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Integers, Pi, and Number Lines.

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RJHayward:
Everyone:
   EITHER Fly, or get out of the Toga Closet.

(Just trying to fit in, by lookinstuupid)

Nominal Animal:

--- Quote from: Vtile on July 05, 2022, 07:36:10 pm ---
--- Quote from: Nominal Animal on June 30, 2022, 06:55:54 pm --- That is, we never, ever try to divide anything by zero there.  Instead, we can think of the process as looking at how the ratio changes as \$h\$ gets closer to zero, and extrapolate the value the ratio would be if we could calculate it at \$h = 0\$.

--- End quote ---
IIRC Newton had this concept of using something almost a zero (taste, feels and quaks like zero) as divisor when forming the concept and ideas of modern calculus. Much more intuitive, but unfortunately not mathematically rigorous.

That was fixed and reformed much later with idea of limits.
--- End quote ---
The way I intuitively grasped it in comprehensive school was by realizing that if one graphed the function
$$g(h) = \frac{f(x+h) - f(x)}{h}$$
(with a specific value of \$x\$, of course), then the limit \$h \to 0\$ is what \$g(0)\$ would be if it could be evaluated.  Thus, extrapolation.

If you start that by drawing a curve, picking a point on that curve, and asking the students how they'd find the tangent of that curve at that point, it's very easily graspable concept.

It becomes even more so, if you tabulate or plot the graph \$g(h)\$ with a logarithmic \$h\$ axis (\$h = 0.1, 0.01, 0.001, 0.0001, 0.00001, \dots\$).


--- Quote from: Vtile on July 05, 2022, 07:36:10 pm ---....In sixties the duck did born again with hyper-reals.
--- End quote ---
Eww.

Well, what else can you expect from mathematicians?  I only use apply the stuff they come up with, to solve problems.

Vtile:
The duck:
https://en.m.wikipedia.org/wiki/Infinitesimal

tggzzz:

--- Quote from: Peter Taylor on July 05, 2022, 05:15:05 pm --- :-DD

Love u all.

--- End quote ---

I'm failing to distinguish you from a conventional troll.

Am I missing something? If so, what?

tooki:

--- Quote from: Peter Taylor on July 05, 2022, 04:41:15 pm ---An integer number (or just number), should be any value that is wholly defined.

An irrational "number" is not a number, so there can be no rational "number".

These terms hark back to the days of the Pythagoreans, when they thought the Earth was the centre of the Universe.

--- End quote ---
You don’t get to decide what you think the definition of a word should mean. They have established, clearly defined meanings. An integer is a number that can be written without a fractional component.

An irrational number most certainly is a number, as are non-integer rational numbers.

You need to shut the fuck up and go back to middle school (it was it primary?) math, rather than trying to tell anyone anything.

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