TOPIC: NUMBERS AND NUMBER SYSTEMS

SUB-TOPIC 1: Classification and Types of Numbers and some basics.

* NUMBERS can be said to be REAL, IMAGINARY or COMPLEX.Complex numbers are of the form (a+ib) where a is the real part and b the imaginary part.

* REAL numbers consists of INTEGERS and DECIMALS.

* INTEGERS are either NEGATIVE integers OR WHOLE numbers.

* negative infinity............, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, infinity are INTEGERS.* 0,1,2,3,4,5,6,.......infinity are WHOLE numbers.* 1,2,3,4,5,6,...............infinity are NATURAL numbers.

Natural numbers can be further classified into:Primes: 2,3,5,7, 11, etc..Composites: 4,6,8,9,10,12, etc.And 1.

* 1 is neither prime nor composite.* Smallest prime number is 2, which infact is the only even prime number.

* An integer is either EVEN or ODD. * Even numbers are of the form 2n and an odd number 2n+1* NOTE: Negative numbers are also either even or odd.

* Real numbers can also be divided into two types: RATIONAL and IRRATIONAL* Rational numbers are those numbers which can be written in p/q form where p,q are integers and q is not equal to 0.

NOTE: Irrational Numbers do not mix.So, whenever you see an irrational number while solving a problem, eliminate the options which are NOT irrational. Like, if you come across root(3) in a problem involving area of equilateral triangle, most of the times the answer option should be irrational until and unless it is being cancelled out during course of solving problem like in the questions involving ratios.

* Now the decimals part; decimals can be of two types: TERMINATING and NON-TERMINATING.TERMINATING: 0.3, 0.25, 0.875NON TERMINATING: 0.33333.. , 0.646464, 0.3454545, 0.2458648515.... etc

NON-TERMINATING decimals can be of two types: RECURRING and NON-RECURRINGRecurring: Pattern is repeated. Like 0.33333.. , 0.646464, 0.3454545,Non-recurring: No repetitive pattern.. like in values of e, pi, rt(2), rt(3)

So, all real numbers are RATIONAL numbers EXCEPT the NON-RECURRING NON TERMINATING decimals.

To summarise the hierarchy:NUMBERS---- REAL, IMAGINARY, COMPLEXREAL ............ INTEGERS, DECIMALSINTEGERS...... NEGATIVE NUMBERS, WHOLE NUMBERSNATURAL NUMBERS.. PRIME, 1 and COMPOSITEDECIMALS.... TERMINATING AND NON TERMINATINGNON-TERMINATING... RECURRING and NON-RECURRING

TASK: Make a hierarchy of numbers and then one should be able to say things like:All integers are rational numbers,All prime numbers are real numbers, etc. This will help.

HOW TO WRITE ANY GIVEN RECURRING DECIMAL into RATIONAL number form?

Example: Convert 0.33333.. in rational number form.

METHOD 1 (Conventional method):

x= 0.33333...10x= 3.33333...Subtract, hence 9x=3so, x= 1/3

METHOD 2 (Faster method):

[ Number (Digits which are not repeated) /As many 9s as the Number of digits repeated followed by as many 0s as the number of digits not repeated ]Here, 3/9 = 1/3

Example: Convert 0.45454545... in rational number form.

METHOD 1 (Conventional method):

x= 0.45454545.....100x= 45.454545.....Subtract, 99x= 45 So, x= 45/99 = 5/11

METHOD 2 (Faster method):

[ Number (Digits which are not repeated) /As many 9s as the Number of digits repeated followed by as many 0s as the number of digits not repeated ]

Here, 45 /99 = 5/11

Example: Convert 0.5424242... in rational number form.

METHOD 1:

x= 0.542424242...10x= 5.42424242...1000x= 542.424242...subtracting last two,990x= 537x=537/990 = 179/330

METHOD 2:

[ Number (Digits which are not repeated) /As many 9s as the Number of digits repeated followed by as many 0s as the number of digits not repeated ]

Here, 542 5 / 990 = 537/990 = 179/330

One more example: x= 0.526484848....1000x= 526.484848...100000x= 52648.484848...Subtracting last two,99000x= 52122x= 52122/99000 (reduce now)

Faster method: [ Number (Digits which are not repeated) /As many 9s as the Number of digits repeated followed by as many 0s as the number of digits not repeated ]

So, here52648 526 / 99000 (as two digits repeated, hence two 9s, and three digits not repeated, hence three 0s).

PERFECT NUMBERS: The numbers whose factors (except the number itself) add up to yield the number are called PERFECT NUMBERS.OrAny even postive number of the form P(P+1)/2 where P is the Mersenne Prime is a Perfect Number.Mersenne prime: Prime number of the form 2^n 1hence, 3,7,31 are some Mersenne primes.Put these values of P to get perfect numbers.

Example of perfect number: 28Factors of 28: 1,2,4,7,14,28Sum of factors except the number itself: 1+2+4+7+14 = 28Hence, perfect number.

Note, 6 is the smallest and only single digit perfect number.28: Only two-digit perfect number.496: Only three-digit perfect number8128: Only four-digit perfect number.

Sum of all factors of perfect number (including the number)= twice of the number.

* TRIANGULAR NUMBERS: 1,3,6,10...This pattern 1,3,6,10 is really important in the world of quants and should strike the moment you see it.This is actually sum of natural numbers.1,1+2=3,1+2+3=6,1+2+3+4=10, and so on..So, a triangular number is of the form n(n+1)/2 where n is a natural number.

Interestingly, Sum of two consecutive triangular numbers is a PERFECT SQUARE.

EXTRA GYAAN: The most common indeterminate forms are denoted 0/0, INFI/INFI, 0 INFI, INFIINFI, 00, 1^INFIand INFI^0.hence, never give wrong answers as 0 or 1 or anything else when you encounter such forms..answer will be indeterminatable unless and until calculas limit is to be applied.

SOME QUESTIONS:1) Find the smallest number which when multiplied to 0.abababab... gives an integer?2) In the question above, find the largest such three-digit number.3) If 0.ababab.... = 5/11 find (a+b) ?4) If n is an odd natural number, what is the highest number that always divides n (n^2 1) ?5) 1 and 8 are first such numbers for which 1+2+3+4+... +n is a perfect square. Which is fourth such number?6) Express 0.124898989... in the rational form (by faster method).

Practice more on the group..

NEXT SUBTOPIC: PRIMES AND COPRIMES (Basics and high-level concepts).