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integral element

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Definition
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When I saw the entry 'integral element' I thought it is connected somehow to integration, and I was quite suprised that it was not the case ;) But nevertheless, is there any connection to integration? And why actually one names such elements as 'integral'?

Thanks in advance.
Regards
Serg.
-------------------------------
knowledge can become a science
only with a help of mathematics

> When I saw the entry 'integral element' I thought it is
> connected somehow to integration, and I was quite suprised
> that it was not the case ;) But nevertheless, is there any
> connection to integration? And why actually one names such
> elements as 'integral'?

In this sense, the word 'integral' is an adjective describing the property of being an integer, or similar to an integer. It is used commonly in algebra and number theory.

The fact that 'integral' in the sense of analysis and in the sense of number theory is the same word is a historical accident.

``Integral (element)'' has in mathematics nothing to do with ``integration''. Ethymologically, both words have their origin in the adjective ``integral'', the basic sense of which is 'whole, entire'.

The origin of the English ``integral'' is Latin ``integer, integra, integrum'' having the initial meaning 'in-tact, un-touched, un-broken, whole' (the Latin prefix ``in'' means 'non', ``teger'' is a derivative of the verb ``tangere'' = 'to touch').

Isn't the meaning of the integration to form something "entire", viz. the antiderivative, from its "shadow", viz. the derivative?

Jussi

The word "integral in the sense of calculus indeed means "whole" or "entire". The original name for differential calculus was "calculus differentialis", which I think means "calculus of differences". At first Leibniz used the term "calculus summatorius" for integral calculus, which I guess means "calculus of summations". Later Jacob Bernoulli came up with the phrase "calculus integralis", which means "calculus of the whole" as opposed to only the differences. This terminology is the one that stuck.

Thanks a lot! Really nice and full explanations!!!

I am actually quite suprised that "integral" and "integer" are in fact related. But your explanations is a perfect motivation for it:

> The origin of the English ``integral'' is
> Latin ``integer, integra, integrum''
> having the initial meaning 'in-tact, un-touched, un-broken, whole'

But that

> It is used commonly in algebra and number theory

I didn't know. Although, I have just remembered that in the entry "integral domain" this misleading in terminology is discussed ;)

But then let me ask another things: why "non-archimedean valuation" actually are called non-archimedian and why they lead to integers. Of cource the last question is connected with the fact that

"The only non-archimedean valuations of $\mathbb{Q}$ are the $p$-adic valuations"

but for me this fact is not obvious. Is there probably some intuitive motivation for it?

Thanks in advance!

Regards
Serg.
-------------------------------
knowledge can become a science
only with a help of mathematics

I have noticed the following thing in the above entry ("integral element"):

there is an expression "$p$-adic valuations", BUT it doesn't get linked to the entry "$p$-adic valuation".

It seems it is connected not with plural form of valuations, since linker doesn't care about plural/possesive, but with math environment $$. May be it is something to do with the following: when one copys the above expression one gets additional space in $$, i.e.:

$ p$-adic valuations

So, may be linker sees the expression with additional space and that's why doesn't link it?

Serg.
-------------------------------
knowledge can become a science
only with a help of mathematics

Please see the Theorem 3 in "ultrametric triangle inequality"; the usual absolute value (e.g. in Q) has the Archimedean property, but no p-adic valuation (e.g. in Q) has it.

Regards,

Jussi

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