Number systems

                                                   NUMBER SYSTEMs

1) NATURAL NUMBERS [N]:{1,2,3,------}

Natural numbers starts with “1” and end with ∞ [infinity] or endless

Ø  In the prehistoric days the people doesn’t know the”0” and fraction etc. The numbers 1 to 9 are used for counting only.

2) WHOLE NUMBERS (W)={0,1,2,3-----}

                        It starts from 0 with ∞ i.e  endless.

 *The Indians found “0” in “876” BC. Then “0” was joined to the natural numbers became whole numbers.

    W=NU{0}

Set of natural numbers is sub-set of whole numbers.

 3) ZERO NUMBES (Z) ={-∞---------5,-4,-3,-2,-1,0,1,2,3-------∞)
The number system having "+”ve and “-“ve,  zero and  end with -∞ and end with +∞ i.e endless on both sides [The symbol ∞ express infinite].The system identifying the -ve numbers as barrows, debts,  opposite directions in geometry.

E.g :If Height from earth = -6m i.e  depth from earth=6m while solving a mathematical problem in heights and distances we doesn’t aware the result whether it is +v or -Ve .  The result can be identified as depth if it is –ve number.

Loss should be as in "-ve“.
 In zero numbering system there are no fractions, rational numbers. They are integers only.
 The zero numbers set consists of natural numbers whole numbers

N⊂W⊂Z

 Set of natural numbers is sub set of set of whole numbers

 Set of whole numbers is sub set of set of zero numbers.

3) Rational numbers [Q]
The set of numbers having numbers of fractional between any two integers and can be written as p/q form [can be written as decimal fractions also like 7.235719].

The number of fraction be found In-between even two integer as numbering scale is “∞” .This is called as density property                                                   

*[N⊂W⊂Z⊂Q]

 The rational numbers  between any two rational numbers
E.g: The rational numbers between 0 to 1

       If the 0 to 1 scale divided into 10 small parts

0---1/10---2/10---3/10---4/10---5/10---6/10---7/10---8/10---9/10---10/10 [i.e is 1]
                                                                                                                  
If the sale is divided in to small points as 100

0--1/100---2/100---3/100---4/100---5/100--------------------------98/100---99/100---100/100 i.e 1

          Similarly the scale between 0&1 can be divided into “n” numbers as small.  So we identify the number of rational numbers between any two rational numbers is infinite. This property is called as density property.

 

Irrational numbers [Q¹]:   √2, √3, √5, √6, √7, √11 etc.

The numbers, whose value with none repeated endless decimal fractions.

^Eg: √3=1.73205080757…….

Eg: √2=1.414213562373…..
 Rationalization Factor: If product of two irrational numbers is a rational number. Then the two irrational numbers are called as rationalization factors each other.
E.g: √3X[-√3]=-3,-3 is rational number. So √3 is rationalization factor of -√3
E.g: [√5-√3]X[√5+√3]=[√5]²-[√3]²⁼5-3=2 identified as[a+b]X[a-b]=a²-b²
 So  rationalization factor of [√5-√3] is  [√5-√3] vice versa.

Real Numbers [R]

The combination or unity of Rational numbers and irrational numbers is the set of real numbers.

N⊂W⊂Q⊂R

3)  Imaginary Numbers [i]: The set of numbers are nth root of “-ve“ numbers when n is even number.

                            i.e √-2, √-3, √-9, √-7, √-25 etc. 

4] Complex numbers [C]:

The numbers, which gives result of addition, subtraction, multiplication and division between imaginary numbers and real numbers. i.e in the form of  or a+√-b=a+ ix     where =I, √b= x the value of b is +ve

a-√-b=a+ ix     

a√-b= iax  

*N⊂W⊂Q⊂R⊂C

Sub-Classification of the integers.

1] Even numbers: the numbers divisible by 2 are called even numbers.

E.g: 2,4, 6, 8, …….

2] odd numbers :numbers not divisible by 2 are called odd numbers.

E.g: 1, 3, 5, 7, …….

3] Composite numbers: the numbers having more than two factors is called composite numbers.

E.g1] : 4   factors of 4 are 1, 2 and 4. The number of factors is 3

E.g2]: 6    factors of 6 are 1, 2, 3 and 6 The number of factors is 4

The numbers consist of only two factors i.e 1 and itself are called prime numbers.

e.g1]: 2 factors of 2 are 1 and 2. The number of factors is 2

*important note: 2 is only one even prime number.

