So, welcome to my course on Calculus of One
Variables. Those who have 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 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 1 2 3 4 5 6 7 8 9 and then everything is
man’s hand evolved once you have that you
can do everything.
But do not ever think that numbers is something
which you didn’t really, not bothered about
it so common. Of course, you know about this
basic numbers. It’s so obvious that I do
not 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 2
oranges and I say about 2 elephants. So, this
same 2 is representing that quantity 2 the
same symbol. So, it doesn’t matter whether
it is orange or 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,
these are some things which contains objects
do not try to define sets as well defined
collection of objects and all those things
because that will get into logical problems.
So, I and you 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, set theory or the theory of sets was first
introduced and studied by 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 1
2 3 4 5 6 7 8 this keeps on going it does
not end, and this is the first time you are
possibly you see in your very basic thing
of life 1 2 3 4 5 6 7 8 9 10 ….. 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 on 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 the infinite was not
really appreciated by his teacher. For his
teacher infinity was just a notional thing
where you take limits and do 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 0 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
2 numbers. So, if you have 2 apples in 1 box
3 apples in another box I can bring them together
make them 5 apples, but there is a issue of
subtraction.
So, if I have 3 apples in a box can I take
away 5 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. In fact Blaise Pascal one
of the very big mathematician, of the 17th
century he; 17th I guess yeah 17th 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 negitive 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 all integers contains 0 contains
1 plus 1 plus 2 plus 3 which is nothing, but
1 2 3 just to separate them from minus 1 minus
2 minus 3 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 4 people
then I have divide into 4 parts and everybody
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 it is not just what
we are trying to say. So, here we are going
to talk about rational numbers. So, we are
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 2 integers
p and q where q is not equal to 0.Of course,
you know very well that division with 0 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 0 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. If you take a right angle triangle
and this is the most famous description away
whose both sides are say 1 centimeter. Then
applying the Pythagoras theorem the hypotenuse
would have a length root 2 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 didn’t not believe there was something
else.
So, when this came out and one of his students
called Hipparchus who actually proved that
root 2 this number root 2 cannot be expressed
in the form p by q. So, we will show all this
things in the third lecture but we are just
going to mention that this root 2 are shown
by one of the Pythagoreans to be not rational
and that Pythagoreans were all hast Hipparchus
was thrown in the sea and killed because he
made such a statement that 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 0 means the whole number set. Then
when negative numbers come in you have the
integer set and every integer can be written
as that integer divided by 1. 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 2, root 2 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 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, you have
the most obvious things this fractions and
standard numbers negative all that you can
do with numbers. So, you will be perfectly
happy just of rational numbers, but unfortunately
geometry which is also very real thing gives
us something very different story because
here you have taken sides whose lengths are
rational numbers one centimeter. So, irrational
number is a reality with which mathematicians
have to put up and they play a 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 called 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 we
can actually find them even irrationals root
2. 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 there can may not be a number which
is both rational and irrational. So, hence
union of these 2 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’s is 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 of a reality in calculus we cannot really
get out of that. So, these natural number
sets are classified into 2 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 2 classification there
is an interesting class of numbers. One of
the see there are 4 arithmetical operations
that you carry on these numbers one is addition
subtraction, multiplication, multiplication
tables of course, have to do with natural
numbers 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 called the
prime numbers and they are very- very important
in the understanding 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 1. So, you can
say I can start with 1, 1 is divisible by
itself and one, but 1 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 2, then 2 is the
only even prime number and then it goes to
3 5 7 and so and so 11.
An important fact about natural numbers is
something called unique prime factorization
which I am writing on the side, this 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 n can be written as say
if you take P1 to a prime number P1 to the
power q1 say p r to the power q r.
You can form of for example, 60, 60 is 15
into 4, 15 is 3 into 5 and 4 is 2 square that
is it. So, all the numbers are prime, so and
this representation is unique now suppose
I have considered 1 to be a prime number then
60 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 do not have a unique prime factorization
that is reason why 1 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 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 mathematical statement involving
the symbol n, so you have to decide whether
the statement given statement is true or false.
So, the method of induction says first check
is P(1) is true and then if P(n) plus 1 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(1) is 1 here and you can put n equal to
1 and see the left side is also 1. So, P(1)
of course, satisfies this. So, now, you take
assume that P(n) is true. So, what is P n
plus 1, in this case the statement P(n) plus
1 is of course, 1 plus 2 plus 3 plus N plus
N plus 1 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 2 into N plus 1 by 2 which
is same as N plus 1 into N plus 1 plus 1 by
2 and hence this holds for N plus 1 also showing
that this result actually holds for all N.
So, combining these 2 features or induction
as well as the fact that 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
also present in Euclid’s element one of
the most famous books in mathematics are all
times. Now we will not do Euclid’s proof
we will do a very modern proof taken from
a book called ‘The Proofs from the book’
actually a the great mathematician Paul Erdos
used to think that God whom we called also
supreme fascist that 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 does not mean. So, mathematical
research does not always consist of the fact
that it you would do always something new
you can have look at old things from very
new angle and get newer 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 Fermat’s 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 2, 2 to the power N plus 1. 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 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 2 to the power 2 to the
power N any power of 2 is an even number.
