Factors in Mathematics

If we multiply to get the number is Factors of that numbers. Factors and multiples are different things. But they both involve the general concept of multiplication.

In another way, a factor is a number that divides into another number exactly and without leaving a remainder.
The Factors of any number are always less than or equals to given Numbers and it is different from Multiples.

Example:-Let us find the factors of 6:-



1     x    6 =   6
2     x    3  = 6

Therefore 1, 2, 3, and 6 are factors of 6.       
                                               INCOMPLETE ANSWER Why ???

 Because multiplying negatives make a positive, −1, −2, −3, and −6 are also factors of 6:

• (−1) × (−6) = 6
• (−2) × (−3) = 6

So ALL the factors of 6 are:

1, 2, 3,  and 6 AND −1, −2, −3,  and −6

Example:- Let us find the factors of 8:-


1     x    8 =   8 or 8 x 1 = 8
2     x    4 =   8 or 4 x 2 = 8

 So factors of 8 are 1, 2, 4, 8. 
                                         INCOMPLETE ANSWER Why ???

Since because multiplying negatives make a positive, −1, −2,  −4, and −8 are also factors of 8:

• (−1) × (−8) = 8
• (−2) × (−4) = 8) 

So ALL the factors of 8 are:

1, 2, 3, 4,  and 8 AND −1, −2, −3, −4,  and −8

Example:-Let us find the factors of 12:-

•  3 and 4 are factors of 12 since 12 = 3 x 4
• Similarly  2 and 6 are also factors of 12 since 12 = 2 x 6,
• And 12 = 1 x 12, Therefore 1 and 12 are factors of 12 also.
• So factors of 12 are 1, 2, 4, 6, and 12. 
                                           INCOMPLETE ANSWER Why ???

 because multiplying negatives make a positive, −1, −2, −3, −4, −6 and −12 are also factors of 12:

• (−1) × (−12) = 12
• (−2) × (−6) = 12
• (−3) × (−4) = 12

So ALL the factors of 12 are: 

Similarly, we can find factors of any numbers in mathematics using the above methods or Prime Factorisation Methods.

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Multiples in Mathematics

In mathematics, Multiples are what we get after multiplying the number by an integer like 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...(not a fraction).
The product of any quantity and an integer are known as Factors. In other words, for the quantities a and b, we say that b is a multiple of a  if  b = na for some integer n like 0, 1, 2, 3, 4, ..., which is called the multiplier If a is not zero, this is equivalent to saying that b/a is an integer since the division of zero is not possible.
The above definition can be also written in many ways like:-
' On  Multiplying numbers the obtained product is called a multiple of that numbers such as 2 x 3 = 6 so 6 is multiple of 2 as well as 3 also'
The Multiples of any number are always greater than or equals to given Numbers and it is different from Factors. Multiples also not to be confused with multiplication generally.

Anyone can write multiples easily with the help of Their tables in such a way ( Here i am writing few examples of multiples like 1, 2, 3, 5, 6, 10, 100, ... ) :-

Multiples of 1

• 1 × 0 = 0, so 0 is a multiple of 1
• 1 × 1 = 1, so 1 is a multiple of 1
• 1 × 2 = 2, so 2 is a multiple of 1
• 1 × 3 = 3, so 3 is a multiple of 1
• 1 × 4 = 4, so 4 is a multiple of 1
• 1 × 5 = 5, so 5 is a multiple of 1
• 1 × 6 = 6, so 6 is a multiple of  1
• 1 × 7 = 7, so 7 is a multiple of 1
• 1 × 8 = 8, so 8 is a multiple of 1
• 1 × 9 = 9, so 9 is a multiple of 1
• 1 × 10 = 10, so 10 is a multiple of 1 
• and so on

Multiples of 2

• 2 × 0 = 0, so 0 is a multiple of 2
• 2 × 1 = 2, so 2 is a multiple of 2
• 2 × 2 = 4, so 4 is a multiple of 2
• 2 × 3 = 6, so 6 is a multiple of 2
• 2 × 4 = 8, so 8 is a multiple of 2
• 2 × 5 = 10, so 10 is a multiple of 2
• 2 × 6 = 12, so 12 is a multiple of  2
• 2 × 7 = 14, so 14 is a multiple of 2
• 2 × 8 = 16, so 16 is a multiple of 2
• 2 × 9 =  18, so 18 is a multiple of 2
• 2 × 10 = 20, so 20 is a multiple of 2 
• and so on