E.g 2]: 3, 7, 11, 13, 17, 19, 23, ……..

* Important note : All prime numbers are odd numbers but all odd numbers are not prime numbers.

Set of prime numbers⊂set of odd numbers

Twin prime numbers
Any pair of prime numbers has difference as 2 is called twin prime numbers.

e.g1]  3,5 difference is 5-3=2

E.g2]  5,7 difference is 7-5=2

Hence 3, 5 are twin prime numbers and 5, 7 are twin prime numbers.
Fractions:all rational numbers except complete integers are  called fractions

these are in the form of p/q.further the numbers are classified as below
a]proper fractions
b]improper fractions
c]mixed fractions

a]proper fractions:These numbers are having numerator is less than denominator always [i.e p<q]
e,g 1/2, 3/4, 5/7, 3/11, 2/7, 4/9, 6/11, 3/10 etc.
The value of every number is less than 1 for always.

b]improper fractions:These numbers are having numerator is greater than denominator always [i.e p>q]
e,g 3/2, 5/4, 7/5, 11/4, 7/2, 9/5, 11/7, 10/3 etc
The value of every number is more than 1 for always.
c]mixed fractions: these numbers are combination of an integer and proper fraction i.e other form of the improper fraction.
e.g 13/4 , 22/3 etc.
The value of every number is more than 1 for always.

Conversion of the improper fraction in to mixed fraction:
Take an improper fraction i.e p/q
Then dived the numerator with denominator identify the quotient and reminder
if the quotient is X and the reminder is y, Take the quotient as complete number, reminder as numerator and denominator of the improper fraction as the denominator of the proper fraction in mixed fraction, finally simply write the mixed fraction equal to the improper fraction given as X y/q
e.g: 7/3 is an improper fraction as 7>3
 divide the 7 by 3, we can obtain a quotient as 2 [2x3=6, reminder =7-6=1]
then mixed fraction= 21/3

Conversion of the mixed fraction  in to improper fraction:

e.g take a mixed fraction as  X y/q
Multiply the complete number with the denominator of  it's proper fraction and add the product to the numerator. write the sum as numerator of the proper fraction and denominator is same
E.g 3 2/7 is mixed fraction. Complete number[X]=3, denominator[q]= 7 and numerator[y]=2
simply as above numerator of the improper fraction[p]=Xxq+y=3x7+2=23
denominator is same
Then finally improper fraction=23/7

Decimal fractions:
The proper improper and mixed fractions can be changed to decimal fractions i.e in the form of 10th parts, 100th parts,1000th parts .......os the 1.

e.g 1/2=0.5
2] 1 [0.5                 2 can not divide 1 i.e can divide 0 times.
     0                       reminder is 1, for further division, reminder indicated by 10 part of 1

     10                    the 2 can divide 5 times.the quotient 5 written after the decimal point.
     10
       0

The types of the decimal fractions.

Some multiplication [product] short cuts or tips:

Some multiplication [product] short cuts or tips:

1] multiplication of the numbers with 5 in ones place;

                                                            1000's and 100's places    10's 1's places

 A] Squares: eg: 1]      35X35=1225   [by 3x[3+1]                             5x5]]

                            2]     75X75=5625   [by 7x[7+1]                              5x5]]

*For n5xn5=n[n+1]25 [n[n+1]is in 1000’s and 100’s place +25 is in 10’s and one’s place]

B] the numbers with 5 in one’s place and differ by 10                                         

Eg:1]  35X45=1575

       2] 65x55=3575

*For [n-1]5xn5=[n²-1]   75

2]the numbers multiplied by 5

   E.g1] 256x5=1280 [i.e [256/2]x10

   E.g2] 445x5=2225 [i.e [445/2]x10+5]]

If an even number n multiplied by 5 the product is [n/2]x10

If odd number n multiplied by 5 the product is [[n-1]/2]x10+5

3] square of the numbers near to 10,100,100…etc.

                                      10,s   one’s

            E.g.1] 9x9=81 [9-1]   1     =81

                                     100’s  one’s 

            Eg. 2] 98x98= [[98-2] [100-98]x[100-98]=9604

i.e 98-[100-98] is in 1000’s and 100’s place further [100-98][100-98] is in 10’s place and one’s place.

 Same way 96x94=[96-[100-94] [100-94]x[100-96]]=9024.

3] product of the numbers near to 10,100,100…etc.

