0:17 hi everyone my name is Larry Prakash and
0:21 welcome to numbers one okay this is the
0:23 first class of numbers numbers then
0:25 starting today right and numbers is a
0:29 super important topic numbers is a super
0:31 important of you for order exam and it
0:34 is like an ocean right let me talk up
0:36 few myths about numbers right how will
0:38 you prove it how the questions are in
0:41 numbers right so numbers are numbers is
0:44 like very very important photo before
0:48 any exam like cats at SSC GMAT right for
0:50 any example take any aptitude exam right
0:54 where quant is required so like in cat
0:57 itself you see around in thirty four
0:59 questions out of thirty four questions
1:01 you see around seven to eight questions
1:03 of numbers maybe more than that also
1:05 right because some questions are in
1:07 direct application of direct concepts of
1:11 numbers okay then it's like an ocean is
1:13 numbers like a ocean right there are so
1:16 many concepts so many concepts in
1:20 numbers right so many concepts right so
1:23 expect promised that I discussed each
1:26 and every concept of numbers and in any
1:29 exam you will give in any exam you will
1:32 give you will not find a single question
1:35 out of my videos in numbers not a single
1:37 question that's my absolute promise to
1:40 you right this is very difficult to say
1:43 that that question came in an Apple
1:45 aptitude exam where it's like a very
1:47 open-ended examination right you cannot
1:50 can be asked anything but I have
1:52 researched it I will turded it I have
1:56 prepared so much right that not a single
1:58 question outside my video can come in
2:00 exam right but for that you have to
2:02 cover all the three parts of your video
2:04 right all the three parts of your video
2:06 right order what are the three parts of
2:10 your videos first part is all the videos
2:13 all the videos right concept videos all
2:16 the concept videos period by making you
2:18 write more than 50 videos are there
2:21 which are concept videos on factors
2:23 factorials remainders
2:27 DVD rules last dessert last to digital
2:30 some basics of numbers everything is
2:32 there not a single you don't need to
2:35 refer a single book if you follow all my
2:37 video you don't need to refer a single
2:40 book right second you need to solve all
2:42 the quizzes you need to solve all the
2:45 quizzes and then see all the quiz
2:48 solution videos quiz solution videos
2:51 have discussed all this solution of
2:54 quizzes right and I have discussed it in
2:57 videos right so any quizzes you have
2:59 right that those solutions also have
3:02 discussion videos there and third part
3:06 is previous here cat previous here cat
3:08 cat
3:12 cat Zach questions right previous ear
3:15 cats add questions right so all these
3:17 add questions we'll discuss later on
3:20 okay all the previous year cat questions
3:22 all the previous year cat questions I
3:25 have made a separate the module for it
3:28 right ear wise right from 1992 right
3:33 from 1990 to 20 2017
3:36 right later to your cat all are
3:39 segregated topic wise so in numbers
3:42 topic it is in numbers more than 50
3:44 questions are there again I made all
3:47 those video solutions for it so you have
3:48 to complete all these three modules
3:51 right it is more than I surround more
3:53 than fifty hours of videos are there or
3:56 numbers only right so if you'll clear
3:59 all those things you not find a single
4:01 doubt in numbers and my challenge you
4:04 cannot form a single you cannot escape a
4:06 single question or not miss a single
4:08 crochet in examination and numbers right
4:11 so that how we have prepared numbers
4:14 numbers is a very point of it and I hope
4:16 you get along with it very nicely any
4:18 doubts you can ping me in doubts forum
4:20 I'm always there at Ruth her Facebook
4:23 page or we have a separate doubts for
4:25 him also that you will be getting that
4:28 you would have caught by now and then we
4:30 have all those solutions I get pushed in
4:34 those Facebook pages also read so follow
4:35 all that and
4:39 write so let's write numbers let's
4:42 subtract numbers right so initially you
4:44 maybe get really good showing try to
4:46 maybe get bored because of I will cover
4:49 every basic in numbers right every basic
