Redefining Infinity '∞'
Infinity(∞) a term used almost by everyday is one of the few terms that limit mans understanding. The word comes from the Latin infinitas or "unboundedness".Any quantity too large to be perceived is termed as infinite.However an idea struck me to partially cover this unboundedness.
If you go ask a person the day after numbers were invented values as large as millions , he would reply that it is undefined , that is infinite .As technology grows , the ability to conceive larger numbers increases.Going back to the previous example , if we give him two numbers , say 2 million and 4 million or 5 billion,all are undefined in his system and hence are equal to infinity.But we know that these values are very different.However , he would have been much more accurate if using 2 million as a comparison he says that 4 million is 2 times larger.
This is precisely the motivation , I had to define this vague quantity.Defining infinity within limits enables us to equate various quantities earlier considered in-equatable , due to factors like division by zero etc.Therefore I started comparing infinity to the first things that came to my mind.I do not argue that infinity can surely be defined in other ways.However it makes no difference as it is similar to the concept of setting different units.
Starting my explanation-
A square was taken of dimension 2*2 units .The area of the square is 4 sq units.Now I stretched this square so that its length was now 4 units and breadth 1 units(now a rectangle).The area is still 4 sq units.Similarly I continued increasing the length and decreasing the breadth proportionally so that the area is a constant .
As lim(b tends to zero) l*b=4, l tends to infinity.(4/0).
Now a square of dimensions 3*3 units.The area of the square is 9 sq units.Similarly the square was stretched such that it's area is a constant.Similarly as b tends to zero, length corresponds to infinity(9/0).But we can observe that if the breadth value was made same for both the squares the length is different.Hence it can be understood that the two values of infinity are different.This shows us that we are the same situation as the person in the beginning of the text and our hypothesis was true.Now we need to make an attempt to solve this problem.
Notation for writing-
8∞2- This can be realised as a square of dimensions 4*4 can be achieved by changing the dimensions of a rectangle of dimensions 8*2.
Value of ∞ of a number -
Now the question arises- What is the use of this notation and concept . Does it solve some of the mathematical paradoxes.Does it give us the value of infinity.Well we cannot get an exact value of infinity but from the square experiment we can decipher that the area remains constant no matter how small breadth becomes.hence since for a given number of decimal points 1 is the smallest value, We can assign the value of breadth when the length tends to infinity as 1*10^-n ,where n is a very large quantity and can be arbitrarily chosen.Hence the length value becomes area/breadth.For the 2*2 square it is 4*10^n which is the value of infinity of 2.General value of ∞ of t is t*10^n.
This enables us to use the laws of exponents for performing mathematical operations on infinity.
For eg: If we have to evaluate 9/0+16/0,conventionally the answer would have been ∞ +∞ =∞.However with our new knowledge we can say 9/0 is ∞ of 3(sorry the editor doesn't provide me a easy way to use subscripts) and 16/0 as ∞ of 4. Also we know ∞ of 3 is 9*10^n and ∞ of 4 is 16*10^n.Therefore we can add them up to get 25*10^n which is nothing but ∞ of 5.This leaves us in a much better situation.This value of n can be used till the real value is discovered(possible through experiments).
Significance-Well now you might ask me ,how will this change anything in mathematics.Well we often come across situations which force us to give up a question.A classic example would be -
We can also extend this idea to all geometric figures.If we try to extend this to a circle, we say that every circle has infinite number of sides.but we know geometrically that from different dimensions of circles the number of sides are different.Therefore we can predict that the number of sides of a circle are ∞ of its radius.
Final Arguement-
Now you may ask me about 3D figures . Indeed a cube of side 1*1*1 does not have same infinity as a square of side 1*1. Infinity is a quantity that depends on the dimensions.Hence we can proceed in the same way to get ∞ of 1 for a cube and hence apply that to all the 3D figures.
Finally we know 1/0 is ∞ of 1.Therefore ∞ of 1 multiplied by 0 is 1.This shows that 0 multiplied by a number is not 0 but a practically insignificant quantity with the value of 1/∞of the number.
