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Numbers 1 || Number Systems || CAT Preparation || Quantitative Aptitude | Rodha | YouTubeToText
YouTube Transcript: Numbers 1 || Number Systems || CAT Preparation || Quantitative Aptitude
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Summary
Core Theme
This content introduces the fundamental concepts of numbers, emphasizing their importance in quantitative aptitude exams and outlining a comprehensive learning approach to master the subject.
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hi everyone my name is Larry Prakash and
welcome to numbers one okay this is the
first class of numbers numbers then
starting today right and numbers is a
super important topic numbers is a super
important of you for order exam and it
is like an ocean right let me talk up
few myths about numbers right how will
you prove it how the questions are in
numbers right so numbers are numbers is
like very very important photo before
any exam like cats at SSC GMAT right for
any example take any aptitude exam right
where quant is required so like in cat
itself you see around in thirty four
questions out of thirty four questions
you see around seven to eight questions
of numbers maybe more than that also
right because some questions are in
direct application of direct concepts of
numbers okay then it's like an ocean is
numbers like a ocean right there are so
many concepts so many concepts in
numbers right so many concepts right so
expect promised that I discussed each
and every concept of numbers and in any
exam you will give in any exam you will
give you will not find a single question
out of my videos in numbers not a single
question that's my absolute promise to
you right this is very difficult to say
that that question came in an Apple
aptitude exam where it's like a very
open-ended examination right you cannot
can be asked anything but I have
researched it I will turded it I have
prepared so much right that not a single
question outside my video can come in
exam right but for that you have to
cover all the three parts of your video
right all the three parts of your video
right order what are the three parts of
your videos first part is all the videos
all the videos right concept videos all
the concept videos period by making you
write more than 50 videos are there
which are concept videos on factors
factorials remainders
DVD rules last dessert last to digital
some basics of numbers everything is
there not a single you don't need to
refer a single book if you follow all my
video you don't need to refer a single
book right second you need to solve all
the quizzes you need to solve all the
quizzes and then see all the quiz
solution videos quiz solution videos
have discussed all this solution of
quizzes right and I have discussed it in
videos right so any quizzes you have
right that those solutions also have
discussion videos there and third part
is previous here cat previous here cat
cat
cat Zach questions right previous ear
cats add questions right so all these
add questions we'll discuss later on
okay all the previous year cat questions
all the previous year cat questions I
have made a separate the module for it
right ear wise right from 1992 right
from 1990 to 20 2017
right later to your cat all are
segregated topic wise so in numbers
topic it is in numbers more than 50
questions are there again I made all
those video solutions for it so you have
to complete all these three modules
right it is more than I surround more
than fifty hours of videos are there or
numbers only right so if you'll clear
all those things you not find a single
doubt in numbers and my challenge you
cannot form a single you cannot escape a
single question or not miss a single
crochet in examination and numbers right
so that how we have prepared numbers
numbers is a very point of it and I hope
you get along with it very nicely any
doubts you can ping me in doubts forum
I'm always there at Ruth her Facebook
page or we have a separate doubts for
him also that you will be getting that
you would have caught by now and then we
have all those solutions I get pushed in
those Facebook pages also read so follow
all that and
write so let's write numbers let's
subtract numbers right so initially you
maybe get really good showing try to
maybe get bored because of I will cover
every basic in numbers right every basic
I'll cover from every basic from every
basic concept to the highest pinnacle
level concept right many unknown
concepts which are not the numbers that
questions also have covered right so
everything is covered in this wind is