4:51 I'll cover from every basic from every
4:54 basic concept to the highest pinnacle
4:56 level concept right many unknown
4:58 concepts which are not the numbers that
5:00 questions also have covered right so
5:02 everything is covered in this wind is
5:04 videos right so if you're a good is
5:06 truant your main get bored with basics
5:08 but doing is in it you may find some
5:10 good points in it if you're a new
5:12 student if for each one is starting from
5:14 absolutely scratch this videos cannot
5:16 better get get better for you right
5:18 everything is there in this videos ok
5:21 Thresh start numbers right numbers so in
5:23 numbers first thing we'll study is
5:26 classification of numbers ok
5:29 classification of numbers the numbers
5:32 are basically classified as real numbers and
5:33 and
5:37 imaginary numbers and imaginary numbers right
5:39 right
5:41 imaginary numbers you don't have to
5:43 imagine about right this is out of the
5:46 syllabus in any Kattegat information
5:49 right but but I suggest you to remember
5:51 few things which will help you in
5:53 problem solving right very few things
5:56 right like biota right whatever is
5:58 remaining remaining numbers right so
6:00 imaginary numbers are all those numbers
6:03 which cannot be represented on the
6:05 number line this is a number line right
6:08 number 10 starts from zero this is this
6:10 side plus infinity this side minus
6:13 infinity right although it is all those
6:15 real numbers right real all real numbers
6:18 can be represented on number line all
6:21 those imagining imaginary numbers cannot
6:23 be represented or number lines only real
6:25 numbers can be this imaginary numbers
6:28 can't be right so like these are
6:30 imaginary numbers right under root 2
6:33 minus 1 it's an imaginary number I'd it
6:35 doesn't exist basically because this
6:37 negative number can't be inside the root
6:41 right in real numbers right but they in
6:44 imaginary number this is called iota in
6:47 iota that is I next I can remember right
6:49 at iota is minus under root minus 1 oh
6:52 ok so if you see I squares what is I
6:54 square you square both side so my square
6:58 becomes minus 1 right what is inh 2 4 is
7:01 2 4 becomes again minus 1 square this is
7:04 I square is square that is minus 1
7:06 square that is 1 so I raise to 4 is 1
7:09 right so only iota is a part you could
7:11 remember from this ok so many problems
7:14 in algebra and numbers also something we
7:16 use this concept to simplify it right so
7:18 very easy to remember where is iota
7:20 under root minus 1 and then root minus 1
7:22 is not defined in real numbers right it
7:25 is only defined in imaginary numbers ok
7:27 so all the real numbers can be
7:29 represented on number line ok these are
7:32 real numbers right now real numbers of
7:34 two parts right right I'll switch on the
7:38 slides ok real numbers of two parts real
7:40 numbers is of two points these are
7:44 rational numbers rational numbers and
7:47 irrational numbers ok and irrational
7:49 numbers two parts rational numbers and
7:52 irrational numbers now what are rational
7:55 numbers rational numbers are all those
7:57 numbers which can be written in the form
8:01 of P by Q where Q is not equal to 0 and
8:06 P Q belongs to in desserts category
8:08 right what are integers integers are all
8:11 those numbers like minus 1 minus 2 minus
8:15 3 0 1 2 3 4 all other word all these are
8:17 integers right involving on it already
8:20 no decimals so P Q are integers right or
8:23 in second where you can write all those
8:26 recurring decimals all those recurring
8:29 decimals having having a fixed pattern
8:31 right I'll discuss this point the
8:34 important point here retained decimals
8:36 having a fixed pattern which pattern
8:39 right so rational numbers are generally
8:41 of the form P by Q where Q is not equal
8:43 to 0 right denominator denominator of
8:45 any number cannot be equal to 0 where P
8:48 and Q belongs to integers category ok
8:51 recreant the recurring decimals having a
8:53 fixed pattern is also a discipline of
8:54 rational numbers right for example
8:59 numbers like this zero point 3 3 3 3 3
9:01 this is a recurring decimal but it's a
9:05 fish pattern right zero point 4 3 4 3 4
9:08 3 again I reckon but the finished