Multiples of 3

• 3 × 0 = 0, so 0 is a multiple of 3
• 3 × 1 = 3, so 3 is a multiple of 3
• 3 × 2 = 6, so 6 is a multiple of 3
• 3 × 3 = 9, so 9 is a multiple of 3
• and so on

Multiples of 5

• 5 × 0 = 0, so 0 is a multiple of 5
• 5 × 1 = 5, so 5 is a multiple of 5
• 5 × 2 = 10, so 10 is a multiple of 5
• 5 × 3 = 15, so 15 is a multiple of 5
• 5 × 4 = 20, so 20 is a multiple of 5
• 5 × 20 = 100, so 100 is a multiple of 5
• 5 × 30 = 150, so 150 is a multiple of 5
• 5 × 400 = 2000, so 2000 is a multiple of 5
• and so on

Multiples of 6

it is not necessary to start from 0 or 1 or 2 or 3 etc , we can write according to our choice.

• 3 × 6 = 18, so 18 is a multiple of 6
• 4 × 6 = 24, so 24 is a multiple of 6
• 5 × 6 = 30, so 30 is a multiple of 6
• 6 × 6 = 36, so 36 is a multiple of 6
• and so on

Multiples of 10

• 10 × 0 = 0, so 0 is a multiple of 10
• 10 × 1 = 10, so 10 is a multiple of 10
• 10 × 2 = 20, so 20 is a multiple of 10
• 10 × 3 = 30, so 30 is a multiple of 10
• 10 × 4 = 40, so 40 is a multiple of 10
• 10 × 5 = 50, so 50 is a multiple of 10
• 10 × 6 = 60, so 60 is a multiple of  10
• ( intentionally I left 70 , which is also multiple)
• 10 × 8 = 80, so 80 is a multiple of 10
• 10 × 9 = 90, so 90 is a multiple of 10
• 10 × 10 = 100, so 100 is a multiple of 10 
• and so on 

We can write multiples of any number in mathematics up to any time according to our requirements.

Multiples of 100

• 100 × 0 = 0, so 0 is a multiple of 100
• 100 × 1 = 100, so 100 is a multiple of 100
• 100 × 2 = 200, so 200 is a multiple of 100
• 100 × 3 = 300, so 300 is a multiple of 100
• 100 × 4 = 400, so 400 is a multiple of 100
• 100 × 5 = 500, so 500 is a multiple of 100
• 100 × 6 = 600, so 600 is a multiple of  100
• 100 × 7= 700, so 700 is a multiple of 100
• 100 × 8 = 800, so 800 is a multiple of 100
• 100 × 9 = 900, so 900 is a multiple of 100
• 100 × 10 = 1000, so 1000 is a multiple of 100 
• 100 × 18 = 1800, so 1800 is a multiple of 100
• 100 × 90 = 9000, so 9000 is a multiple of 100
• 100 × 100 = 10000, so 10000 is a multiple of 100
• and so on we can write so many times...

Basic Properties of Multiples

(1)   0  ( Zero) is a multiple of every number. 

Such as:-

• 1 × 0 = 0, so 0 is a multiple of 1
• 2 × 0 = 0, so 0 is a multiple of 2
• 3 × 0 = 0, so 0 is a multiple of 3
• 10 × 0 = 0, so 0 is a multiple of 10

(2) Every number is a multiple of 1 ( One). 

Such as:-

• 1 × 0 = 0, so 0 is a multiple of 1
• 1 × 1 = 1, so 1 is a multiple of 1
• 1 × 2 = 2, so 2 is a multiple of 1
• 1 × 3 = 3, so 3 is a multiple of 1  etc    

(3)  Multiples of a number are always infinite (we can write to them easily).