E.g1]                           [two places left to 10’s]          two places left from start

1000’s, 100’s                           10’s 1’s 

Eg. 98x97=9506         [98-[100-97]]                    [100-98]x[100-97] as the number of digits in every number is ‘’2’’

E.g2] Can you imagine the product of the two numbers 99994x99997

5places                                                5 places          

Simple: 99994-[1000000-99997]    [1000000-99994]x[100000-99997]=9999100018.

The number of digits in product=2xthe number of digits in each number.

So we have obtained the product by multiplying the each number differences  to 100000 is 6x3=18 it should be in first 5 digits so that is 00018. Next, the difference between the one of the numbers to Nearest Round number 100000-another number]

Product of any two numbers which are near to 50, 500, 5000 etc

Two places       Two places

E.g1: 47X48=[47-[50-48]]/2      [50-47]x[50-48] here the left side number should be multiplied by 50/100 i.e ½

                    =[47-2]/2                3x2

                    =45/2                       06

                    =22.5                       06

                                 =2256

The fraction 0.5 in 100,s place= direct number 5 in 10’s place [0.5x10=5]

Two places       Two places

Eg2] 44x46= [44-[50-46]]/2         [50-44]x[50-46]

                    =[44-4]/2                   6x4

                   = 40/2                          24

                    =20                             24

            The product=2024

                            Three places                three places

E.g3] 498x493=[498-[500-493]]/2      [500-498]x[500-493]

                        =[498-7]/2                   2x7

                        = 491/2                        014

                        =245.5                         014

                        =245514

The fraction 0.5 in 1000’s place =direct integer in 100’s place.

Product of any two numbers which are near to 25, 250, 2500 etc

Two places       Two places

Eg1] 23x24= [23-[25-24]]/4    [25-23]x[25-24] ,  the left side number should be multiplied by 50/100 i.e ¼

e.g2]           = [23-1]/4             2x1

                    =22/4                   02 as two places

                    =5.5                     02

                                  =552

The fraction 0.5 in 100’s place = direct integer 5 in 10’s place.

The all remaining product types are to be practiced as the tips applied for some special cases to easy to solve.

The above particular cases only, the multiplication tips applied.

For remaining, the normal method of practice is sufficient.

 3] The 9 is a magic number can remember 9^th table easily.

A]  9^th Table:                          Ten’s place      one’s place

                        9x1=9              0                                  [9-0]

                        9x2=18            1                                  [9-1]

                        9x3=27            2                                  [9-2]

                        9x4=36            3                                  [9-4]

                        9x5=45            4                                  [9-4]

                        9x6=54            5                                  [9-5]

                        9x7=63            6                                  [9-6]

                        9x8=72            7                                  [9-7]

                        9x9=81            8                                  [9-8]

                        9x10=90          9                                  [9-9]

                                                 10's place                    1's place

                        9xn=                [n-1]                            [10-n]

If we/ remind the 9 table up to 9x10=90 we can solve the many multiplications by 9

The remainder when a number divided by 9 with out division

1] 95÷9 quotient is 10 and remainder 5

2] 782÷9 quotient is 86 and remainder 8

In each event we find out quotient for finding out remainder.

But we can find out remainder individually just by adding the digits of the number up to 1digit.

E.g. 78932185÷9 then remainder 7+8+9+3+2+1+8+5=43 then 4+3=7 so the remainder is 7.

By eliminating the 9,s from sum obtained ≥ 9 while calculating sum,   we can get remainder easily

[7+8]-9+9-9+[3+2+1+8-9]+5=6+5+5 again-9=7

Add upto 9 and eliminate

6+[1+8]+9+[3+2+[1+8]]+5 after eliminating the red colour sums we get  =6+5+5=[6+3]+2+5 again after eliminating the red colour sums we get =7

2] By The same way if  a number divided by 3, we can find the remainder, without knowing quotient. 

E.g 57÷9, the remainder is 5+7=12, further 1+2=3 but the 3 is divisible by 3 so remainder is 0

E.g 43÷9 , the remainder is 4+3=7,  the 7 is not divisible by 3, so remainder is 7-3x2=1

All the best wishes for competitive examination and aware of mathematics. [ a well known subject means ‘’can be expressed the subject in mathematics’’]

[Next we approach to provide best understanding on divisibility and number systems i.e types of numbers] 

1] multiplication of the numbers with 5 in ones place;

                                                            1000's and 100's places    10's 1's places

 A] Squares: eg: 1]      35X35=1225   [by 3x[3+1]                             5x5]]

                            2]     75X75=5625   [by 7x[7+1]                              5x5]]