videos right so if you're a good is
truant your main get bored with basics
but doing is in it you may find some
good points in it if you're a new
student if for each one is starting from
absolutely scratch this videos cannot
better get get better for you right
everything is there in this videos ok
Thresh start numbers right numbers so in
numbers first thing we'll study is
classification of numbers ok
classification of numbers the numbers
are basically classified as real numbers and
and
imaginary numbers and imaginary numbers right
right
imaginary numbers you don't have to
imagine about right this is out of the
syllabus in any Kattegat information
right but but I suggest you to remember
few things which will help you in
problem solving right very few things
right like biota right whatever is
remaining remaining numbers right so
imaginary numbers are all those numbers
which cannot be represented on the
number line this is a number line right
number 10 starts from zero this is this
side plus infinity this side minus
infinity right although it is all those
real numbers right real all real numbers
can be represented on number line all
those imagining imaginary numbers cannot
be represented or number lines only real
numbers can be this imaginary numbers
can't be right so like these are
imaginary numbers right under root 2
minus 1 it's an imaginary number I'd it
doesn't exist basically because this
negative number can't be inside the root
right in real numbers right but they in
imaginary number this is called iota in
iota that is I next I can remember right
at iota is minus under root minus 1 oh
ok so if you see I squares what is I
square you square both side so my square
becomes minus 1 right what is inh 2 4 is
2 4 becomes again minus 1 square this is
I square is square that is minus 1
square that is 1 so I raise to 4 is 1
right so only iota is a part you could
remember from this ok so many problems
in algebra and numbers also something we
use this concept to simplify it right so
very easy to remember where is iota
under root minus 1 and then root minus 1
is not defined in real numbers right it
is only defined in imaginary numbers ok
so all the real numbers can be
represented on number line ok these are
real numbers right now real numbers of
two parts right right I'll switch on the
slides ok real numbers of two parts real
numbers is of two points these are
rational numbers rational numbers and
irrational numbers ok and irrational
numbers two parts rational numbers and
irrational numbers now what are rational
numbers rational numbers are all those
numbers which can be written in the form
of P by Q where Q is not equal to 0 and
P Q belongs to in desserts category
right what are integers integers are all
those numbers like minus 1 minus 2 minus
3 0 1 2 3 4 all other word all these are
integers right involving on it already
no decimals so P Q are integers right or
in second where you can write all those
recurring decimals all those recurring
decimals having having a fixed pattern
right I'll discuss this point the
important point here retained decimals
having a fixed pattern which pattern
right so rational numbers are generally
of the form P by Q where Q is not equal
to 0 right denominator denominator of
any number cannot be equal to 0 where P
and Q belongs to integers category ok
recreant the recurring decimals having a
fixed pattern is also a discipline of
rational numbers right for example
numbers like this zero point 3 3 3 3 3
this is a recurring decimal but it's a
fish pattern right zero point 4 3 4 3 4
3 again I reckon but the finished
pattern right these are rational numbers
okay we'll come back to this point again
right now irrational numbers irrational
numbers right what are irrational
numbers irrational numbers are out the
form are of the form any power okay I
should write it like this but I rational
numbers first thing is you say okay
directly the note of the form P by Q
okay any number which cannot any number
which cannot be written written as P by
Q form as P by Q from right to the
irrational numbers okay for example for
example under root 2 you can't write
under root 2 as an irrational as an
pyq form right or
order of domain mr. order of 2 means
what it is 2 raise to 1 by 2 this under
H 2 under root 2 right
Q root of 5 this is what is this 5
minute to 1 by 3 so these are all over
these are all irrational numbers they
can't be written in the terms of 5 P by
Q and the second in terms of rating
decimals recurring decimals
recurring decimal is having having no
fixed pattern having no fixed pattern
okay so reckoning decimals having no
fixed pattern are also called irrational
