Set of Quantum Numbers

Now imagine a number so small that it is the difference between 1 and the number 0.99999...

But isn't the number 0.9999… equal to 1?

The proof is simple, multiply the number 0.9999… by 10, you will get 9.9999…. Subtract 0.9999… from this number, the remaining number is 9!

If you do the same thing with 1, 10 times 1 is 10, subtracting 1 leaves 9.

So the number 0.9999… is equal to 1. Doesn't math say that?

Okay, with this simple calculation we seem to be able to prove that the number 0.9999… is equal to 1, but one is still the number 0.9999… and the other is the number 1.

So, in my opinion, if they were equal, they would not be numbers in two separate representations.

I think this is also a fact!

Also, in this proof process, I am not sure whether we have actually proven that the number 0.9999… is equal to 1?

Think about it, after all, we start with the number 0.9999…, but first we multiply it by 10, and then we subtract the number 0.999… from it.

I think what we actually proved is that the difference between 1 and 10 is equal to the difference between 9.999… and 0.999…, both differences being 9 anyway.

As if this does not prove to us that the numbers 1 and 0.999… are equal!

Okay, this is mathematics, and this is how proofs are done in mathematics, that much I can understand.

However, personally I am not satisfied enough with this proof.

In my opinion, there is still a tiny difference between the two numbers, even if it extends to infinity!

If it wasn't so, we wouldn't show the two differently!

We would pass on both of them. But look, since we are trying to prove their equality, they are actually different numbers.

This is simple Aristotelian logic!

***

Yes, let's accept from the beginning that there is such a tiny difference between the two numbers and try to analyze the situation that way.

By the way, the concept of infinity, like many other concepts in mathematics, is based on a presupposition.

Infinity x number = infinity

Infinity - number = infinity

There are many presuppositions like this regarding the concept of infinity.

In other words, just as we have presuppositions on some subjects in mathematics, the issue of infinity is the same. When we multiply a number whose fraction extends to infinity by 10, we perform the above proof calculation with the assumption that its fraction remains the same as the fraction of our first number.

However, we are multiplying an infinitely long fractional number. I don't think we can be that sure whether the fraction will change or not.

***

Anyway, here's the equation I have in mind that could be used for the proof:

I say let's assume from the beginning that there is a very small number called Lambda=1-0.9999….

Let this number be a number like 0.0000…..001.

Let the digits to the right after the comma be infinite in number, and let the last number be 1 at the last point at that infinite distance.

How will that work? We call it an infinitely long step and also put the number one at the end as if it has an end!

Both infinite and finite? Is this possible?

Why not?

Didn't we say that mathematics is based on certain presuppositions?

For example, a negative number does not have a square root, but a whole set of complex numbers is considered with the square root of negative one (-1).

This "square root minus one" number, whose mathematical representation is "i", that I mentioned is a completely hypothetical number!

It is not possible for such a number to exist in the world of real numbers in mathematics.

But by definition, this number exists in the world of complex numbers in mathematics, and quite complex formulas have been developed using this number, with which we make many useful calculations in the world of science.

So my presupposition is that this tiny number, which I call lambda, exists.

I think the existence of lambda is not contrary to mathematics!

***

If you think it is suitable, let's continue our process now:

0.999…. In this case, the number becomes "1 - lambda".

Mathematically, if we call the number 0.9999… as “x”, our equation is:

x = 1 - lambda

is happening.

***

What was our proof method?

We were multiplying both sides of the equation by 10 and then subtracting "x" from both sides, right?

In this case, 9.x remains on the left side of the equation. If we calculate, 10.x - 1.x = 9.x.

On the right side: [10 - 10th place - x] will remain.

At this point we have something to decide!

Can we now call the [10thlamda] value 0.0000….001, that is, “lamda”?

Normally, 10 times the value is 0.0000….010, but there is a rule that the zeros after the last digit have no effect on the digits after the comma!

On the other hand, one of the infinite number of digits in between is missing, but since "infinity minus one" is still "infinity", we can assume that the number of digits has not changed.

This is also a presupposition!

Presupposition of the concept of infinity!

In this case, the number 0.000….001 seems to equal the number 0.000….010.

After all, they both have an infinite number of digits in between, that is, zeros, and both have the number 1 at the end.

Since the subject is the digits after the comma, the zero at the end of the number 0.000….010 has no meaning!

Here we come to confusion again!

If we follow the definition of multiplication with these assumptions, we need to take ten times the "lamda" according to our proof method, and 10 times the lambda number apparently gives us the lambda number itself!

I'm not entirely sure if this conclusion is correct, but apparently we have to accept that it is.

Let's see where this account will end up!