*For n5xn5=n[n+1]25 [n[n+1]is in 1000’s and 100’s place +25 is in 10’s and one’s place]

B] the numbers with 5 in one’s place and differ by 10                                         

Eg:1]  35X45=1575

       2] 65x55=3575

*For [n-1]5xn5=[n²-1]   75

2]the numbers multiplied by 5

   E.g1] 256x5=1280 [i.e [256/2]x10

   E.g2] 445x5=2225 [i.e [445/2]x10+5]]

If an even number n multiplied by 5 the product is [n/2]x10

If odd number n multiplied by 5 the product is [[n-1]/2]x10+5

3] square of the numbers near to 10,100,100…etc.

                                      10,s   one’s

            E.g.1] 9x9=81 [9-1]   1     =81

                                     100’s  one’s 

            Eg. 2] 98x98= [[98-2] [100-98]x[100-98]=9604

i.e 98-[100-98] is in 1000’s and 100’s place further [100-98][100-98] is in 10’s place and one’s place.

 Same way 96x94=[96-[100-94] [100-94]x[100-96]]=9024.

3] product of the numbers near to 10,100,100…etc.

E.g1]                           [two places left to 10’s]          two places left from start

1000’s, 100’s                           10’s 1’s 

Eg. 98x97=9506         [98-[100-97]]                    [100-98]x[100-97] as the number of digits in every number is ‘’2’’

E.g2] Can you imagine the product of the two numbers 99994x99997

5places                                                5 places          

Simple: 99994-[1000000-99997]    [1000000-99994]x[100000-99997]=9999100018.

The number of digits in product=2xthe number of digits in each number.

So we have obtained the product by multiplying the each number differences  to 100000 is 6x3=18 it should be in first 5 digits so that is 00018. Next, the difference between the one of the numbers to Nearest Round number 100000-another number]

Product of any two numbers which are near to 50, 500, 5000 etc

Two places       Two places

E.g1: 47X48=[47-[50-48]]/2      [50-47]x[50-48] here the left side number should be multiplied by 50/100 i.e ½

                    =[47-2]/2                3x2

                    =45/2                       06

                    =22.5                       06

                                 =2256

The fraction 0.5 in 100,s place= direct number 5 in 10’s place [0.5x10=5]

Two places       Two places

Eg2] 44x46= [44-[50-46]]/2         [50-44]x[50-46]

                    =[44-4]/2                   6x4

                   = 40/2                          24

                    =20                             24

            The product=2024

                            Three places                three places

E.g3] 498x493=[498-[500-493]]/2      [500-498]x[500-493]

                        =[498-7]/2                   2x7

                        = 491/2                        014

                        =245.5                         014

                        =245514

The fraction 0.5 in 1000’s place =direct integer in 100’s place.

Product of any two numbers which are near to 25, 250, 2500 etc

Two places       Two places

Eg1] 23x24= [23-[25-24]]/4    [25-23]x[25-24] ,  the left side number should be multiplied by 50/100 i.e ¼

e.g2]           = [23-1]/4             2x1

                    =22/4                   02 as two places

                    =5.5                     02

                                  =552

The fraction 0.5 in 100’s place = direct integer 5 in 10’s place.

The all remaining product types are to be practiced as the tips applied for some special cases to easy to solve.

The above particular cases only, the multiplication tips applied.

For remaining, the normal method of practice is sufficient.

 3] The 9 is a magic number can remember 9^th table easily.

A]  9^th Table:                          Ten’s place      one’s place

                        9x1=9              0                                  [9-0]

                        9x2=18            1                                  [9-1]

                        9x3=27            2                                  [9-2]

                        9x4=36            3                                  [9-4]

                        9x5=45            4                                  [9-4]

                        9x6=54            5                                  [9-5]

                        9x7=63            6                                  [9-6]

                        9x8=72            7                                  [9-7]

                        9x9=81            8                                  [9-8]

                        9x10=90          9                                  [9-9]

                                                 10's place                    1's place

                        9xn=                [n-1]                            [10-n]

If we/ remind the 9 table up to 9x10=90 we can solve the many multiplications by 9

The remainder when a number divided by 9 with out division

1] 95÷9 quotient is 10 and remainder 5

2] 782÷9 quotient is 86 and remainder 8

In each event we find out quotient for finding out remainder.

But we can find out remainder individually just by adding the digits of the number up to 1digit.