numbers right also called irrational
numbers okay littering that some of this
one's having no fish pattern like this
like no to put in root 2 so root 2 is if
you see it is 1.4142 is goes on right
there's no fish pattern your neck here
Oh point 4 3 4 3 4 3 or point 3 3 3 3
here absolutely no 5th pattern ok or a
number of this form let's say some
number is point 2 1 3 8 5 4 going on so
this thing about these are irrational
numbers they're so recurring decimal
that means non terminating terminating
decimals ok so non terminating decimals
with no fixed pattern right so I should
better write in piece of reckoning is
also right increase of recurring you can
also write non terminating decimals it's
a good
actually I should write it non
terminating decimals okay and here you
can hear also you can write non
terminating decimals okay you know so I
can write non terminating decimals okay
right so these are rational irrational
am i right now we'll get on this point
we'll get them to this point non
terminating decimals having a fished
pattern okay go to next line so we
called okay we called rational numbers
are also those numbers which are non
terminating which are non terminating
decimals with with fixed pattern okay
with fixed pattern how to solve it see
so that means for example if I'm calling
zero point two three two three two three
like it is written like this it's a
number if I am calling it a rational
number that means it can wait in the
form of P by Q right so how to write
this in terms of P by Q how to write
this in terms of P by Q so we can write
this in terms of P by Q C assume
assume
this X is equal to point two three two
three two three it goes and going on red
now since it is repeating after two
point try this pattern is repeating
after every two point so what I can
write it as
x - 0 multiplying by two zero means
multiply by 100 so just like equation
one and you get a second equation if you
multiply equation 1 by 100 Y hundred
because 100 contains two zeros two zeros
Y 2 0 2 y 2 zeros because there are
groups of two do you video repeating so
100 X is what 23 point 2 3 2 3 2 3 again
till infinity this second equation
correct now if we're to solve it I
didn't solve this two equation simply
subtracted if I simply subtract it will
see decimal parts in both the part is
gone because both decimal parts are same
0.23 to 3 to infinity here also 0.23 to
3 till infinity this have cancelled it
cancelled white what is left 23 - 0 is
23 and 100 X - X is what 19
so we can say okay therefore X is equal
to 23 divided by 99 right so that means
this value point 2 3 2 3 2 3 right is
what is pyq 23 by 99 right right so this
is a way to do it okay take one more
example take one more example let's say
what is the value of x is equal to let
us say
0.33 3 3 3 till infinity right since it
is repeating after one point after one
part so that will make it efficient one
first okay now since it is repeating
after every group of 1 1 since I can say
all needs are same right this is
repeating after in after a 1-1 after
everyone does it here it repeats after
every 2 good is it okay so x 1 0 1 is it
repeating means x 1 0 that means Penix
right so by now you should know it why
am i applying by 10 or 800 because just
to get the decimal parts in multiplying
by any means was this decimal part
becomes same so that if I subtract
decimal part becomes cancelled okay here
also heroes or 10x is equal to 3 point 3
3 3 again till infinity is equation 2
again multiplied by 10 so that decimal
part is no same subtract both the
equations ok again decimal part gets
cancelled so decimal part gets cancelled
now what is left 10x minus X is 9x and 3
minus 0 is 3 therefore X is equal to 3
by 9 that is equal to 1 by 3 so this
value right therefore I can write here
0.333 3 didn't fit to do that what is
the value it is 1 by 3 right so this
very important point about non
terminating decimal right so now you can
generalize here that a number eights
suppose a number of the form point
eh-eh-eh-eh infinity it is what it is
basically a by nine right 0.333 over it
is 3 by 9 it is a by 9 a number of the
perform point a b a.b a.b till
infinities of is what you know the form
a B by 99 9 it is always it anyway why
because we multiply by hundred decimal
part to get same so 100 X 100 minus 500
X minus a sin 99 X so always that 19 and
99 will come in denominator right
similarly if it is like point a be say 0
point a b c ABC ABC like this it is what
it should be ABC upon triple nine it
should be ABC upon triple nine right can
sense it you see X is equal to point ABC
ABC ABC
now to equal the decimal part it is
