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Integers, Pi, and Number Lines.

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Peter Taylor:
In classic education, we are taught that a number line contains an infinite number of irrational numbers between each integer. I will show that it should contain only integers.

dy / dx is never evaluated because dx is infinitely small and zero.

When working with Fourier Transforms for example, the terms are introduced into a formulae as a pair to help solve it, and then disappear at the end without ever being evaluated.

The imaginary number "square root -1" is never evaluated, but introduced into a formulae at the start to help solve it, and disappears at the end.

Pi also is not evaluated, but remains the ratio of the circumference of a circle to its radius.

"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.

If we remove all these non-numbers, or questions, or ratios, or any unevaluated formulae from our number line, we are left only with integer numbers.

So, a number line should contain only integer numbers.

QED.

aeberbach:
Similarly you can take an opaque tube and put a ball bearing in it, and seal the ends. Then because you can no longer know the exact position of the bearing, it is no longer there! (Ignore that rattling noise.) ;)

John B:

--- Quote from: Peter Taylor on June 29, 2022, 11:01:29 pm ---These are termed irrational "numbers", having no exact value.

--- End quote ---

But they do have exact values, or positions on the number line (complex numbers excluded), just they can't be expressed as the ratio of 2 integers.

RJHayward:
Actually, 'Infinitely small' and 'zero' are not the same.
In CALCULUS mode, you need a really good school instructor.  You see; Calculus being a methodology, a new method of approach to arriving at usable numeric result.  It's NOT instinctual, and so it seems impractical.  For example, the mind-excercise that says things like; "dx goes to zero, as you repeat forever..."
But that is only one fragment.  With cleverness, it can be seen, that as that 'dx' portion moves a certain way, that another 'fragment', 'dy' also, in turn, is moving some characteristic way.
It's almost as nebulous a concept as saying, for example, one WIZARD shrinks (you) to half size, and then another WIZARD makes you 4X bigger, with end result; you are twice original size.  The little, localized exchanges make no real sense, but combined the two actions DO make a nice, clean result.
A lackluster, dopey but competent MATH teacher might not be able to teach this essence.
Now you've got me wondering about Issac Newton and his thoughts on inventing The Calculus...

Someone:
Some nonsense to reply to the nonsense:

--- Quote from: Peter Taylor on June 29, 2022, 11:01:29 pm ---"Square root 2" likewise is not evaluated, but remains a question: "What value multiplied by itself equals 2 ?".

These are termed irrational "numbers", having no exact value.

But these are not numbers, or values, but questions, or formulae.
--- End quote ---
Only because people tend to think in units of integers, other problems are thought of in other units:
https://www.explainxkcd.com/wiki/index.php/1292:_Pi_vs._Tau