E.g. 78932185÷9 then remainder 7+8+9+3+2+1+8+5=43 then 4+3=7 so the remainder is 7.

By eliminating the 9,s from sum obtained ≥ 9 while calculating sum,   we can get remainder easily

[7+8]-9+9-9+[3+2+1+8-9]+5=6+5+5 again-9=7

Add upto 9 and eliminate

6+[1+8]+9+[3+2+[1+8]]+5 after eliminating the red colour sums we get  =6+5+5=[6+3]+2+5 again after eliminating the red colour sums we get =7

2] By The same way if  a number divided by 3, we can find the remainder, without knowing quotient. 

E.g 57÷9, the remainder is 5+7=12, further 1+2=3 but the 3 is divisible by 3 so remainder is 0

E.g 43÷9 , the remainder is 4+3=7,  the 7 is not divisible by 3, so remainder is 7-3x2=1

All the best wishes for competitive examination and aware of mathematics. [ a well known subject means ‘’can be expressed the subject in mathematics’’]

[Next we approach to provide best understanding on divisibility and number systems i.e types of numbers] 

Small difference between mathematic operator and sign

            Most of the students could not find the deference between mathematical operator like subtraction, addition, multiplication and division and sign +ve and –ve and fails in competitive mathematics with mistake. To avoid the mistake, please read the following examples.

Find out operator and sign

E.g 1:   2+3=5 in this equation + is operator i.e addition and the sign of two numbers is +ve [the numbers without sign means +ve numbers] simply addition of two +ve numbers.

E.g2:    7-3=4 here – is an  operator of subtraction and the two numbers 7 and 3 are +ve

            The same can be written as 7+[-3] here operator is + i.e addition but sign of the 3 is –ve i.e addition of one +ve number 7 and one –ve number -3.

E.g3:    8-[-3]=11 here operator is 1^{st “} –“ and sign of 8 is +ve and sign of 3 is –ve

E.g4:    -8/[-16]=1/2 here operator of the two numbers is / sign of the numbers is –ve.

Finding of unknown number ‘X’

E.g1:    if X/[-2]=8 then the X=8x[-2] the sign of the denominator cannot be changed from RHS to LHS to find out value of the X only operator  / can be changed i.e the number -2 is dividing X on LHS and to find  X valve the -2 multiply 8 one RHS without changing of it’s sign.

E.g2:    if X-2=8, then X=8+2 here 2is subtracted from X on LHS and to find out X, 2 is to be added on RHT without changing it’s sign +ve. [as X-[+2]=8].

E.g 3:   If Xx[-2]=8 then X=8/ -2  here -2 is multiplying the X on LHS and to find out the X the number -2 is dividing the 8 on RHS without changing it,s sign –ve.

E.g4:    If X/[-2]=8 then X=8x[ -2]  here -2 is dividing the X on LHS and to find out the X the number -2 is multiply the 8 on RHS without changing its sign –ve.

Conclusion:

The operator is different from sign. The operation between two numbers changes from LHS to RHS while finding out of unknown values, without changing its sign of the number other than the number to be findout on LHS.

[Note : Aim of the blog is to make the students bright in mathematics and make the parents as teachers.]

Addition of the numbers

Addition of the two +ve numbers gives a +ve number.

+ve number+ +ve number=+ve number

e.g : 3+5=7

Addition of the two –ve numbers gives a –ve number

-ve number+ -ve number=-ve number

-7+[-3]=-10

Addition of one –ve number and one +ve number vice versa gives the difference between the numbers with the bigger numbers sign.

-ve number+ +ve number= Sign of big number [difference of the numbers]

e.g: -3+2=-1 here the sign of the big number is –ve and difference between the numbers 3 and 2 is 1.

e.g: -7+10=3 here the sign of the big number is +ve and difference between the numbers 10 and 7 is 3.  

Conclusion

*Addition of the similar sign numbers gives the some of the numbers with same  sign.

*Addition of the different sing numbers give the difference of the numbers with bigger number sign.

Multiplication

Multiplication or division between two +ve numbers give a +ve number

e.g: 3x5=15

Multiplication or division of between two –ve numbers gives +ve number

E.g: -3x[-5]=15

Multiplication or division between one –ve number and one +ve number vice versa gives –ve number.

E.g: -3x5=-15

E.g: -15/3=-5

E.g: 3x[-5]=-15

E.g: 15/[-5]=-3

 Conclusion

*Multiplication or division between the similar sign numbers gives always +ve number.

*Multiplication or division between the different sign numbers gives always –ve number.