As I mentioned in another thread, I'm finally opening a thread to talk about a concept so simple and elementary that I actually find it hard to believe that, to the best of my knowledge, it appears to remain "yet to be discovered" by our advanced mathematicians and technologically sophisticated society up to this day (even though in a certain way it is already in use).

As a side comment, it seems to me that modern-day mathematicians are so absorbed by the thrill of developing new and sophisticated concepts, that they've forgotten to care about what's still lacking in the most elementary areas of mathematics. For example, the lack of a standard system to properly notate the accuracy of a number; at a minimum, to make it clear whether we are representing an exact quantity ("further-calculation-safe") or instead an inaccurate (chopped/rounded) one (that can potentially mangle and invalidate the results of further calculations). If I write 0.66666 and I really mean 0.66666 (i.e., 66666/100000) and not 0.66666... (i.e., 2/3 chopped at the fifth decimal), how do I make that clear in a standard notational (non-verbal) way (since 0.66666 is more than likely to be understood as the latter rather than the former)? All it would take to fix this potentially severe shortcoming would be to introduce a simple symbology, such as 0.66666¬ to indicate exact (i.e., a terminating 0.666660000000.... with a chopped tail of recurring 0s after the last given digit) and 0.66667~ to indicate rounded, in addition to 0.66666... (or better 0.66666···) to indicate chopped; and further extensions could be added for more fine-grained indications (such as the type of rounding).

Now, on to the topic of the thread, there exists a well-known nuisance with our current year-numbering system that keeps getting people confused, and even to inspire absurd heated discussions and rants. If you go to a public forum and ask "What year did you celebrate the beginning of the 21st century?", you're likely to get two irreconcilable replies (the "scientific" one vs. the "common-sense" one) and probably end up with a flamefest. All because of our society not having yet properly recognized this most simple concept of positional ordinals.

(Note that here I will talk exclusively about ordinal numbers in the common linguistic sense of it: first, second, third..., and not about what Cantor chose to call "ordinal numbers"—actually, "ordinal sets"—which are quite a different thing in many respects, such as including a zeroeth ordinal.)

Our current system for ordinal numbers is actually an ancillary system to our cardinal number system. While we use an independent numbering system to represent cardinal numbers (currently a system of the positional or "tiered" type, so as to facilitate the representation of arbitrarily large quantities, which would become quite convoluted using a sign-value system like that of the Romans or a system of the "linguistic" type such as that of Chinese numbers, both of which require the addition of more and more new symbols for the system to be able to "scale", whereas a place-value system has a fixed, finite set of symbols from the start), our usual system to represent ordinal numbers is really one of "positional-cardinal ordinals", essentially dependent on the former. We represent ordinals by referencing a cardinal (counting how many elements we've traversed in the series so far), representing this cardinal in the cardinal way and then adding to it a mere suffixed marker to indicate "ordinality" (i.e., to indicate that the number is meant to represent the relative place of an element in an ordered series, instead of the amount of something).

It is this "numerical miscegenation" between cardinals and ordinals that is precisely the crux of the matter with the above calendrical issue, because cardinal and ordinal numbers are quite different mathematical beasts (measuring quantity vs. indicating relative position in a series), with several mutually-incompatible characteristics and rules. For example, zero is a most prominent member of the cardinal numbers, whereas the concept of an ordinal zeroeth is inherently nonsensical (the usage of the neologism "zeroeth" to name the first member of a zero-based indexed series, or the first ordinal set of Cantor's—the empty set of zero members—being quite another matter).

The source of all the fuss about year numbering is the fact that the series of cardinals is "odd" or "centred-symmetrical" (so to speak; I don't really know what the proper nomenclature for this concept is), whereas the ordinal series is "even" or "centreless-symmetrical". That is, the integer cardinals go like this (in binary):

... , −101 , −100 , −11 , −10 , −1 , 0 , +1 , +10 , +11 , +100 , +101 , ...

They are "centred" on element zero, a sui generis member of the cardinals with some rules of its own (e.g., the concept of sign doesn't apply to it, because zero is actually at the centre of symmetry with respect to sign). Whereas the series of integer ordinals doesn't have any "central" element: the first member in the forward series ("positive ordinals") is immediately preceded by the first member in the backward series ("negative ordinals"), with no unsigned zeroeth member in between (the series is centred not at one central element, but between its two centremost elements):

... , −101th , −100th , −11rd , −10nd , −1st , +1st , +10nd , +11rd , +100th , +101th , ...

Because, a quantity can be reduced to zero and still be conceptually a quantity (the amount of stuff you have when you have no stuff left), but if a position is "reduced to zero" it ceases to exist as a position: a "no place" is no longer a place; you must have at least one position, one place in a series (its first) for the concept of position (relative place in an ordered series) to apply.

