Division in Mathematics

                      

                The beauty of Mathematics depends on basic four arithmetic Operations along with some functional operation:-

1. Addition  ( + )
2. Subtraction ( - )
3. Multiplication  ( x )
4. Division (÷)                                                                                                                                                                                                                                                                  So, we shall discuss the fourth operation Division.

DIVISION

                           Generally, Division symbol  (÷)having a horizontal line a dot below and another at above , OR  " /  " ( Read as a fraction  Bar or Solidus) is often used to indicate mathematical division from ancient times.  The result is expressed with equals " = " symbols.

                          it is a basic arithmetic operation that represents the operation of repeated subtraction of objects from a given collection. The result of the division is also called a  Repeated Subtraction.

                      The division is considered as Reciprocal of multiplication and can be learned as follows:-  

                
4  ÷ 1 = 4 means how many times one can be subtracted from 4 to get Remainder 0.    
 4
-1                            (  Once subtracted )
------------
 3
-1                              (  Twice subtracted )
-------------
 2
-1                                ( Thrice subtracted )
------------
 1
-1                               (  Fourth times subtracted )
------------
0  ( Remainder)
-------------------             
              Since 1 is repeatedly subtracted FOUR times so answer of 4  ÷ 1 is 4.

           The simplest way of viewing division is in terms of QUOTIENT: from the quotient perspective, 15 / 5 means the number of 5s that must be added to get 15. In terms of partition, 15 / 5 means the size of each of 5 parts into which a set of size 15 is divided. For example, 15 apples divide into five groups of THREE apples, meaning that fifteen divided by five is equal to Three. This is denoted as 15/ 5 = 3, or 15/5 = 3. 
The number / Polynomial which divided is known as Dividend and the obtained result is called Quotient, The number from divided is called as Divisor. In the example, 15 is the dividend, 5 is the divisor, and 3 is the quotient, and Remainder is zero.

15

-5                           (  Once subtracted )

------------

 10

-5                              (  Twice subtracted )

-------------

5

-5                                ( Thrice subtracted )

------------

0  ( Remainder)   Therefore 15 /5 = 3.

• BY DIVIDING  NUMBERS ZERO ( 0) 

                     1  ÷ 0 = ????                        means how many times ZERO can be subtracted from 4 to get Remainder 0.  

1

-0                          (  Once subtracted )

------------

1 ( Remainder)

-0                              (  Twice subtracted )

-------------

1 ( Remainder)

-0                             ( Thrice subtracted )

------------

 1 ( Remainder)

-0                              (  Fourth times subtracted )

------------

1  ( Remainder)

-0

---------

1 ( Remainder)

                             IN ALL CASES REMAINDER IS SAME TO THE DIVIDEND 1 SO IF WE USE THIS PROCESS UNCOUNTED NUMBER OF TIMES THEN Remainder IS ALWAYS 1 ( DIVIDEND ), SO DIVISION OF ZERO IS NOT POSSIBLE OR RESTRICTED.

1  ÷ 0 = ????           ANSWER -  NOT POSSIBLE.

By Division of zero to any positive numbers, is not possible.

•   BY DIVISION OF   ONE (1) 

  1 ÷ 1  = 1 

   2÷1 = 2 - (1 + 1 ) = 2   ( Twice subtracted)

        3  ÷1 =  3 - (1 + 1 + 1 ) = 3 ( Thrice subtracted)

      4   ÷ 1=  4 - (1 + 1 + 1 + 1  ) = 4

  5 ÷1=  5 - (1 + 1 + 1 + 1 + 1 ) = 5 

6  ÷ 1= 6 - ( 1 + 1 + 1+ 1 + 1 + 1 ) = 6    

 7   ÷1=  7 - (1 + 1 + 1+ 1 + 1 + 1 + 1  )= 7

 8   ÷1 =  8 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 ) = 8
 9 ÷ 1 =  9 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1)  = 9
 10    ÷ 1= 10 - (1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1 +1 )  = 10  ( Ten times subtracted)

Any number  (x  ) ÷ 1 = any number ( x)

 ...

By multiplication of ONE  to any numbers, results are always the same number so ONE ( 1 )  Iis known as an identity for the division.

BY MULTIPLYING  DIFFERENT NUMBERS                                                                                        

            8  ÷ 2 = 4   

        12  ÷ 2 = 6

    18 ÷ 3 = 6 

  100 ÷ 10 = 10 

          21 ÷ 7  = 3 

                                           similarly, more results can be verified.

--------------------

Multiplication in Mathematics

                The beauty of Mathematics depends on basic four arithmetic Operations along with some functional operation:-

1. Addition  ( + )
2. Subtraction ( - )
3. Multiplication  ( x )
4. Division (÷)                                                                                                                                                    so we shall discuss the third operation multiplication.

