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

Summary

In this introductory class, Larry Prakash emphasizes the significance of numbers in various exams like CAT and GMAT. He assures comprehensive coverage of numerical concepts through concept videos, quizzes, and previous year question solutions. Prakash debunks myths about numbers, explaining the vastness of the topic. He categorizes numbers into real and imaginary, focusing on real numbers, which include rational and irrational. Rational numbers, expressible as fractions, are further divided into integers and recurring decimals. Irrational numbers, non-recurring decimals, are also discussed. Prakash provides methods to convert recurring decimals into fractions, promising to tackle more complex problems in subsequent classes.

Takeaways

• 📚 The class 'Numbers' is crucial for exams like CAT and GMAT, with a significant portion of questions related to numerical concepts.
• 🌊 The topic of numbers is vast, often compared to an ocean, indicating the depth and breadth of concepts involved.
• 🔍 The instructor promises comprehensive coverage of numerical concepts, aiming to ensure no exam question is outside the scope of the class.
• 📈 The class is divided into three parts: concept videos, quizzes, and solutions, and previous year CAT questions, emphasizing a structured learning approach.
• 📊 Real numbers, which can be represented on a number line, are contrasted with imaginary numbers, which cannot.
• 📐 Rational numbers are those that can be expressed as a fraction p/q, where q ≠ 0, and include integers and recurring decimals with a fixed pattern.
• 🔢 Irrational numbers cannot be expressed as fractions and include non-terminating, non-recurring decimals, such as the square root of 2.
• 🔄 A method to convert recurring decimals to fractions is demonstrated, involving algebraic manipulation to eliminate the decimal part.
• 📉 The instructor provides a shortcut for converting specific types of recurring decimals to fractions, simplifying the process.
• 📝 The classification of numbers into rational and irrational is further broken down into subcategories like integers, whole numbers, and natural numbers.

Q & A

What is the main focus of the 'Numbers' class introduced by Larry Prakash?

The main focus of the 'Numbers' class is to prepare students for competitive exams like CAT, SSC, GMAT, and others that require aptitude tests, specifically in the area of quantitative aptitude.

How does Larry Prakash describe the scope of the 'Numbers' topic?

Larry Prakash describes the 'Numbers' topic as vast and ocean-like, indicating that it encompasses a wide range of concepts and is crucial for success in various aptitude exams.

What is the promise Larry Prakash makes to his students regarding the comprehensiveness of his 'Numbers' class?

Larry Prakash promises that after covering all the concepts discussed in his videos, students will not encounter a single question in any exam that is outside the scope of his teachings.

What are the three parts of Larry Prakash's video series on 'Numbers'?

The three parts of the video series are concept videos, quiz solution videos, and previous year CAT question videos, all of which are designed to cover the entire syllabus and prepare students thoroughly.

What is the significance of the number 'i' in the context of the 'Numbers' class?

In the 'Numbers' class, the number 'i' represents the imaginary unit, where i^2 equals -1. It is used to explain the concept of imaginary numbers, which are not covered in the syllabus for exams like CAT but are helpful for problem-solving.

How does Larry Prakash define real numbers in his class?

Real numbers are defined as all the numbers that can be represented on the number line, including both rational and irrational numbers.

What is the difference between rational and irrational numbers as explained by Larry Prakash?

Rational numbers are those that can be expressed as a fraction of two integers (P/Q), where Q is not zero. Irrational numbers, on the other hand, cannot be expressed in this fraction form and include non-terminating, non-repeating decimals.

How does Larry Prakash explain the concept of recurring decimals?

Larry Prakash explains that recurring decimals are a type of rational number where a sequence of digits repeats indefinitely after the decimal point, such as 0.333... which can be expressed as 1/3.

What is the method Larry Prakash uses to convert a recurring decimal into a fraction?

The method involves setting up an equation where the recurring decimal is represented by a variable, then manipulating the equation to eliminate the decimal part and solve for the variable in terms of a fraction.