One might think of cardinal numbers in terms of points and distances: a cardinal is translated into a point and the quantity represented by it is the distance from this point along the real line to the point representing the central cardinal zero; a distance that gets reduced to zero in the case of zero itself. Whereas ordinals may be better thought of in terms of adjacent segments: an ordinal is translated into a segment of a certain size and the position represented by the ordinal is that of the relative place that this segment occupies in the series of segments on the line, with the first elements in the forward and backward series being the ones immediately adjacent to the central point where the positive-first and the negative-first segments meet. Graphically:

An empty series of elements (an empty series of segments in the graphical representation) simply has no elements in any position for a relative order to be established and referenced; it has no "places" in it. Thus, the zeroeth element of a series cannot exist, since that would be an inexistent element to occupy an inexistent place in the series. Which is why the concept of an ordinal zeroeth is essentially nonsensical.

Even though some (particularly, computer geeks and set-theory mathematicians) have tried to argue otherwise, because of a conceptual confusion between the idea of an ordinal zeroeth (a inherent nonsense) and the linguistic label "zeroeth" that has been applied to other unrelated concepts, such as:

– The label (index) applied to the first element of a zero-based indexed series (such as the first element of an array in computer programming). Note that an empty array of 0 elements would have no element to be labelled its "zeroeth", and the use of numerical indices to label the elements in a series is a quite different thing from the real concept of an ordinal. One could instead use letters or names for the index labels as well, and numerically indexed series can start at whatever number one chooses (not necessarily at one or at zero). The use of numbers in indexed series should really be called something like "indicial numbers", and one should be careful not confuse them with real ordinals, which they are not.

– The label sometimes applied to an out-of-series element, such as to the introductory chapter of a book, or to the ground floor of a building (a floor which, from the usual point of view in Europe and Latin America, belongs neither to the series of floors with a positive elevation above ground, nor to the series of floors with a negative elevation above ground, but is the singleton element in the series of floors at a zero elevation above ground; however, from the point of view of North Americans and several others, floors are considered to be above or below ground according to whether the height span between that floor and its upper floor is above or below ground, rather than according to the elevation at the point of the floor level, so the ground floor is considered the first floor in the series of floors above ground, rather than the singleton floor at precisely zero-elevation).

– The nomenclature Cantor chose for his "ordinal number" sets, naming as "the zeroeth ordinal" what actually is the first member in his series of so-called "ordinal numbers" (a different concept from the common linguistic concept of ordinal numbers that I'm talking about here). Because, even if his "zeroeth" ordinal set corresponds to the empty set with zero elements within it, the very presence of this empty set itself already raises the count of elements in the series of Cantor's ordinals to one (1) element. That is, his so-called "zeroeth" set is actually the first (1st) set of his ordinal set series, no matter how many elements that one "zeroeth" set contains. Since if you had zero (0) elements in the ordinal series, you would simply have no element whatsoever to properly label the zeroeth (0th) element of it. Cantor's mistake here was to confuse and intermingle the concept of the cardinality of a set (the number of elements it contains) with the concept of the relative position occupied by that set in a necessarily non-empty ordered series of sets (its real ordinal number), by calling the set of cardinality 0 in this series his "ordinal number zeroeth", the set of cardinality 1 his "ordinal number first", etc., thus trying to force a parallelism between cardinals and ordinals that actually does not exist (as explained and graphically represented above, the series of cardinals and ordinals are actually essentially incompatible and "do not match", and this incompatibility is precisely the source of the ages-long year-numbering confusion). So Cantor's ordinals are actually not the same thing as the common-language ordinals, but quite another matter (despite what set-theory mathematicians seem to think).

[to be continued...]

Nice post! We sort of touched on this issue in the pitch notation thread, as well.

I always thought of the difference as a row of muffins side to side. :wacko: You can count either by the muffins are the cracks between the muffins.

The circles and the carets...

Counting "the cracks between the muffins" is such a good idea that several programming languages have implemented it. In Python,

What's "out-of-series" about European floor numbering?

------------
4th floor
------------ 4 floors above ground
3rd floor
------------ 3 floors above ground
2nd floor
------------ 2 floors above ground
1st floor
------------ 1 floor above ground
0th floor
============ GROUND
-1st floor
------------ 1 floor below ground
-2nd floor
------------ 2 floors below ground

Is a nice clear system in which the distance between any two floors (in both senses of "floor") can be determined by subtracting the floor numbers. The American system does not share this property, and thus causes people to get the statement "you walked up N flights of stairs" wrong most of the time.

