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I think that the word integral in calculus is nothing to do with integer or integer numbers.

But why is integral is chosen for integration? In algebra, integral means related to integers, and this is exactly the same as the word integral in calculus with a very different (?) meaning; are they connected? If not why from millions of words are they same?

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    $\begingroup$ I'm hoping for a better answer than this, but both of these concepts have to deal with the idea of "wholeness". Taken from the entry of integer at dictionary.com "Latin: untouched, hence, undivided, whole =in- in-3 + -teg- (comb. form of tag-, base of tangere to touch) + -er adj. suffix]" Maybe someone can expand on this concept of "wholeness" and how it applies to integrals. It is easy to see how it applies to the word integer. $\endgroup$ – Paddling Ghost Jul 27 '15 at 20:02
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    $\begingroup$ Integers are used to count whole things like 3 sacks of wool. 2.5 sacks mean that one sack is not whole. One can make it whole, e.g. if it is only 1/3 full, you may add to it some more wool like another 1/3, then another 1/4 then another 1/12, so that you get the whole sack at the end. 1=1/3+1/3+1/4+1/12 is called integration (making whole from parts). Identical thing happens in calculating an integral (whole) area from small parts (e.g. rectangles). $\endgroup$ – A.Γ. Jul 27 '15 at 20:46
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    $\begingroup$ This is a weakness of english, too far from latin which made some words collapse into each other. In french instead of integral for calculus, integral for algebra we use integrale for calculus et entier for algebra. Same roots, but very diff words. Entier is also the word for integer, btw. $\endgroup$ – Pierre Jul 28 '15 at 7:49
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They share a Latin root corresponding the the concept of 'wholeness'. In the context of integers, this would be interpreted as 'whole numbers' - i.e. numbers with no fractional part. In the context of integration, this would correspond to 'summing up to create a whole' in the sense of the integral representing a continuous sum or area. I think that's why they use similar words.

I'm no expert in such things, but I've included some links to an online etymology dictionary that appears to agree with me.

1: Integer: From the Latin adjective integer, "whole, complete"
2: Integrate: "to render something whole"/"to put together parts or elements and combine them into a whole"
3: Integral: "of or pertaining to a whole"

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