Basic of Number System and Real Number P3

         Basic Of  Number System

First let's  remind Some basic Definitions:

What exactly Number is? 

In Hindu-Arabic System, we have Ten Digits, namely 0,1,2,3,4,5,6,7,8,9  called Zero, One ,Two, Three, Four, Five, Six, Seven, Eight, Nine respectively.

A Number is denoted by Group of Digits, called Numeral.

Note: If you want to denote a Numeral then use Place-Value  chart.

In the Chart give Number is  701050008.

Let's take an easy Example to Understand The Place-value Chart.

say,  752= 7×100 (Hundred)+ 5×10  (Tens)   +     2×1(Once)

         852= 8×100 (Hundred) +  5×10  (Tens)   +     2×1(Once)

        

         952= 9×100 (Hundred) +  5×10  (Tens)   +     2×1(Once)

Means, In a Number 

                 1.  If digit is at Unit Place just Multiply with One (As name  Unit Place)

               2.  If digit is at Tens Place just Multiply with Ten(As name  Tens Place)

               3.  If digit is at Hundred Place just Multiply with Hundred (As name  Hundred Place)

                            

                                ad so on....

Note: So By using this concept you can write any Number into Equivalent form.

Type of Numbers:   

Natural Number:  We can define it in  one sentence like, Counting numbers are called Natural Number.

example: 1,2,3,4,5,6,7.... all are natural Numbers.

Whole Number: All counting numbers and 0 (zero)  are called Whole Number.

example: 0,1,2,3,4,5,6..... all are whole Numbers.

Note1:    Every Natural Number is  a Whole Number.

Note2:    Zero(0) is a Whole Number not a Natural Number.

Integer Number: All counting Numbers and Zero with Negative Numbers are called Integers.

example:  .......-4,-3,-2,-1,0,1,2,3,4,........

Case1: Set of Positive Integers.

{1,2,3,4,5,6....}

Case2: Set of Negative Integers.

{-1,-2,-3,-4,-5,-6....}

Case3: Set of All Non-Negative Integers.

{0,1,2,3,4,5,6....}

Case4: Set of  All Non-Positive Integers.

{......-4,-3,-2,-1,0}

Even Number: A counting Number divisible by 2 is called an Even Number.

example: 0,2,4,6,8,....etc.

Odd Number: A counting Number not divisible by 2 is called Odd Number.

example: 1,3,5,7,9.....etc.

Prime Number: A counting number is called a Prime Number if it has exactly two factors, namely itself and 1 (one). 

All Prime Number less than 100.

example: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Trick to find out whether given Number is  Prime Number or Not?

let  'x'  is given Number and let  'n' be the smallest counting Number such that  𝑛^2 >=x

read like (n square greater than equals to x.)

1.  let's take a practical example: 137, check prime or not?

Solution:

Step1:

here given,     x=137; 

Step2:

you think ,       n=?

How? :  Think a number whose Square just greater than x.

here,       n=12. (since Square of  12= 144,  that is just greater than 137)

Step3:

since  n=12 (we got)

so we can write Prime Number up to 12.

 Prime Numbers are: 2,3,5,7,11

Step4:

Now we check the give Number (x=137) is Prime or Not  By Dividing each Prime Number  here,

 Prime Numbers based on Question : 2,3,5,7,11.

1. If Remainder is Zero means the Given Number 137 is Not a Prime Number.

2. If Remainder is Not Zero means the Given Number 137 is a Prime Number.

so look,  137/2, 137/3, 137/5, 137/7, 137/11  anyone has Remainder not Zero, so It is Prime Number.

2.  let's take a practical example: 437, check prime or not?

n=?  ;    think:  21*21=441 ( just greater Give Number here 437)

so   n=21.

Now writing all Prime Number up to 21.

2,3,5,7,11,13,17,19

Now divide the given Number by Each Prime Number, just we wrote above.

437/2, 437/3, 437/5, 437/7, 437/11,437/13,437/17,  any has Remainder not Zero,  but Look 437/19 has Remainder Zero. So it(437) is Not a Prime Number.

Note:  If the given Number is divide by Any Specific Prime Number then The Given Number is Not a Prime Number.