^^ I mean "out of the ordinal series". Look carefully at your own illustration. The ground floor belongs neither to the ordinal series of floors above ground (it's not "the first floor above ground", if we consider "above ground" to mean "at an elevation greater than 0") nor to the ordinal series of floors below ground (it's not "the first floor below ground" either).

Of course you can perfectly choose to call it the "0th" floor, but that wouldn't be referencing its true ordinal number. That "0th" label would be an indicial number (i.e., "0th" there means: "in some way related to cardinal number 0"). You're using the floor elevation value (a quantity, which can be zero) as an index (a label or tag) to identify that floor in the series of floors. You yourself make it clear that such floor numbers are actually of a cardinal (and not of an ordinal nature) when you mention that you can obtain the distance (a quantity, not a position) between floors by simply subtracting their floor numbers (i.e., by subtracting their indicial numbers, which here are based on each floor elevation value, a quantity, not a position)

The true ordinal corresponding to an element immediately preceding the first, is the first going backwards (i.e., negative first). It's never the zeroeth, which in a truly ordinal sense is a nonsensical concept, as I've been trying to explain.

That is, if you consider that above ground refers to a positive (non-negative non-zero) elevation, then in this series the ordinal number corresponding to the the ground floor is not the "zeroeth" floor above ground, but the minus-first floor above ground; i.e., the first floor going backwards (instead of forward) from the point where the series starts (which in this case is the elevation point of the first floor with a positive elevation and not the point of zero elevation).

Here's an illustration:

Note that Cantor's nomenclature of his so-called "ordinal numbers" is also of an indicial rather than truly ordinal nature, since his "ordinal" sets are identified not by their real ordinal position in the series (the "ordinal" at the first position is not called the "first") but by indexing each set with its cardinality value: the ordinal set of cardinality 0 is called the "0-th" (even though it is actually the first of the series), the one of cardinality 1 is called the "1st" [should rather be called the 1-th] (even though it is in fact the second), etc.

Another example that hopefully will clarify the conceptual non-existence of ordinal zeroeth (as opposed to the perfectly possible indicial "0-th").

Think of a person standing at a certain point on a line. This person now begins to walk following this line in a forward direction. We can now quantify the distance walked ("the person has walked three steps"), using cardinal numbers. Or, since the steps are performed in an orderly manner, we can identify each individual step using ordinal numbers ("the third step was longer than the previous.").

Now, suppose that instead of starting to walk forward from the starting point, this person starts to walk backward. Again, we can quantify the distance walked using cardinal numbers, and additionally we can numerically express the fact that the person is now moving in the opposite (backwards) direction, by introducing the concept of sign: −3 steps (readable as "the person has retroceded 3 steps" or "has walked 3 steps backwards"). Similarly, we can identify each individual backward step, and distinguish it from the forward steps, by using negative ordinals: the −3rd step (readable as "the third step backwards"). Note that, even though it seems totally unusual to write "−3rd", the concept of negative ordinals is in widespread use since long ago. Only that, instead of simply notating them by means of a sign, we complicate our lives with some other various adhoc means, such as verbally saying before, as in year numbers: 2 CE (second year of the common era = +2nd year of the era), 2 BCE (second year before the common era = −2nd year of the era).

As you may have observed in this example of a person walking from an initial point of rest, there is no room whatsoever for the ordinal concept of a zeroeth step, even though the cardinal concept of zero steps is perfectly sound. If the person has moved 0 steps, he hasn't walked any step yet, so there is no step to be labelled "the zeroeth", and once he performs one step in either direction, he is performing the (positive or negative) first step in that direction. Both ordinal series of steps, the forward and the backward ones, are immediately adjacent in order; there's no gap that could be filled by a zeroeth step between them.

I'd like to make sure that the issue of the non-existence of an ordinal zeroeth has been clearly understood before continuing, because this is a central concept that distinguishes ordinals from cardinals, and the key to solving the year-numbering problem by means of true positional ordinals instead of the current intermingled "cardinal ordinals" (ordinals expressed in an underlying cardinal way).

Well, yes and no. What you're actually telling the program to do is "give me the characters of this string from indices 0 (inclusive) to 4 (exclusive)". It works the same, for example, in Java ("alpha.substring(0,4)", "alpha.substring(4,7)").

It's become kind of a standard to use half-open intervals of the kind [a,b) in programming languages. The reasoning being that this way you can concatenate them ( [a,b) + [b,c) → [a,c) ) without leaving out ( (a,b) + (b,c) → (a,c) − {b} ) nor repeating ( [a,b] + [b,c] → [a,c] + {b} ) any internal element of the concatenated interval. So, for example, the statement "alpha.substring(0,4) + alpha.substring(4,7)" in Java is equivalent to "alpha.substring(0,7)" (note that Java overloads the operator + to mean concatenation in the case of strings). Another reasoning is that this way you can compute the size of the interval by subtracting the values of both ends (so "alpha.sustring(0,4)" has a length of 4−0=4, and "alpha.substring(4,7)" has a length of 7−4=3).