Can you provide an example of how Larry Prakash converts a non-terminating decimal with a repeating pattern into a fraction?

For example, to convert 0.333... into a fraction, set x = 0.333..., then multiply by 10 to get 10x = 3.333.... Subtracting the original equation from this new equation eliminates the decimal part, leaving 9x = 3, which simplifies to x = 1/3.

Outlines

00:00

📚 Introduction to Numbers

Larry Prakash introduces the topic of 'Numbers', emphasizing its significance in various exams like SSC, GMAT, and aptitude tests. He clarifies misconceptions, noting that while the subject may seem daunting, it's crucial for success. Larry promises to cover every concept in numbers, ensuring no question in any exam will be outside the scope of his videos. He outlines three parts of his video series: concept videos, quiz solutions, and previous year question analyses. He asserts that by completing these, students will master numbers and be well-prepared for exams.

05:02

🔢 Classification of Numbers

The script delves into the classification of numbers, distinguishing between real and imaginary numbers. Real numbers, which include integers and fractions, can be plotted on a number line, while imaginary numbers cannot. Larry explains that imaginary numbers are not part of the syllabus for exams like CAT but are useful for problem-solving. He introduces the concept of 'i', the square root of -1, and its properties. The paragraph transitions into a discussion of rational and irrational numbers, with rational numbers being those that can be expressed as a fraction of two integers, while irrational numbers cannot.

10:04

📉 Understanding Rational and Irrational Numbers

Larry further elaborates on rational numbers, which include integers and recurring decimals with a fixed pattern. He provides examples and explains how to convert recurring decimals into fractions. Irrational numbers, on the other hand, are numbers that cannot be expressed as fractions and include non-terminating, non-repeating decimals. The instructor gives examples of irrational numbers like the square root of 2 and explains how to identify them. He also discusses the conversion of certain non-terminating decimals into a form of P/Q, illustrating the process with detailed examples.

15:05

🔍 Converting Recurring Decimals to Fractions

The script provides a method for converting recurring decimals to fractions. Larry uses algebraic manipulation to demonstrate how to eliminate the decimal part by creating equations that, when subtracted, result in a whole number. This process involves recognizing the repeating pattern in the decimal and using it to form equations that can be solved for the fraction representation. He gives several examples, including converting 0.333... to 1/3 and 0.25656... to a fraction, highlighting the importance of recognizing the pattern and the steps to eliminate it.

20:06

📐 Further Exploration of Non-Terminating Decimals

Larry continues to explore non-terminating decimals, focusing on those with specific repeating patterns. He introduces a shortcut method for converting such decimals into fractions by identifying the repeating sequence and using it to form a numerator and denominator. The instructor explains how to handle extra digits in the decimal part by adjusting the multiplication factor and subtraction to isolate the repeating sequence. He also generalizes the method for decimals with different repeating patterns, providing a systematic approach to converting them into fractions.

25:09

🔢 Subcategories of Integers

The script concludes with a discussion on the subcategories of integers, including whole numbers, natural numbers, odd numbers, and even numbers. Larry clarifies the definitions and distinctions between these types of numbers, noting that whole numbers are non-negative integers, natural numbers are positive integers, and both odd and even numbers are non-negative. He provides formulas for generating odd and even numbers and emphasizes their properties, setting the stage for further exploration in subsequent videos.

Keywords

💡Numbers

Numbers are the fundamental building blocks of mathematics, used to count, measure, and perform calculations. In the context of the video, numbers are emphasized as a crucial topic for various exams like the CAT, SSC, and GMAT, where quantitative aptitude is tested. The instructor promises to cover every concept related to numbers, suggesting that mastering this topic is essential for excelling in these exams.

💡Aptitude Exam

An aptitude exam is a type of test designed to measure a candidate's natural abilities, particularly problem-solving skills and logical reasoning. The video script mentions aptitude exams as a context where a strong understanding of numbers is vital, with specific reference to the CAT exam, which contains a significant number of questions related to numerical concepts.

