When we say $1,2,3...$ are natural numbers, why don't we include rational and irrational numbers?

Isn't $\pi$ something natural?

Shouldn't we say all real numbers the Natural numbers?

Shouldn't we, just, say them "Counting Numbers"?

Maybe I don't know the correct reason for which the word "natural" is prefixed. So, please tell me IYK.

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    $\begingroup$ The "Natural Number" entry on the Earliest Known Uses of Some of the Words of Mathematics page may be of some interest for context. Apparently, the term dates back to the 1400s. $\endgroup$ – Blue May 31 '15 at 17:46
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    $\begingroup$ @SufyanNaeem It is not very natural to cut an apple into half when you think about it !! hahahaha $\endgroup$ – alkabary May 31 '15 at 17:47
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    $\begingroup$ Let us not forget $e$ the base of the natural logarithm. $\endgroup$ – quid May 31 '15 at 17:49
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    $\begingroup$ "This question as asked makes no good fit for this site as it asks for discussion." Is this not clear enough as an explanation for a downvote. $\endgroup$ – quid May 31 '15 at 17:57
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    $\begingroup$ It's not wise to ask "why" when it comes to naming conventions in mathematics. For instance: Whose idea was it to name something a quandle? $\endgroup$ – Milo Brandt May 31 '15 at 17:59

In the formative days of modern mathematics, there was some debate as to what the natural numbers should be considered. Grassman even suggested that the natural numbers are a result of a recursive definition, and as such should not be considered natural. Later, the Bourbakis decided that zero should be included in the naturals; differing conventions exist to this day.

The decision of the nomenclature "naturals" largely became one of a convention settling to an almost-stable state -- if any numbers are to be called naturals, it is either $\{0,1,2,\ldots\}$ or $\{1,2,\ldots\}$.

As this set of numbers can be used to wholly-construct the reals, one might say that the naturals are the bottom-most foundation of the real numbers.

Another example is to consider anthropological evidence. It is known that many civilizations in antiquity separately came upon the concept of counting systems. Some included zero, some included zero as a placeholder digit, some even excluded the number one. Because these conclusions were established more or less in parallel between societies that had no contact, one might think that the naturals are the most fundamental natural consequence of human intellect.

Coming full circle, however, to support Grassman's standpoint on the issue is the fact that other societies don't have a concept of numbers beyond "a few." This is evidenced by many languages, some surviving today in tribal cultures, that don't have counting words, instead having words for "one," "two," and "many." Indeed, as it is known that language affects neural plasticity, perhaps the natural numbers are not natural at all, as there are groups of living humans who, in the entire history of their lineage, have never known the concept of 87.392.


I'm unsure about the historical etymology of "natural numbers". You should note, though, that the word natural has many meanings. To me, in this context it isn't meant as existing in or derived from nature, but rather as coming instinctively to a person.

To be more precise, neuroscientists have shown that humans (and other animal species, too) innately recognise a few numbers, notably 1, 2, and 3 (for example, see this talk). The formal operation of counting allows us to assign a cardinality (i.e. a size) to finite but larger sets of objects, and this gives rise to the natural numbers.

With this in mind one could argue that $\pi$, being defined as the ratio between a circumference and it's diameter, isn't at all "natural" but rather quite contrived. This despite being an omnipresent constant!


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