9:11 pattern right these are rational numbers
9:13 okay we'll come back to this point again
9:16 right now irrational numbers irrational
9:17 numbers right what are irrational
9:20 numbers irrational numbers are out the
9:24 form are of the form any power okay I
9:26 should write it like this but I rational
9:28 numbers first thing is you say okay
9:29 directly the note of the form P by Q
9:33 okay any number which cannot any number
9:39 which cannot be written written as P by
9:41 Q form as P by Q from right to the
9:45 irrational numbers okay for example for
9:49 example under root 2 you can't write
9:51 under root 2 as an irrational as an
9:54 pyq form right or
9:56 order of domain mr. order of 2 means
9:59 what it is 2 raise to 1 by 2 this under
10:01 H 2 under root 2 right
10:05 Q root of 5 this is what is this 5
10:07 minute to 1 by 3 so these are all over
10:08 these are all irrational numbers they
10:11 can't be written in the terms of 5 P by
10:13 Q and the second in terms of rating
10:16 decimals recurring decimals
10:22 recurring decimal is having having no
10:26 fixed pattern having no fixed pattern
10:30 okay so reckoning decimals having no
10:33 fixed pattern are also called irrational
10:34 numbers right also called irrational
10:36 numbers okay littering that some of this
10:38 one's having no fish pattern like this
10:41 like no to put in root 2 so root 2 is if
10:44 you see it is 1.4142 is goes on right
10:46 there's no fish pattern your neck here
10:50 Oh point 4 3 4 3 4 3 or point 3 3 3 3
10:52 here absolutely no 5th pattern ok or a
10:54 number of this form let's say some
10:59 number is point 2 1 3 8 5 4 going on so
11:00 this thing about these are irrational
11:02 numbers they're so recurring decimal
11:05 that means non terminating terminating
11:08 decimals ok so non terminating decimals
11:11 with no fixed pattern right so I should
11:13 better write in piece of reckoning is
11:15 also right increase of recurring you can
11:19 also write non terminating decimals it's
11:21 a good
11:23 actually I should write it non
11:26 terminating decimals okay and here you
11:28 can hear also you can write non
11:31 terminating decimals okay you know so I
11:35 can write non terminating decimals okay
11:38 right so these are rational irrational
11:39 am i right now we'll get on this point
11:41 we'll get them to this point non
11:43 terminating decimals having a fished
11:45 pattern okay go to next line so we
11:50 called okay we called rational numbers
11:53 are also those numbers which are non
11:55 terminating which are non terminating
12:00 decimals with with fixed pattern okay
12:03 with fixed pattern how to solve it see
12:07 so that means for example if I'm calling
12:09 zero point two three two three two three
12:12 like it is written like this it's a
12:14 number if I am calling it a rational
12:17 number that means it can wait in the
12:20 form of P by Q right so how to write
12:23 this in terms of P by Q how to write
12:25 this in terms of P by Q so we can write
12:28 this in terms of P by Q C assume
12:30 assume
12:34 this X is equal to point two three two
12:37 three two three it goes and going on red
12:40 now since it is repeating after two
12:43 point try this pattern is repeating
12:45 after every two point so what I can
12:46 write it as
12:49 x - 0 multiplying by two zero means
12:52 multiply by 100 so just like equation
12:54 one and you get a second equation if you
12:57 multiply equation 1 by 100 Y hundred
12:59 because 100 contains two zeros two zeros
13:02 Y 2 0 2 y 2 zeros because there are
13:04 groups of two do you video repeating so
13:08 100 X is what 23 point 2 3 2 3 2 3 again
13:10 till infinity this second equation
13:12 correct now if we're to solve it I
13:15 didn't solve this two equation simply
13:18 subtracted if I simply subtract it will
13:20 see decimal parts in both the part is
13:22 gone because both decimal parts are same
13:26 0.23 to 3 to infinity here also 0.23 to
13:28 3 till infinity this have cancelled it
13:31 cancelled white what is left 23 - 0 is
13:35 23 and 100 X - X is what 19
13:39 so we can say okay therefore X is equal
13:43 to 23 divided by 99 right so that means
13:47 this value point 2 3 2 3 2 3 right is
13:51 what is pyq 23 by 99 right right so this
