Challenging Infinity

Infinity

variables and fundamentals

Let’s denote the number of decimal numbers between 0 and 1 as D and the number of positive integers as N. For integers n and n + 1 where 𝑛 ≥ 1, there are also D decimal numbers between n and n+1. Therefore, the total number of decimal numbers between 1 and N is (N−1)×D.

Function

f(x) = 1/x

For all x>1, this function maps x to a number between 0 and 1. The function f(x)= 1/x ​ is indeed one-to-one for all x>1.

Problem

For all decimal numbers greater than 1, f(x) gives output between 0–1. So, a total of (N−1)×D inputs gives D outputs where f(x) is a one-to-one function. However, how is this possible if f(x) is one-to-one? Also, remember that (N−1)×D is a huge number compared to D.

In simpler terms, imagine your teacher says, ‘Students, if you have 25 crore distinct pens, then when you reverse them, you will now have 17 crore pens.’ You will ask the teacher, ‘How is this possible? If I have 25 crore pens, after reversing them, it should still be 25 crore, shouldn’t it?’ But your teacher starts giving answers like, ‘Because there are so many pens, and we are continuously adding more.’ So, will you be satisfied with this answer? Of course not. I believe the same goes for my problem as well.

I know this might seem strange, but this question is more complicated than it looks. There might be hidden layers to it. This question arose in my mind when I was in 8th or 9th grade. The more I think about it and the more I learn, the more surprised and thoughtful I become about infinity. Whenever I start thinking, I feel like my mind will explode if I delve too deep. Many answers I got claim that since n = ∞ and d = ∞, then

(N-1)*D = (∞-1)*∞ = ∞*∞ = ∞ = D.

Then I asked, “How does ∞−1=∞ and ∞*∞=∞? ” He replied that this is the nature of infinity and that infinity is a concept.

However, this explanation seems illogical to me. For any question related to infinity, the answer shouldn’t always be “infinity is a concept.” It feels like infinity is a concept used as a shield to conceal many answers that people don’t know.

First, I am confused about what infinity is.

Different Perspectives on Infinity:

1) Infinity as a Point:

Some argue that infinity is a point where we assume there’s no number greater than that number. However, if this were true, there wouldn’t be a number in the universe where x−1=x, making the statement ∞-1=∞ incorrect.

2) Infinity as a Collection:

Infinity might simply be a set of numbers, where no matter how big a number you think of, it’s still part of infinity. But if we view infinity as a set, saying “Infinity — 1 = infinity” is incorrect because infinity is a set, not a single number which you can use in an arithmetic equation.

Just imagine E as the set of even numbers. Throughout, we know that adding two to any even number results in another even number. However, we can’t say E + 2 = E because E is a set, not a number. Let’s assume ‘e’ is an element in the set of even numbers. Still, saying ‘e + 2 = e’ is incorrect because the moment we select ‘e’ from the set, it becomes a definite number, and ‘e + 2’ can never be equal to ‘e.’ Instead, you can say that ‘if e is a member of the even number set, then e + 2 is also a member of the same set where e belongs.’ However, ‘e + 2 ≠ e.’

The same principle applies to infinity. If you select one element from infinity, you can say that “x -1” is part of the same large set as x. But it’s incorrect to say that “x - 1 = x” because when you select one element from infinity, it’s not infinity anymore, and “x — 1” isn’t the same as x.

3) Infinity as a Direction

Another perspective considers infinity as a direction rather than a destination. In this view, infinity represents an unboundedness or a limitless extension in a particular direction, such as the endlessness of the number line extending to positive or negative infinity. And if you consider infinity as a direction, then the concept of infinity becomes irrelevant to my question.

Personally speaking, I find the idea of infinity as a direction more logical than any other interpretation.

Yet, none of these definitions make my question wrong. No number, whether integer, decimal, rational, real, imaginary, or fractional, possesses the property of x−1=x. It seems people want to say x−1=x, which is impossible in any condition. So, they invented the concept of infinity and then used it as an answer to every question, claiming “infinity is a concept.”

I asked my classmate for clarification. He gave an example of a right-angled triangle where the angle opposite the perpendicular is 90°. He asked me about the height of the triangle, which becomes tan(90°), resulting in infinity. I argued that the answer isn’t infinite; it’s undefined because lines that never meet don’t equate to infinity. Therefore, using infinity in this context makes no sense.

Feedback

Please share your views on this article. Alternatively, we can connect on LinkedIn/ankitjha2603. As a passionate physics and math enthusiast, I’m always open to being proven wrong. Your insights are highly valued.