happening of the three digit right so
multiplied by three zeros that is
thousand X so thousand x is equal to ABC
point ABC ABC women right now if you
subtract these two equations so here
decimal part gets cancelled because both
equations have same decimal part a B C
minus 0 is what it is ABC thousand X
minus X is what 9 triple nine x you can
see right what is X is equal to so X is
equal to ABC by triple nine right so at
this point ABC ABC is of the form ABC
triple nine right similarly you can
write point ABCD ABCD
going on like this it is what it is ABCD
by four times nine
so I can Delta Li right now I can
directly write now yeah sure I think
it's fine now next part C now a bit
variety in this I'll add right suppose
is a question point two
five six five six five six is equal to
some P by Q
so what is the value of P by Q
right this is a question you okay so
0.25 six 0.25 six is like in this soy
point two five six five six five sorry
two is not ribbiting mind it okay so
this P by Q through the value of P by Q
right means I have to write this this
whole part in terms of some rational
numbers autism in the form of P by Q
okay so we can write it we can write it see
see
again I write okay let some X is equal
to 0.25 six five six five six he'll
infinity this equation one now to a coil
t decimal part see directly i cannot
equate it as well but because this
decimal part doesn't have a kind of same
design see right we do not find six five
six so I can multiply 100 right to is
extra here to eliminate these two first
I shall multiply by ten X so 10 X is
equal to what 10 X is equal to I can
write two point five six five six five
six now it's okay we get equation two
and now since not noid becomes common
right because it is in decimal part is a
fixed pattern now five six five six five
six right so I should multiply two
equally decimal part so that again
subtract and eliminated okay I have to
multiply by two more zeros because is he
reading of a 2 so 2 moves into that
means 10 and two more zeros it becomes
thousand right so multiplying that means
multiplying equation two multiplying
equation two by hundred that visual
means this okay so multiplying equation
to my hundred victims thousand X is
equal to 256 point five six five six and
so on this equation three okay so you
can solve equation two and three that
subtracted to eliminate decimal part 256
minus two is equal to 250 4000 X minus
10 X is equal to nine ninety X therefore
X is equal to
254 by nine ninety you can cancel it 127
by four ninety five right therefore X is
equal to
127 by four ninety five that is equal to
P by Q this is the answer right so
converted 0.25 six five six five six in
terms of pyq for disaster 127 by 495
right so only single catch point here
was here two was extra in decimal part
right otherwise it was a fixed pattern
okay so since is a fixed fixed pattern
they have it has to be rational number
eight but two was explained here so we
can eliminate two first multiplying my
10 it becomes normal come to normal now
and now multiply again by seeing the
pattern of digits right so see now we
can directly donate such kind of
questions we can just read it actually
do we don't need to solve this much
right directly do it suppose the
question you find it okay
like suppose okay point two five six
five six five six five six okay so what
you can do is whenever this you can
write like this point two five six and
you can put a bar on five see what a bar
means Barmes recurring that means bar
that bar miss reckoning that means five
six is occurring five six is occurring
repeatedly okay so like five six five
six five six right the entire shortcut
ride this integral part is this integral
part this is the integral part in
decimal part decimal part integral part
rate so integral part is zero plus here
right 256 here it is the whole part on
whichever number bar is more there X
subtract that - - okay subtract 8 minus
2 divided by no 9 and 3 times 9 which is
alike which is bar on the same number of
digits right that means here 5 6 bar is
on 5 6 right so bar is on 5 & 6 so I
should put - or two times since PI's on
two limits so I should put nine two
times right and now 1 0 I should put for
this to having naught bar right so one
day I'd have not having bad for that for
that I'll put 0 okay so you can find
this C what is the answer it could be
done 254 by 9 is 0 right same thing so
simple thing register - point 0.25 6 bar
right whole 256 here subtracted is it
which is not having bar and divided by
as many nines which are as as many hex
are having him bar that mean is he put
here select two digits are having more