13:53 is a way to do it okay take one more
13:56 example take one more example let's say
13:58 what is the value of x is equal to let
13:59 us say
14:03 0.33 3 3 3 till infinity right since it
14:05 is repeating after one point after one
14:07 part so that will make it efficient one
14:09 first okay now since it is repeating
14:11 after every group of 1 1 since I can say
14:13 all needs are same right this is
14:15 repeating after in after a 1-1 after
14:18 everyone does it here it repeats after
14:22 every 2 good is it okay so x 1 0 1 is it
14:25 repeating means x 1 0 that means Penix
14:28 right so by now you should know it why
14:31 am i applying by 10 or 800 because just
14:33 to get the decimal parts in multiplying
14:35 by any means was this decimal part
14:38 becomes same so that if I subtract
14:41 decimal part becomes cancelled okay here
14:43 also heroes or 10x is equal to 3 point 3
14:46 3 3 again till infinity is equation 2
14:49 again multiplied by 10 so that decimal
14:51 part is no same subtract both the
14:54 equations ok again decimal part gets
14:56 cancelled so decimal part gets cancelled
15:00 now what is left 10x minus X is 9x and 3
15:04 minus 0 is 3 therefore X is equal to 3
15:08 by 9 that is equal to 1 by 3 so this
15:11 value right therefore I can write here
15:14 0.333 3 didn't fit to do that what is
15:17 the value it is 1 by 3 right so this
15:19 very important point about non
15:22 terminating decimal right so now you can
15:25 generalize here that a number eights
15:27 suppose a number of the form point
15:29 eh-eh-eh-eh infinity it is what it is
15:33 basically a by nine right 0.333 over it
15:37 is 3 by 9 it is a by 9 a number of the
15:42 perform point a b a.b a.b till
15:44 infinities of is what you know the form
15:48 a B by 99 9 it is always it anyway why
15:51 because we multiply by hundred decimal
15:54 part to get same so 100 X 100 minus 500
15:58 X minus a sin 99 X so always that 19 and
16:00 99 will come in denominator right
16:03 similarly if it is like point a be say 0
16:08 point a b c ABC ABC like this it is what
16:13 it should be ABC upon triple nine it
16:16 should be ABC upon triple nine right can
16:19 sense it you see X is equal to point ABC
16:21 ABC ABC
16:26 now to equal the decimal part it is
16:27 happening of the three digit right so
16:28 multiplied by three zeros that is
16:32 thousand X so thousand x is equal to ABC
16:36 point ABC ABC women right now if you
16:38 subtract these two equations so here
16:40 decimal part gets cancelled because both
16:43 equations have same decimal part a B C
16:46 minus 0 is what it is ABC thousand X
16:49 minus X is what 9 triple nine x you can
16:51 see right what is X is equal to so X is
16:56 equal to ABC by triple nine right so at
16:58 this point ABC ABC is of the form ABC
17:01 triple nine right similarly you can
17:05 write point ABCD ABCD
17:08 going on like this it is what it is ABCD
17:11 by four times nine
17:13 so I can Delta Li right now I can
17:16 directly write now yeah sure I think
17:19 it's fine now next part C now a bit
17:21 variety in this I'll add right suppose
17:23 is a question point two
17:29 five six five six five six is equal to
17:32 some P by Q
17:36 so what is the value of P by Q
17:39 right this is a question you okay so
17:45 0.25 six 0.25 six is like in this soy
17:47 point two five six five six five sorry
17:50 two is not ribbiting mind it okay so
17:52 this P by Q through the value of P by Q
17:55 right means I have to write this this
17:57 whole part in terms of some rational
17:58 numbers autism in the form of P by Q
18:01 okay so we can write it we can write it see
18:02 see
18:05 again I write okay let some X is equal
18:09 to 0.25 six five six five six he'll
18:14 infinity this equation one now to a coil
18:16 t decimal part see directly i cannot
18:18 equate it as well but because this
18:21 decimal part doesn't have a kind of same
18:23 design see right we do not find six five
18:26 six so I can multiply 100 right to is
18:29 extra here to eliminate these two first
18:31 I shall multiply by ten X so 10 X is
18:33 equal to what 10 X is equal to I can
18:35 write two point five six five six five
18:39 six now it's okay we get equation two
18:42 and now since not noid becomes common
18:44 right because it is in decimal part is a
