Numbers 1 || Number Systems || CAT Preparation 2024 || Quantitative Aptitude

0:00 [Music] 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

0:48 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

1:35 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

2:21 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

3:08 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

3:53 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

4:39 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

5:26 classification of numbers the numbers are basically classified as real numbers and imaginary numbers and imaginary numbers 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

6:13 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

6:58 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

7:44 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

8:29 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

9:16 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

10:01 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

10:46 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

11:35 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

12:20 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 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

13:08 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

13:53 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

14:41 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

15:27 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

16:13 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

16:58 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

17:45 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 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

18:31 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

19:18 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

20:07 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

20:53 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

21:41 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

22:29 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

23:17 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

24:03 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 so non-negative in tennis means what

24:48 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

25:35 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

26:22 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

27:09 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

27:55 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]