MULTIPLICATION

                           Generally, "X " ( Popularly Read as a cross) OR  'Dot ' . ' symbol is used from ancient times. The subtraction is written using anyone from X OR. OR *  between the terms; that is, in symbolically. The result is expressed with equals " = " symbols.

                          it is a basic arithmetic operation that represents the operation of repeated addition of objects from a given collection. The result of the multiplication is also called a  Repeated addition.

                       multiplication can be learned as follows:-  

 The multiplication of numbers is another way of   REPEATED ADDITION; that is, the multiplication of two numbers is equivalent to adding as many times one of them, the multiplicand, as the value of the other one, is known as the multiplier. 
The multiplier can be written first traditionally and multiplicand second These are called factors.

For example, 2 multiplied by 3 (often written as 3 X 2 OR 2 X 3 and spoken as "3 times 2" OR 

" 2 times 3")  can be calculated by adding 3 copies of 2 together:

 3  +  3  =  Think about result ( 2 x 3  = 6 )

OR 

 adding 2 copies of 3 together:

2 X 3 = 2 + 2 + 2 = 6

For example, 4 multiplied by 3 (often written as ${\displaystyle 3\times 4}$ and spoken as "3 times 4") can be calculated by adding 3 copies of 4 together:

${\displaystyle 3\times 4=4+4+4=12}$

Therefore  3 and 4 are known as factors of 12.

•                                       If in ANOTHER way that we have four  fruits  and we have got four more  then Total fruits are such as 

                                       4 x 2 = 4 + 4 = 8 
OR
                                       2 X 4 = 2  +  2 + 2 +  2 = 8

BOTH results are the same it is an important property of multiplication. 

• BY MULTIPLYING SAME NUMBERS ZERO ( 0) 

 Zero MULTIPLIED by  ZERO  always equals  to ZERO 
  0 X 0 = 0 
 0 X 1 = 0 

   0 X 2= 0 + 0 = 0

     0 X 4 = .... Think and Write as addition as four zeroes together

  0 X 5= 0 + 0 + 0 + 0 + 0 = 0 

0 X 6 = .... Think and Write as addition as six zeroes together

 ...

If we multiply zero to any number result is always Zero. ( Important Fact of Mathematics)

•   BY MULTIPLYING SAME NUMBERS  ONE (1) 

  1 X 1 = 1 

  1X 2= 1 + 1 = 2

       1 X 3 = 1 + 1 + 1 = 3

     1X 4 = 1 + 1 + 1 + 1 = 4

 1 X 5= 1 + 1 + 1 + 1 + 1 = 5 

1 X 6 = 1 + 1 + 1+ 1 + 1 + 1 = 6    

1 X 7 = 1 + 1 + 1+ 1 + 1 + 1 + 1 = 7

1 X 8 = 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 = 8
1 X 9 = 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1  = 9
1 X 10= 1 + 1 + 1+ 1 + 1 + 1 +1 + 1 + 1 +1 = 10

 ...

By multiplication of ONE  to any numbers, results are always the same number so ONE ( 1 )  Iis known as multiplicative identity.

• BY MULTIPLYING  DIFFERENT NUMBERS 

                                            7  X  3 = 7 + 7 + 7  = 21   (" SEVEN multiplied by THREE equals  TWENTY ONE")   

THIS CAN BE ALSO WRITTEN AS

                                                      7                                                                                             X  3                                                                                             ___
                                   21__

--------------------

Subtraction in Mathematics

                        

                The beauty of Mathematics depends on basic four arithmetic Operations along with some functional operation:-

1. Addition  ( + )
2. Subtraction ( - )
3. Multiplication  ( x )
4. Division (÷)                                                                                                                                            so we shall discuss second operation subtraction.

SUBTRACTION

                          Generally, - ( Read as "MINUS " ) symbol is used from ancient times. The subtraction is written using the MINUS "-" between the terms; that is, symbolically. The result is expressed with equals " = " symbols.

it is a basic arithmetic operation of removing/ deleting objects from a given object. The result of subtraction is also called a difference.

Subtraction can be learned as follows:-

If we consider a general example that we have three mangoes and one have rotten then remaining mangoes are such as

3 - 1 = 2 is read as (" Three minus one equals TWO")

If in ANOTHER way that we have three mangoes and ALL have rotten then remaining mangoes are such as

3 - 3 = 0 is read as (" Three minus three equals ZERO")

BY SUBTRACTING SAME NUMBERS ZERO ( 0)

0 - 0 = ...? we have nothing/nil objects so our answer is also No objects so Zero it  is read as (" zero minus ZERO equals ZERO")
1 - 0 = 1 is read as(" one minus ZERO equals ONE")
2 - 0 = 2 is read as(" Two minus ZERO equals TWO")
3 - 0 = 3 ...???? ( obvious answer is 3 ) is read as(" Three minus ZERO equals THREE")
4 - 0 = 4 is read as (" Four minus ZERO equals FOUR")
5 - 0 = ...???? ( obvious answer is 5 )is read as (" Five minus ZERO equals FIVE")
6 - 0 = 6 is read as(" Six minus ZERO equals SIX")
....