18:46 fixed pattern now five six five six five
18:49 six right so I should multiply two
18:51 equally decimal part so that again
18:53 subtract and eliminated okay I have to
18:55 multiply by two more zeros because is he
18:57 reading of a 2 so 2 moves into that
18:59 means 10 and two more zeros it becomes
19:02 thousand right so multiplying that means
19:05 multiplying equation two multiplying
19:09 equation two by hundred that visual
19:11 means this okay so multiplying equation
19:13 to my hundred victims thousand X is
19:18 equal to 256 point five six five six and
19:21 so on this equation three okay so you
19:23 can solve equation two and three that
19:27 subtracted to eliminate decimal part 256
19:31 minus two is equal to 250 4000 X minus
19:34 10 X is equal to nine ninety X therefore
19:36 X is equal to
19:42 254 by nine ninety you can cancel it 127
19:45 by four ninety five right therefore X is
19:47 equal to
19:50 127 by four ninety five that is equal to
19:54 P by Q this is the answer right so
19:57 converted 0.25 six five six five six in
20:01 terms of pyq for disaster 127 by 495
20:03 right so only single catch point here
20:07 was here two was extra in decimal part
20:08 right otherwise it was a fixed pattern
20:11 okay so since is a fixed fixed pattern
20:13 they have it has to be rational number
20:15 eight but two was explained here so we
20:17 can eliminate two first multiplying my
20:20 10 it becomes normal come to normal now
20:22 and now multiply again by seeing the
20:25 pattern of digits right so see now we
20:26 can directly donate such kind of
20:28 questions we can just read it actually
20:29 do we don't need to solve this much
20:31 right directly do it suppose the
20:32 question you find it okay
20:35 like suppose okay point two five six
20:38 five six five six five six okay so what
20:42 you can do is whenever this you can
20:44 write like this point two five six and
20:47 you can put a bar on five see what a bar
20:49 means Barmes recurring that means bar
20:51 that bar miss reckoning that means five
20:53 six is occurring five six is occurring
20:55 repeatedly okay so like five six five
20:58 six five six right the entire shortcut
21:01 ride this integral part is this integral
21:02 part this is the integral part in
21:05 decimal part decimal part integral part
21:08 rate so integral part is zero plus here
21:11 right 256 here it is the whole part on
21:13 whichever number bar is more there X
21:16 subtract that - - okay subtract 8 minus
21:22 2 divided by no 9 and 3 times 9 which is
21:24 alike which is bar on the same number of
21:27 digits right that means here 5 6 bar is
21:30 on 5 6 right so bar is on 5 & 6 so I
21:33 should put - or two times since PI's on
21:35 two limits so I should put nine two
21:38 times right and now 1 0 I should put for
21:41 this to having naught bar right so one
21:43 day I'd have not having bad for that for
21:46 that I'll put 0 okay so you can find
21:48 this C what is the answer it could be
21:51 done 254 by 9 is 0 right same thing so
21:53 simple thing register - point 0.25 6 bar
21:57 right whole 256 here subtracted is it
21:59 which is not having bar and divided by
22:03 as many nines which are as as many hex
22:05 are having him bar that mean is he put
22:06 here select two digits are having more
22:09 so two lines he put here one is it not
22:10 having one so you put a zero for that
22:13 right so ii will do directly this one we
22:17 do directly now okay like suppose it is
22:21 0.42 3 2 3 2 3 like this what is the
22:24 value that you should do it it is what
22:29 it is point 4 - 3 bar is on only 22 23 because
22:30 because
22:32 two or three to three is repeating so
22:34 put like this what is the value it is
22:39 like zero plus four 23 minus 4 subtract
22:42 4 minus 4 right for not knowing bar not
22:45 to read heading bad so two nines one
22:47 minute having not bar one day not having
22:49 bar so put a zero for the answer answer
22:51 is four one nine upon nine nine zero
22:54 this is the P by Q form of this non
22:57 terminating decimal here it's fine okay
23:00 this is the way you do it fine so okay
23:02 now we know now the verticals we know is
23:05 we know real numbers all the numbers
23:07 which can be represented on the number
23:10 nine are real numbers right
