Beginning
of infinities big and small
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Numbers were "invented" by man a very, very long time ago. The purpose of numbers is to indicate the quantity or size of something. Numbers "appeared" when precision was required. When adjectives "big", "small", etc. were not enough.Β
Very special number - zero - was also invented, but a little later than the other (usual) numbers. Its peculiarity is not to denote the quantity of something, but the complete absence of ... something.
With the development of mathematics, the specifics of zero became more evident. Its unique features were found out: when multiplying any number by it, the result will always be the same - zero. And to divide by zero is just prohibited...
Another pole that stands out from other numbers has become ... not even a number, but a concept denoting an uncountable number: infinity.
If zero is still somehow βtangibleβ (not difficult to imagine it), then with infinity difficulties arise. Probably because infinity does not appear in everyday lifeπ.
Let's fantasize. Let's take an apple. With a sharp knife, it is easy to cut it in half. Then - to cut one of the halves in half. Then one of the quarters - again in half ...
The longer our fantasy continues, the greater the number of apple slices will be produced.
Some one of the ancients suggested that such a "cutting" could not continue indefinitely. After certain "cut", the new parts will not be parts of apple. This is how the concept atom arose. (And, probably, engineers and physicists came from here π).
Philosophers continued the thought experiment, repeat "cutting" again and again, forever. Resulting in an infinite number of apple slices and an infinitely small size of apple slices (for an infinite number of slices).
It is especially interesting that (in our example with an apple) the sum of an infinite number of terms (slices) is finite - this is one whole apple.
(But this task is for mathematicians then πβ¦)
The following is about some properties and features of
infinitely large and infinitely small (numbers, sums, ...)
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