gets it will get repeated
so from there the pattern would start repeating
again so you can check this out by dividing
for example i will give you an exercise just
to check its you can take as a homework and
i click quite a good amount of fun to understand
what actually happens we really understand
that two need not start repeating here so
so some other reminder can start repeating
and so the factor would just go from that
point in this case p by q this can be represented
you just can do the division and just see
so it is a second reminder which starts ah
repeating here and this is what happens the
very interesting fact is that that is given
in the notes i will not do the proof here
because of time is that if you give me any
repeated decimal i can always represented
by a rational number so any decimal which
is a periodic representation like this can
always be represented as a decimal this will
be in the notes so you can see it from the
web so in the notes we have also spoken about
real numbers and what are the rules of handling
combining real numbers because those rules
or rules through which you do addition subtraction
multiplication for example you will see a
rule called distribution of multiplication
over addition this is exactly the way when
you multiply two digit number with the two
digit number or three digit number with a
two digit number that is exactly whats happening
that rule is actually followed you learned
at school obviously you do not know that there
is a very very basic rule that has been followed
so we have now some broad idea about numbers
how to handle them and what are the properties
you see this prime factorization is such a
crucial idea and thats that is why it was
introduced and with this i stop for today
and tomorrow we are going into the world of
countability uncountabilities of finite and
the infinite
thank you very much
