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A 'simple' Physics postulation...

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Ed.Kloonk:

GlennSprigg:
O.P. here... I was intrigued & overwhelmed by some of the mathematical/physics  comments here!   :-+
By far the largest 'problem' is defining what 'stop' means, but the following might 'negate' the concern for such thoughts??  :phew:
We generally agree, that to 'stop', should mean to be a rest for a known period of time...  However!!!... Consider ALL the other speeds
that the object was traversing through while initially decelerating and then re-accelerating through???  It was NEVER at ANY of those speeds
for ANY amount of time!!  So in the same manner, it was never at say 5 m/s or 10 m/s for any time greater than zero!! if you know what I mean!   8)

On a similar note, I had an argument with my maths teacher when I was young, about right-angled triangles....  2 sides, 1x1, hypotenuse = Root-2.
Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points, as you can keep adding digits to the end,
making it realistically ever so further, even though microscopic!!  However, I went on to say that if you DO draw such a triangle, with theoretically
zero line thickness, then there the hypotenuse lies...  exactly between those 2 actual physical points!!!  If you grasp what I mean!?   :phew:

vad:

--- Quote from: GlennSprigg on July 11, 2021, 11:29:57 am ---Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points
--- End quote ---
This statement is false.

Square root of 2 is irrational number. The only thing that distinguishes irrational numbers, from other real numbers (from rational numbers), is that the irrational number cannot be expressed as a/b, where a and b are integers, while the rational number can be expressed that way. Nowhere it follows that if you cannot express distance between two points as a/b, such distance is somehow forbidden.

Contrary to that, the distance can be irrational number. You can prove this by contradiction:

Suppose distance between two points must always be a rational number.
Take a right triangle with both legs of size 1.
The distance between two vertices of hypotenuse of such triangle is square root is 2, according to Pythagorean theorem.
It can be proven elsewhere that the square root of 2 is not a rational number.
Therefore the initial assumption that the distance between two points must always be a rational number must be false.
:)

Brumby:

--- Quote from: vad on July 11, 2021, 01:02:44 pm ---
--- Quote from: GlennSprigg on July 11, 2021, 11:29:57 am ---Now Root-2 goes on for ever, so 'technically' can not be drawn as a physical distance between to points
--- End quote ---
This statement is false.
--- End quote ---
Yes, it is absolutely false.

Just because you can't define a value to a known number of decimal places, does NOT mean you can't draw an object with that value.  I can offer several examples without even trying.


--- Quote ---Square root of 2 is irrational number. The only thing that distinguishes irrational numbers, from other real numbers (from rational numbers), is that the irrational number cannot be expressed as a/b, where a and b are integers, while the rational number can be expressed that way. Nowhere it follows that if you cannot express distance between two points as a/b, such distance is somehow forbidden.

Contrary to that, the distance can be irrational number. You can prove this by contradiction:

Suppose distance between two points must always be a rational number.
Take a right triangle with both legs of size 1.
The distance between two vertices of hypotenuse of such triangle is square root is 2, according to Pythagorean theorem.
It can be proven elsewhere that the square root of 2 is not a rational number.
Therefore the initial assumption that the distance between two points must always be a rational number must be false.
:)

--- End quote ---

Absolutely correct!   :-+

TimFox:
Again, modern mathematics as taught in freshman analysis courses rigorously defines real numbers, including simple irrationals like root-2 and transcendental numbers like e and pi.
There is no need to re-fight 19th century battles.

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