So, this is an odd number now I will take
2 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 ,Right. So, there are no other prime numbers
which are a common dividers to this 
and every prime number is bigger than 1. So,
there cannot be no 2, no prime number which
is a common divisor to 2 different Fermat’s
number. So, that is why they are called relatively
prime or co prime.
So, if I have a number is that 2 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,
I have infinite number of Fermat’s number.
So, this Fermat’s number is infinite, infinite
numbers. There be infinite number of Fermat’s
numbers, while I will show that give me any
pair of Fermat’s number they are co prime.
So, there cannot be a given common prime which
is a factor to both of them. So, for each
of the Fermat’s number there should be different
primes. See Fermat’s number infinite so
are the number of primes that is the whole
idea.
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 1. Now assume that
so Fk, here k is of course, strictly less
than n this k is 10 to the power n minus 1.
So, let q be a divisor of F k and F n. So,
you are talking about common divisors now
if it is. So, if it divides all the Fk’s
and Fn. So, there is the common divisor right.
So, we have a, 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 I 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 Fn is also divisible by q which
means 2 also has to be divisible by q which
means q this would imply let q = 1 or q = 2,
Right.
Now q cannot be 2 because if q is 2 if q,
but an q divides Fk then Fk must be even 2
only divides even number 2 does not divide
odd number. So, q cannot be 2 this is a conclusion
since F k is odd which means what? q is equal
to 1. So, only common divisor possible between
any F k and F n here is 1. So, for any pair
you take k less than n for any pair F k F
n that you chose or any F k 1, k 2 where my
for any pair F k F n that you chose does not
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 something lesser. So,
whatever k you chose from here the only common
factor is 1. So, then which means we have
proved that for any 2 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
1 I have to add 1 here. So, I come to this
simple fact k equal to 0 to n minus 1. 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 Fn what is what is the
structure of Fn you know that this F n is
given as this 1. So, use that and finally,
prove that this is nothing, but F of n plus
1 minus 2 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 10, Right, or division through
multiples of tens. So, that when we give representation
through multiples of 10 that gives raise to
decimal representation of numbers.
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 0 number
or then p by q is same as r p by r q this
is something you have to understand. Now decimal
representation means that you can represent
there is something called a 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 0 and 1
the left hand side of the decimal point where
you have the whole numbers actually that part
would be 0. So, you have already done some
decimal 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 2
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 a smaller
number by a bigger number you are 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 0.5, but luckily your number 1
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 9 the distance between
1 and that number would go to 0. Here also
is hidden the stamp of the infinite which
we will come in more detail later on. For
example, one-third would be 0.33333.
So, this is the terminating decimal this is
the example of a termination decimal these
are examples of non terminating decimals.
Take this for example, this nice 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
that is a question. When will it give me a
terminating decimal?
So, consider this example 
this is nothing, but 8625 that is you have
learnt in school divided by 10000 as a 4 digit.
So, of one after one you put 4 zeros and this
is same as 69 by 80 you observe that the greatest
common divisor between 8625 and 10000 is 125.
A greatest common divisor or HCF whatever
you want to call GCD between 8625 and 10 to
the power 4 is 125, so if we divide, we have
basically divided both of both the sides by
the greatest common divisor and if you look
at 80.
So, this is the terminating decimal actually
now we have brought it. into a rational number
form. Now this look at this 80 and look at
its prime factorization its prime factorization
gives me what is, what is this it is you can
write it as 20 to 4, so 5 into 4 into 4 5
into 2 square into 2 square, so its 5 into
2 to the power 4, 16 into 5 that is it. So,
this is this is this prime factorization.
So, what is happening that when you have a
terminating decimal it looks like as if the
lower, 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, in this case what happens, what is 80
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 80 is this which is multiple
of this number. So, you have observe that
because 10 is equal to 2 into 5 the only prime
factors of this number is 2 into 5 and so
only prime factor of this number will also
be 2 and 5.
So, at decimal can be a terminating decimal
if you have a rational number and if Q is
only prime factors are 2 and 5 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,
so this is our terminating decimal 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
5 by 11. 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 0.454545 get s repeated.
So, what is happening? So, for example, you
take 3097 by 9900 you have 0.31282828. So,
initially 2 things are not repeated then again
it is repeated. So, the shortened of writing
this is 0.45 bar on the shortened of writing
this is 0.3128 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 2 by 7 ,7
is a prime number. So, it cannot have 2 and
5 that is prime factors, Right. The only factor
are 7 and 1. So, this means it cannot be a
terminating decimal it must be a non terminating
decimal. Let us see if I divide 2 by 7 what
would happen. So, here is a 7, 2 can be written
as 2.0000000. So, first is of course, 0 then
you put a point any of 20.
So, you to 7 2s 14, 6 it at 60 and I am just
writing it down I am writing down answer which
you can verify because this is 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 2; 20. So, again you start
over from this 2 and once you start from this
20, see if it is 7 what could be the reminders,
reminders will less than 7 it has to be 6,
5, 4, 3, 2, 1. 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, 2 by 7 is nothing, but 0.285714. 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, it is you can take
as a homework and I click quite a good amount
of fun to understand what actually happens
we really understand that 2 need not start
repeating here. 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 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 2 digit number with the 2 digit
number or 3 digit number with a 2 digit number
that is exactly what is happening that rule
is actually followed. For eg. 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 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.