Such as:-

• 100 × 0 = 0, so 0 is a multiple of 100
• 100 × 10 = 1000, so 1000 is a multiple of 100
• 100 × 20 = 2000, so 2000 is a multiple of 100
• 100 × 18 =1800, so 1800 is a multiple of 100
• 100 × 19 = 1900, so 1900 is a multiple of 100
• 100 × 10 = 1000, so 1000 is a multiple of 100 
• 100 × 18 = 1800, so 1800 is a multiple of 100
• 100 × 90 = 9000, so 9000 is a multiple of 100
• 100 × 1000 = 100000, so 100000 is a multiple of 100

and so on we can write so many times...
(4)  The product of any two or more factors is the multiples of each factor. 
Such as:-

• 2 x 5 x 1 = 10,

       Therefore, 10 is the multiple of both 1, 2, and 5.

• 3 x 1 = 3,

       Therefore, 3 is the multiple of both 3 and 1.

• 30 = 2 x 3 x 5 x 1,

       Therefore, 30 is the multiple of 1, 2, 3 and 5.

• 36 = 2 x 2 x 3 x 3 x 1 ,

       Therefore, 36 is the multiple of 1, 2, 3, 4, 6, 9, 12, 18 and 36.

(5)  Every number is a multiple of Itself. 

Such as:-

• 1 × 1 = 1, so 1 is a multiple of 1
• 3 × 1 = 3, so 3 is a multiple of 3
• 2 × 1 = 2, so 2 is a multiple of 2
• 5 × 1 = 5, so 5 is a multiple of 5
• 10 × 1 = 10, so 10 is a multiple of 10
• 100 × 1 = 100, so 100 is a multiple of 100
• 2012 × 1 = 2012, so 2012 is a multiple of 2012
• 6666 × 1 = 6666, so 6666 is a multiple of 6666

Note :- In general, when a and b are both integers, and b is a multiple of a, then a is called a DIVISOR of b.We can also says that a divides b. 

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What is Mathematisation & Why ?

What is mathematisation???????

 If we consider a very uncommon example may help us work out some sort of answer to this type of question. It is an example of counting objects /articles /things. We may count our belongings the number of students, the number of fingers in our hand, and the number of anything we want to think about more precisely, more clearly and of course with counting numbers we get to quantify things, do arithmetic. In common  the word 'mathematisation' to refer to the mental processes which produce mathematics from ancient times. 

Mathematisation is a word that strikes a chord among most of the students, especially those who have been tensed by the void traditional or mechanical manner in which mathematics has been taught from ancient times and has been used by most of us to capture what they would consider as the heart of the mathematical enterprise – the thinking and the reasoning process of learner.

Therefore in the precious mathematization of a problem or area of study consists of applying mathematical concepts/innovative ideas to that problem or field so as to think more precisely or clearly about things.

Firstly, French-born economist and mathematician cum Nobel Laurate Gerard Debreu introduces an account of events that relates to the origins of interdisciplinary in economics. In the analysis of Market trend, Economics or commercial activity somehow allocate numbers to products/object/things for purchase/sale or in other words for transactions. The rules of the business may not always be clear but numbers ( mathematical symbols ) are somehow assigned to costs and revenues to optimize their gain or loss.

Not only in recent times but from decades spread of mathematised economic (economical) theory was helped even by its esoteric character in business and management. Since its messages cannot be deciphered by modern welfare economists like Nobel laurates Amartya Sen who do not have the proper key, their evaluation is entrusted to those who have access to the code. But acceptance of their technical expertise also implies acceptance of their values of welfare and development economics in the era of mathematisation.

The beauty of mathematics is that it is clear thinking with a logical approach. A person is tempted to add, however, that mathematics is clear thinking plus the mental tools used to clarify our thinking. So algorithms, mental tools or structures are created like number systems, Decimal system which makes sorting out things and ideas about things much more precise.

In recent times a variety of scientific fields have seen increasing mathematisation throughout the last four decades. Driven by the construction of large-scale data processing facilities in the All Engineerings and Branches of Natural sciences (including Medical science also) and the introduction of digital tools and models in the human and social sciences, contemporary research practices are increasingly dependent on mathematics.