23:13 then real 100e a real number it comes
23:17 rational numbers okay then also
23:19 irrational numbers rational numbers all
23:21 the numbers which can be converted to
23:24 the form pyq he rational numbers all
23:26 those numbers which can't be converted
23:28 to the form P by Q now under rational
23:30 numbers again we'll see a category
23:33 called what all the integers non integer
23:35 than negative integers positive integers
23:37 right so you should write ok what is
23:40 water integers so you know what are
23:42 integers integers are
23:45 not having decimals right that means all
23:51 those are my leg - 4 - 3 - 2 - 1 0 1 2 3
23:54 4 till infinity right all these are
23:56 integers okay all these are integers
23:59 from minus infinity to plus infinity it
24:03 is all into this red water now we should
24:09 note here 0 is 0 is a non-negative 0 is
24:13 a non-negative as well as as well as non
24:17 positive integer K 0 is a non-negative
24:21 as well as a non positive integer ok so
24:23 0 is a non-negative as well as non
24:26 positive integer ok this is that into
24:28 this right now
24:31 under intense category the invisi then
24:32 the numbers will come like we know now
24:35 what are whole numbers what are whole
24:38 numbers so whole numbers are all those
24:39 positive numbers and whole numbers are
24:41 always non-negative should write whole
24:44 numbers are always all the morning integers
24:45 integers
24:48 so non-negative in tennis means what
24:53 non-negative integers 1 0 1 2 3 4 till
24:55 infinity these are whole numbers right
24:58 now natural number of natural numbers so
25:01 natural numbers are all the positive
25:04 integers all the positive integers right
25:10 that means 1 2 3 4 5 10 infinity these
25:12 are whole numbers and natural numbers
25:16 right now next is what are all numbers
25:19 so odd numbers are any numbers of the
25:22 form to K minus 1 they form for number 2
25:24 K minus 1
25:28 right see it depends actually it is 2 K
25:33 minus 1 or 2 k plus 1 if you take if
25:35 starting from 1 so if you're taking 2 K
25:39 minus 1 K value I can take a 0 here here
25:43 K sorry K value I should take as 1 in
25:46 this case I am in reverse ok so K value
25:48 I should take as 1 2 K minus 1 means K
25:51 value has to be 1 2 k plus 1 means K
25:53 value has to be 0 starting if you are
25:56 starting with it so K should be 0 so
25:59 that it just starts with 1 right so if K
26:01 it depends on it depends totally on your
26:04 right when you assume or 3/4 two odd
26:07 numbers here assumed 2 k plus 1 and 2 K
26:10 minus 1 or 2 k plus 1 2 k plus 3 so all
26:12 the odd numbers idea of the form 2 K
26:15 minus 1 so that basically means what if
26:17 you take 2 K minus 1 so if I put K is
26:22 equal to 0 it becomes minus 1 so my odd
26:24 numbers are never negative odd numbers
26:26 and even numbers right odd numbers and
26:30 even numbers these are always always
26:32 positive number right so odd numbers and
26:35 even numbers always positive or
26:37 non-negative because you know you also
26:39 considered as an even number so I should
26:43 write odd numbers and
26:47 even numbers are
26:50 non-negative right there non-negative
26:52 there won't be they want to be negative
26:55 ok so tart from k equal to if I start
26:58 from k equal to 0 or HR from equal to 1
27:01 depending wish on which one you are
27:03 degrade you take in this form start k
27:05 equal to 1 T in this form we start k
27:06 equal to zero
27:09 so if i put k equal to 1 year here is it
27:12 1 if you were k equal to 2 you get 3 if
27:15 you get k equal to 3 we will get 5 that
27:20 is a hard all over 1 3 5 7 9 11 13 and
27:23 so on now all the even numbers what are
27:27 you numbers like 0 2 4 6 8 right so even
27:30 numbers are of the form are of the form
27:33 again 2 kids even numbers are of the
27:35 form 2 k you start from k equal to 0
27:40 here ok so 0 2 4 6 8 k equal to 0 to k
27:43 equal to 1 2 sorry K equal to 0 0 k
27:46 equal to 1 to k equal to 2 4 right so
27:49 all the 0 2 4 6 8 are on even numbers
27:51 right and odd and even numbers are all
27:55 this non-negative numbers right now
27:58 so in the next video I discuss few
28:00 questions related to or relating to
28:02 order even numbers and good questions
28:05 will discuss ok thank you [Music]