Since the subtraction of zero in any numbers, results are similar to the number so ZERO is known as SUBTRACTIVE IDENTITY.

BY SUBTRACTING SAME NUMBERS ONE (1)

• 1 - 1 = 0  is read as(" One minus one equals ZERO")
  
  2 - 1 = 1  is read as(" Two minus one equals ONE")
  
  3 - 1 = 2  is read as(" Three minus one equals TWO")
  
  4 - 1 = 3  is read as(" Four minus one equals THREE")
  
  5 - 1 = 4  is read as(" Five minus one equals FOUR")
  
  6 - 1 = 5 is read as (" Eight minus one equals FIVE ")
  
  7 - 1 = 6 is read as(" Seven minus one equals SIX")
  
  ...

• BY SUBTRACTING DIFFERENT NUMBERS

• 7 - 4 = 3is read as (" SEVEN minus FOUR equals THREE")
  
  THIS CAN BE ALSO WRITTEN AS
  7 

          - 4 ___

       3__



The numbers or the objects ( Like Pencils, Pens, Fruits, etc ) to be subtracted in general subtraction are collectively referred to as the terms, the differences.

__________________________________________________________________-

Addition in Mathematics

The beauty of Mathematics lies on the four basic arithmetic operations:-

1. Addition  ( + )
2. Subtraction ( - )
3. Multiplication  ( x )
4. Division (÷) 

ADDITION

     Generally, +  ( Read as "PLUS "  ) symbol is used from ancient times. The addition is written using the PLUS "+"  between the terms; that is, symbolically. The result is expressed with equals " = " symbols.

Addition can be learned as follows:-  

•   BY ADDING SAME NUMBERS  ONE (1) 

  0 + 1 = 1  is read as (" zero plus one equals ONE")   

 What about 1 + 1 = ???  ( Answer is always 2)

      2 + 1 = 3 is read as  (" Two plus one equals THREE")   

      3 + 1 = 4  is read as (" Three plus one equals FOUR")   

  4 + 1 = 5 is read as  (" Four plus one equals FIVE")   

5 + 1 = 6  is read as (" Five plus one equals SIX")   

    6 + 1 = 7  is read as (" Six plus one equals SEVEN")   

 ...

• BY ADDING SAME NUMBERS TWO (2) 

  0 + 2 = 2 read as  (" zero plus two equals TWO")   

     1 + 2 = 3 = 2 + 1   is read as (" one plus two equals THREE")   

    2 + 2 = 4 read as (" Two plus two equals FOUR")   

     3 + 2 = 5 read as (" Three plus two equals FIVE")   

  4 + 2 = 6 read as (" Four plus two  equals SIX")   

        5 + 2 = 7  read as(" Five plus two  equals SEVEN")   

• BY ADDING SAME NUMBERS ZERO ( 0) 

    0 + 0 = 0  read as  (" zero plus ZERO equals ZERO")   

  1 + 0 = 1  read as (" one plus ZERO  equals ONE")   

   2 + 0 = 2  read as(" Two plus ZERO equals TWO")   

         3 + 0 = 3 read as(" Three plus ZERO  equals THREE")   

     4 + 0 = 4  read as(" Four plus ZERO  equals FOUR")   

  5 + 0 = 5  read as(" Five plus ZERO  equals FIVE")   

6 + 0 = 6 read as (" Six plus ZERO  equals SIX")   

 ...

Since the addition of zero in any numbers, results are similar to the number so ZERO is known as ADDITIVE IDENTITY.

• BY ADDING DIFFERENT NUMBERS 

  1+ 2 = 3 is read as  (" ONE plus two equals THREE")   

       4 + 3 = 7  is read as (" FOUR plus three equals  SEVEN")   

     3 + 5 = 8 is read as   (" THREE plus five equals EIGHT")   

  7 + 2 = 9 is read as   (" SEVEN plus two  equals NINE")   

        5 + 6 = 11 is read as  (" FIVE plus six  equals ELEVEN")   

    6 + 6 = 12 is read as   (" SIX plus six equals TWELVE")   

 ...

The numbers or the objects  ( Like Pencils, Pens, Fruits, etc ) to be added in general addition are collectively referred to as the terms, the addends, or the summands.

Method of addition

Standard algorithm (Procedure) for adding any multidigit number is to arrange (align) the addends (Numbers to be added) vertically (In initial Stage) and add the columns starting from ones column on the right, such as;

   14

+ 23

____

   37

____

NOTE:-

 In above procedure, if sum of the column exceeds by nine then the extra digit is " transferred " into the just next column. The transferred digit is popularly called "carried" and the method is known as addition by carry.

   

  1

   47

 +16

____

   63

____