The fields such as Artificial Intelligence in computer science, cognitive neuroscience, linguistics, Aerospace Engineering, Climate science & Weather forecast, Economics, and network science, research is being mediated through the development, integration, and application of mathematical models on the basis of mathematisation. Although, this dependency on applied mathematics (Like Statistics & Operation Research, etc) and more broadly on computation and digitalisation has so far received little attention by scientists as well as philosophers of science along with mathematisation.

In nutshell, we may hope that a serious, ingenious, study of mathematisation will help all of us in mainly two respects: it will help to connect the inner mathematical experience with its outer objective form and, in the classroom, it will help us to handle the individuality and spontaneity of students who are coming to terms with the most impersonal and rigorous subject we ask them to learn. 

Mathematisation is a word that strikes a chord among most of the students, especially those who have been tensed by the void traditional or mechanical manner in which mathematics has been taught from ancient times and has been used by most of us to capture what they would consider as the heart of the mathematical enterprise – the logical thinking and the reasoning along with positive attitude, or More precisely, mathematisation is the act of putting a structure onto a structure.

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Order of Operations ( BODMAS OR PEDMAS )

We have a popular Phrase about Mathematics is that "Mathematics is the mother of civilization and culture. " Therefore the Order of Operations ( BODMAS OR PEDMAS ) is a well-defined collection of rules / Instructions based on Observations about which standard operation/procedure to perform first in order to evaluate given mathematical expressions in mathematics so the solution is unique ( Not Different).

"Operations" mean things like addition, subtraction, multiply, divide, squaring, Parentheses, etc. BODMAS is in mathematics an operation/procedure to solve problems. For orders of operation following rule is commonly followed.

'BODMAS'  is an acronym used in mathematics and in it  'B' stands for Bracket, 'O' stands for Of, 'D' stands for Division, 'M' represents Multiplication, 'A' Represents Addition, and
'S' represents Subtractions. The 'BODMAS' is used to explain or solve the order of operation of a mathematical expression. 

             

    In some area (regions) of the world, the 'BODMAS' is also known as 'PEDMAS' which 'P' stands for stands for Parentheses, 'E' stands for Exponents, 'D' stands for Division, 'M' represents Multiplication, 'A' Represents Addition, and

'S' represents Subtractions.

Illustrations for  Order of Operations ( BODMAS OR PEDMAS )

•         For solving any problems, Do things in Brackets First, if more than one bracket is present than the following order may be followed, -, ( ), { }, [ ].

		2 × (4 + 5)	=	2 × 9	=		18
		2 × (4 + 5)	=	8 + 5	=	13	(wrong)  WHY ????

      Since 2 is not multiplied by both terms in brackets so the second term is 5 written instead of  10 so the answer is wrong. 

• Exponents(Powers, Roots) before Multiply, Divide, Add or Subtract

		5 × 22	=	5 × 4	=	20
		5 × 22	=	102	=	100	(wrong)

                                                                               WHY ????

• Multiply or Divide before you Add or Subtract on solving

		12 + 5 × 3	=	12 + 15	=	27
		12 + 5 × 3	=	17 × 3	=	51	(wrong)

                                                                               WHY ????

•  Otherwise, just go left to right

		45 ÷ 5 × 3	=	9 × 3	=		27
		45 ÷ 5 × 3	=	45 ÷ 15	=	3	(wrong)

                                                                        WHY ????

            For example,  Thus, the expression 2 + 3 × 4 is interpreted to have the value 2 + (3 × 4) = 14, not (2 + 3) × 4 = 20.

             if more than one bracket is present than the following order may be followed,-, (  ), { }, [ ].

    The technique for learning or Remember is such as, How Do I Remember It All ... ? BODMAS ! It's so simple......

B	Brackets first
O	Orders (i.e. Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

       

How Do I Remember It All ... ? PEDMAS ! It's so simple.....

P	Parenthesis first
E	Exponents(i.e. Powers and Square Roots, etc.)
DM	Division and Multiplication (left-to-right)
AS	Addition and Subtraction (left-to-right)

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