***

Anyway, in this case our equation is with the “x” we subtracted from both sides:

9.x = 10 - lambda - x

It turns into shape.

Since x is “1 - lambda”, when we enter this value instead of “x” on the right side of the equation, our equation is:

9.x = 10 - lambda - (1 - lambda)

is happening.

If we rearrange our equation, our equation is:

9.x = 10 - 1 + lambda - lambda

In this case, the (+/-) lambdas will cancel each other, leaving:

9.x = 9

remains.

From here we reach the value x = 1.

Hooray, yes we proved it! The number 9.99999..., that is, "x" is equal to 1!

***

But there still seems to be a problem?

Since we initially said "1 - lambda" for "x", our equation becomes "1 = 1 - lambda" and from here we reach the value "lamda = 0".

Let's decide now, is it equal to zero? Or lambda = 0.000…01?

I'm a little confused.

I'm not that convinced by this result!

Let's change the method we use for proof a little and instead of multiplying both sides of the equation by 10, let's multiply by 9.

At least we can make sure, let's try this way.

In this case, the result of your first step is:

9.x = 9 - 9.lamda

It will happen.

In this case, the number called [9th lambda] will be 0,000…09! We are doing multiplication after all!

Then we were subtracting an “x” from both sides of the equation, right?

In this case, our equation is:

8.x = 9 - 9.lamda - x

and if we enter the value “1 - lambda” instead of “x” on the right side:

8.x = 9 - 9.lamda - (1 - lambda)

is happening. And from here:

8.x = 8 - 8.lamda

We reach the value.

I won't make it too long, the result we will get from here is again:

x = 1 - lambda

will be 0.9999… and our lambda value will be 0.0000….01 as in the beginning.

We had already given the value Lambda = 0.000…01 from the beginning.

In this case, the number 1 and 0.9999…. The number is not equal!

***

If we go one way, we find lambda = 0, if we go the other way, lambda = 0.0000…001.

So what are we gonna do?

In some cases, we can prove that the number 0.9999… is equal to 1, but in some cases, it is not possible to prove this equality.

In fact, in the second attempt above, that is, in the case where we multiply the equation "x=1-lamda" by 9, if we add the number "x" from both sides of the equation instead of subtracting it from both sides in our second operation, we can prove that the number 0.9999... is equal to 1.

Because even though the [9thlamda] value (0,000…09 from 0,000….01 or is equal to .

In fact, this situation is locked in the definition of the concept of infinity, and partly in our ability to operate in decimal order.

I said from the beginning how it will work, we say infinity and we also say there is 1 at the end.

So, on the one hand, we accept that our number has an end, but on the other hand, we think that our number has infinite digits.

Could this be the reason for the confusion?

Wrong premise, wrong conclusion!

Is this the conclusion we will reach?

In the world of real numbers, there is no such number as 0.000….01! Are we reaching this conclusion after all?

I'm actually confused!

***

Well, okay.

However, in the world of real numbers, there cannot be a number called "i"!

But in mathematics, there is a number called "i", the world of complex numbers! This number exists in the world of complex numbers!

It is determined by a presupposition that the number "i" exists, and many operations are performed with "i".

If we determine the existence of a number lambda = 0.000…01 as a presupposition in the above operations and say that in some cases the value of this number is 0 (zero) and in some cases it is equal to 0.000…01, we will have stepped into a completely different world of mathematics!

This means that we can establish the existence of the lambda number as a presupposition and create another set of numbers such as complex numbers!

***

This situation reminded me of the behavior of subatomic particles in the quantum world.

You know, we say that an electron is both there and here at the same time.

That's right, in the world of such small numbers, a number can have the value of both zero and 0.000…01!

***

Oh my God, I found the qubit in mathematics!

In the computer world, a qubit means a bit of information that has the value of both zero and one.

(The “lice” here is not the insect lice, but the lice in the computer world! Just like a mouse!)

Quantum computers work with qubits, not bits.

One bit of information is either "zero (0)" or "one (1)".

A qubit has the value of both "zero" and "one", (0/1).

From the calculations above, I have proven that our lambda number can be both zero (0) and 0.000…01.

I need to focus on this subject and improve the calculations. Maybe serious formulas can be developed by creating a set of "quantum numbers" like the set of "complex numbers" with this method!

Is it obvious that I have really broken new ground in mathematics!

***

Anyway, let me check the internet and see if anyone has ever thought of what I said. Maybe someone figured out quantum numbers before me! I shouldn't be too happy just yet!

I end the article by saying stay with mathematics. The language of the universe is mathematics, and mathematics can be very interesting sometimes. Look, there are even quantum numbers in the world of mathematics.

With love and respect to everyone from Moscow.