Finally I would like to sum up the whole explanation by a simple theorem(copyrighted © Sabyasachi)
" Infinity has different values in different cases and dimensions and these values can be related to different numbers in the numeric system with the appropriate dimension consideration"
Sabyasachi
If you go ask a person the day after numbers were invented values as large as millions , he would reply that it is undefined , that is infinite .As technology grows , the ability to conceive larger numbers increases.Going back to the previous example , if we give him two numbers , say 2 million and 4 million or 5 billion,all are undefined in his system and hence are equal to infinity.But we know that these values are very different.However , he would have been much more accurate if using 2 million as a comparison he says that 4 million is 2 times larger.
This is precisely the motivation , I had to define this vague quantity.Defining infinity within limits enables us to equate various quantities earlier considered in-equatable , due to factors like division by zero etc.Therefore I started comparing infinity to the first things that came to my mind.I do not argue that infinity can surely be defined in other ways.However it makes no difference as it is similar to the concept of setting different units.
Starting my explanation-
A square was taken of dimension 2*2 units .The area of the square is 4 sq units.Now I stretched this square so that its length was now 4 units and breadth 1 units(now a rectangle).The area is still 4 sq units.Similarly I continued increasing the length and decreasing the breadth proportionally so that the area is a constant .
As lim(b tends to zero) l*b=4, l tends to infinity.(4/0).
Now a square of dimensions 3*3 units.The area of the square is 9 sq units.Similarly the square was stretched such that it's area is a constant.Similarly as b tends to zero, length corresponds to infinity(9/0).But we can observe that if the breadth value was made same for both the squares the length is different.Hence it can be understood that the two values of infinity are different.This shows us that we are the same situation as the person in the beginning of the text and our hypothesis was true.Now we need to make an attempt to solve this problem.
Notation for writing-
Let us set a standard for writing various value like 4/0,9/0 etc.We shall use a subscript with the lemniscate( ∞).For eg:In the above cases the infinity for the square of length 2 is ∞2
We shall also use the superscript notation to simplify calculations .For eg infinity of 4 i.e 16/0 can be written as- 8∞2- This can be realised as a square of dimensions 4*4 can be achieved by changing the dimensions of a rectangle of dimensions 8*2.
Value of ∞ of a number -
Now the question arises- What is the use of this notation and concept . Does it solve some of the mathematical paradoxes.Does it give us the value of infinity.Well we cannot get an exact value of infinity but from the square experiment we can decipher that the area remains constant no matter how small breadth becomes.hence since for a given number of decimal points 1 is the smallest value, We can assign the value of breadth when the length tends to infinity as 1*10^-n ,where n is a very large quantity and can be arbitrarily chosen.Hence the length value becomes area/breadth.For the 2*2 square it is 4*10^n which is the value of infinity of 2.General value of ∞ of t is t*10^n.
This enables us to use the laws of exponents for performing mathematical operations on infinity.
For eg: If we have to evaluate 9/0+16/0,conventionally the answer would have been ∞ +∞ =∞.However with our new knowledge we can say 9/0 is ∞ of 3(sorry the editor doesn't provide me a easy way to use subscripts) and 16/0 as ∞ of 4. Also we know ∞ of 3 is 9*10^n and ∞ of 4 is 16*10^n.Therefore we can add them up to get 25*10^n which is nothing but ∞ of 5.This leaves us in a much better situation.This value of n can be used till the real value is discovered(possible through experiments).
Significance-Well now you might ask me ,how will this change anything in mathematics.Well we often come across situations which force us to give up a question.A classic example would be -
- 1/0 is ∞
- 2/0 is ∞
- 1/0 is ∞ of 1
- 2/0 is ∞ of √2
We can also extend this idea to all geometric figures.If we try to extend this to a circle, we say that every circle has infinite number of sides.but we know geometrically that from different dimensions of circles the number of sides are different.Therefore we can predict that the number of sides of a circle are ∞ of its radius.
Final Arguement-
Now you may ask me about 3D figures . Indeed a cube of side 1*1*1 does not have same infinity as a square of side 1*1. Infinity is a quantity that depends on the dimensions.Hence we can proceed in the same way to get ∞ of 1 for a cube and hence apply that to all the 3D figures.
Finally we know 1/0 is ∞ of 1.Therefore ∞ of 1 multiplied by 0 is 1.This shows that 0 multiplied by a number is not 0 but a practically insignificant quantity with the value of 1/∞of the number.
Finally I would like to sum up the whole explanation by a simple theorem(copyrighted © Sabyasachi)
" Infinity has different values in different cases and dimensions and these values can be related to different numbers in the numeric system with the appropriate dimension consideration"
Sabyasachi
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