so two lines he put here one is it not
having one so you put a zero for that
right so ii will do directly this one we
do directly now okay like suppose it is
0.42 3 2 3 2 3 like this what is the
value that you should do it it is what
it is point 4 - 3 bar is on only 22 23 because
because
two or three to three is repeating so
put like this what is the value it is
like zero plus four 23 minus 4 subtract
4 minus 4 right for not knowing bar not
to read heading bad so two nines one
minute having not bar one day not having
bar so put a zero for the answer answer
is four one nine upon nine nine zero
this is the P by Q form of this non
terminating decimal here it's fine okay
this is the way you do it fine so okay
now we know now the verticals we know is
we know real numbers all the numbers
which can be represented on the number
nine are real numbers right
then real 100e a real number it comes
rational numbers okay then also
irrational numbers rational numbers all
the numbers which can be converted to
the form pyq he rational numbers all
those numbers which can't be converted
to the form P by Q now under rational
numbers again we'll see a category
called what all the integers non integer
than negative integers positive integers
right so you should write ok what is
water integers so you know what are
integers integers are
not having decimals right that means all
those are my leg - 4 - 3 - 2 - 1 0 1 2 3
4 till infinity right all these are
integers okay all these are integers
from minus infinity to plus infinity it
is all into this red water now we should
note here 0 is 0 is a non-negative 0 is
a non-negative as well as as well as non
positive integer K 0 is a non-negative
as well as a non positive integer ok so
0 is a non-negative as well as non
positive integer ok this is that into
this right now
under intense category the invisi then
the numbers will come like we know now
what are whole numbers what are whole
numbers so whole numbers are all those
positive numbers and whole numbers are
always non-negative should write whole
numbers are always all the morning integers
integers
so non-negative in tennis means what
non-negative integers 1 0 1 2 3 4 till
infinity these are whole numbers right
now natural number of natural numbers so
natural numbers are all the positive
integers all the positive integers right
that means 1 2 3 4 5 10 infinity these
are whole numbers and natural numbers
right now next is what are all numbers
so odd numbers are any numbers of the
form to K minus 1 they form for number 2
K minus 1
right see it depends actually it is 2 K
minus 1 or 2 k plus 1 if you take if
starting from 1 so if you're taking 2 K
minus 1 K value I can take a 0 here here
K sorry K value I should take as 1 in
this case I am in reverse ok so K value
I should take as 1 2 K minus 1 means K
value has to be 1 2 k plus 1 means K
value has to be 0 starting if you are
starting with it so K should be 0 so
that it just starts with 1 right so if K
it depends on it depends totally on your
right when you assume or 3/4 two odd
numbers here assumed 2 k plus 1 and 2 K
minus 1 or 2 k plus 1 2 k plus 3 so all
the odd numbers idea of the form 2 K
minus 1 so that basically means what if
you take 2 K minus 1 so if I put K is
equal to 0 it becomes minus 1 so my odd
numbers are never negative odd numbers
and even numbers right odd numbers and
even numbers these are always always
positive number right so odd numbers and
even numbers always positive or
non-negative because you know you also
considered as an even number so I should
write odd numbers and
even numbers are
non-negative right there non-negative
there won't be they want to be negative
ok so tart from k equal to if I start
from k equal to 0 or HR from equal to 1
depending wish on which one you are
degrade you take in this form start k
equal to 1 T in this form we start k
equal to zero
so if i put k equal to 1 year here is it
1 if you were k equal to 2 you get 3 if
you get k equal to 3 we will get 5 that
is a hard all over 1 3 5 7 9 11 13 and
so on now all the even numbers what are
you numbers like 0 2 4 6 8 right so even
numbers are of the form are of the form
again 2 kids even numbers are of the
form 2 k you start from k equal to 0
here ok so 0 2 4 6 8 k equal to 0 to k
equal to 1 2 sorry K equal to 0 0 k
equal to 1 to k equal to 2 4 right so
all the 0 2 4 6 8 are on even numbers
right and odd and even numbers are all
this non-negative numbers right now
so in the next video I discuss few
questions related to or relating to
order even numbers and good questions
will discuss ok thank you [Music]
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