💡Concept Videos

Concept videos are educational resources that focus on explaining fundamental ideas or principles in a subject. The instructor mentions creating over 50 concept videos on numbers, covering topics like factors, factorials, remainders, and number theory. These videos aim to provide a comprehensive understanding of numerical concepts without the need for additional study materials.

💡Quizzes

Quizzes are short tests or exercises designed to assess a student's understanding of a particular subject. In the script, the instructor encourages students to solve quizzes to reinforce their learning and then watch quiz solution videos for a deeper understanding of the concepts. This approach helps students to apply what they've learned and identify areas for improvement.

💡Previous Year Questions

Previous year questions refer to past exam questions that are often used for practice and to understand the exam pattern and difficulty level. The video script highlights the importance of solving previous year CAT questions, which are segregated topic-wise and provided with video solutions, to familiarize students with the types of questions that may appear in actual exams.

💡Real Numbers

Real numbers include all the numbers that can be represented on a number line, encompassing both rational and irrational numbers. The script touches on the classification of numbers, with real numbers being a primary category. Understanding real numbers is fundamental to grasping more complex numerical concepts discussed in the video.

💡Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator. The video explains that rational numbers include integers and recurring decimals with a fixed pattern, which can be written in the form of P/Q. This concept is integral to understanding how numbers operate within equations and real-world applications.

💡Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions. Examples given in the script include the square root of 2 and the square root of 5. These numbers are important in higher mathematics and various scientific fields where precise measurements are required.

💡Integers

Integers are numbers without fractions or decimals, including positive integers (1, 2, 3, ...), negative integers (-1, -2, -3, ...), and zero. The script clarifies that integers are a subset of rational numbers because they can be expressed as a ratio where the denominator is 1 (e.g., 5 can be written as 5/1). Integers are used extensively in calculations and mathematical proofs.

💡Whole Numbers

Whole numbers are non-negative integers, including 0 and all positive integers. The video script distinguishes whole numbers from other types of numbers, emphasizing their use in counting and arithmetic. Whole numbers are fundamental in basic mathematics education and everyday applications.

💡Odd and Even Numbers

Odd numbers are integers that are not divisible by 2, while even numbers are those that are divisible by 2. The script explains that odd numbers can be represented by the formula 2k+1 and even numbers by 2k, where k is an integer. Understanding the properties of odd and even numbers is essential for solving various mathematical problems and has practical applications in fields like computer science and engineering.

Highlights

Introduction to the importance of numbers in various exams like CAT, SSC, GMAT, etc.

Promise to cover every concept of numbers to ensure no questions are left out of the syllabus.

Emphasis on the comprehensive nature of the course, including concept videos, quizzes, and previous year question solutions.

Explanation of the three parts of the course: concept videos, quizzes, and solutions to previous year CAT questions.

The claim that with complete coverage, students will not miss a single question in the examination.

Introduction to the classification of numbers: real and imaginary numbers.

Definition and explanation of imaginary numbers and their representation.

Introduction to iota (i) and its properties in imaginary numbers.

Differentiation between rational and irrational numbers within real numbers.

Definition and examples of rational numbers, including integers and recurring decimals.

Explanation of how to convert recurring decimals into a rational number format (P/Q).

Introduction to irrational numbers and examples, such as the square root of 2.

Discussion on non-terminating decimals with no fixed pattern, which are classified as irrational numbers.

Method to convert a specific type of non-terminating decimal into a rational number format using algebraic manipulation.

Shortcut method for converting repeating decimals to a fraction form using bars to denote repeating sequences.

Categorization of integers into positive integers, negative integers, and zero.

Definition and examples of whole numbers as non-negative integers.

Explanation of natural numbers as a subset of whole numbers, specifically positive integers.

Introduction to odd and even numbers, their definitions, and formulas for generation.

Transcripts

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 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 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

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 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

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

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 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