Composite Number: The natural Numbers which are not Prime, are called Composite Numbers.

example: 222, 246,  6668.etc

Co-Prime Number: Two natural Numbers x, y are said to be Co-Prime  if there HCF

 (Highest Common factor)  is   1    (One).

example: (2,3), (4,5), (7,9), (8,11) ....etc. are  Co-Prime Number.

Checking Divisibility of Numbers

Divisibility By 2:   Any number is divisible by 2 if and only if its Unit place has value 0,2,4,6,8.

example:     757575757757570,      454546,      9999999999999992,      1234567890 etc.

All are divisible by 2. Since contain  0,  6,  2,  0  at Unit place in above examples.

Divisibility By 4:   A number is divisible by 4 if the number formed by its last Two Digit is divisible by 4.

Example: 1234516,  5656720, 77777777777728,  99999999932 etc.

here in example  16, 20, 28, 32  are last Two digit and divisible by 4, so whole number is Divisible by 4.

Divisibility By 8:   A number is divisible by 8 if the number formed by Hundred's, Ten's and Unit's Digit of the given Number is divisible by 8.

Example: 765756352, 8578969352 etc.

here in example  352  is at Hundred's, Ten's and Unit's place and divisible by 8, so whole number is Divisible by 8.

Divisibility By 5:   A number is Divisible by 5 only when it's Unit Digit is 0 or  5.

Example: 574747470,  456464445,  9888990  etc.

here in example  0 or 5 is at  Unit's place and divisible by 5, so whole number is Divisible by 5.

Divisibility By 10:   A number is Divisible by 10 only when it's Unit Digit is  0.

Example:  646464890, 44445770, 81123450  etc.

here in example  0 is at  Unit's place and divisible by 10, so whole number is Divisible by 10.

Divisibility By 3:   A number is Divisible by 3 only when the Sum of Digits is Divisible by 3.

Example: 123,  4563, 555 etc.

Here Sum of Digit:  1+2+3=6  and we know 6 is Divisible by 3 so 123 is also Divisible by 3.

Here Sum of Digit:  4+5+6+3=18  and we know 18 is Divisible by 3 so 4563 is also Divisible by 3.

Divisibility By 9:   A number is divisible by 9 only when the Sum of it's Digit is Divisible by 9.

Example: 4563, 123453 etc.

Here Sum of Digit:  4+5+6+3=18  and we know 18 is Divisible by 9 so 4563 is also Divisible by 9.

Here Sum of Digit:  1+2+3+4+5+3=18  and we know 18 is Divisible by 9 so 123453 is also Divisible by 9.

Divisibility By 11:   A number is Divisible by 11 if the Difference between the Sum of its at Odd Places and the Sum of its digit at Even Places is Either 0 or a number is Divisible by 11.

Example: 29435417 etc.

(Sum of its Digit at Odd Places) -    (Sum of Digits at Even Places)

(7+4+3+9)                                   -    (1+5+4+2)  =  11.  

here Difference is Divisible by 11 so whole Number is also Divisible by 11.

Basic Of Real Number

What exactly Real Number is? 

Real Number:  Real numbers are the numbers which include both rational and irrational numbers. 

Rational numbers such as integers (-1, 0, 1), fractions(1/5, 12.5). Irrational numbers such as √7, π(22/7), etc., are all real numbers..

Note: √-7 is Imaginary Number(Due to Negative Sign within square root), you read it in another Chapter name Complex Number.

Let's Elaborate Rational Number.

Terminating decimal expansion:  If any real number is in p/q  form  and their Quotient is terminated at certain time, called Terminating decimal expansion.

Example: 3/8=0.375,  13/125=0.104,   7/80=0.0875,  14588/625=23.3408   etc.  look their Quotient after certain time Terminated.

Non-Terminating Repeating:  If any real number is in p/q  form  and their Quotient is non-terminated and repeating form,  called Non-Terminating Repeating.

Example: 1/7= 0.1428571428571............Non-Terminated and Repeating.
 
same as  1/3=0.333333333333333.........Non-Terminated and Repeating.

Thanks :)

happy Learning.