The downside is that this practice, combined with the also computer-widespread practice of using 0-based indicial numbers to identify the elements of arrays, strings, etc., can easily lead to confusing situations if one is not very careful, since it is completely at odds with the usual way of thinking and verbally expressing things (e.g., a statement phrased "alpha.substring(4,7)" can easily mislead programmers into thinking that the substring will contain from the 4th to the 7th characters of the original string, or if one remembers that intervals are being expressed as half-open intervals of the kind [a,b), to think that it will contain from the 4th to the 6th, when in fact it will contain from the 5th [index 4] to the 7th [index 6] character; a confusion that only grows larger when dealing with multiple strings being successively substringed and/or concatenated).

Yes, but this is a linguistic distinction, not a mathematical one. There are at least four reasonable ways we could derive ordinal numbers from cardinal numbers:

[DoHTML]
<dl>
<dt>Forward Inclusive</dt>
<dd>Calling X the "Nth" item in a list means that there are N items before and including X. E.g., November is the "11th" month of the year: The "11" counts January, February, ..., November. This is the traditional system.</dd>
<dt>Forward Exclusive</dt>
<dd>Calling X the "Nth" item in a list means that there are N items before but excluding X. E.g., November is the "10th" month of the year: The "10" counts January, February, ...., October. This is the approach used for array indices in most programming languages.
<dt>Backward Inclusive</dt>
<dd>Calling X the "Nth" item in a list means that there are N items after and including X. E.g., November is the "2nd" month of the year: The "2" counts November and December. This is what negative array indices mean in Python.
<dt>Backward Exclusive</dt>
<dd>Calling X the "Nth" item in a list means that there are N items after but excluding X. E.g., November is the "1st" month of the year: The "1" counts December.
</dl>
[/DoHTML]

Assuming that there is at least one item in the list, inclusive ordinals start at one while exclusive ordinals start at zero. Thus, before there was a symbol for zero, languages had a natural tendency to favor inclusive ordinals over exclusive ones. However, this does not mean that exclusive ordinals are less valid as a concept, any more than "<" is less valid than "[DoHTML]&le;[/DoHTML]".

Unfortunately, we lack a way of expressing this concept. Backward ordinals are distinguished from forward ordinals with terms like "Nth-to-last", but there is no parallel for exclusive vs. inclusive ordinals. And this is the cause of the confusion.

No. My point is that ordinals are an independent mathematical concept (the concept of relative order), one that is not derived from, nor necessarily dependent on cardinals (the concept of quantity). The relative order of an element in a series is exactly the same no matter if you choose to express it in a "forward inclusive" or "backward exclusive" cardinal-derived way.

It is precisely because of the way our language names ordinals (in most cases —barring first and second in English— as a morphological derivation from the names of cardinals) that we think of ordinals in terms of cardinals and represent them in an underlying cardinal way (we first represent a cardinal quantity and then merely add to it an "ordinal suffix"). And it is precisely this practice (the representation of ordinals in terms of cardinals) that is the source of all the fuss about year numbering. Ordinals and cardinals are separate mathematical entities, each with their own rules; which is why if do arithmetic with ordinals using a cardinal-based representation and cardinal arithmetic, the results will likely be incorrect (especially when the involved ordinals differ in sign).

In a certain way, there is a similar conceptual confusion regarding rational numbers. People tend to think of them in terms of fractions (many even equating them with fractions), and thus there is even a convoluted mathematical definition of rational number that goes like "a rational number is an equivalence class of fractions". In fact, each rational number can be expressed in a unique way (its prime factorization) that is independent of any fractional representation of it. For example, the rational number underlying the "equivalence class of fractions 3/4 ~ 6/8 ~ 9/12 ~ ..." can be more elegantly expressed in a unique factorized way as 2^−2 · 3^1.

As I will explain later, ordinals can also be expressed in their own terms, without resorting to an underlying cardinal representation. That's really what this thread is about: that ordinals are not an ancillary derivation dependent on cardinals, but an independent class of mathematical entities in their own right, and with their own characteristics and arithmetic rules.

No. Languages had, and still have, a "natural tendency" to favor what you call "inclusive ordinals" over "exclusive ordinals" (actually, zero-based indicials) for the simple reason that the concept of an ordinal zeroth is nonsensical, as I have already explained with several quite clear examples.

Sure you can choose to label the first element of the series its "zeroth" if you like (Cantor did that). But then you will have two distinct "zeroths": positive "zeroth" and negative "zeroth". As explained and graphically shown in the first post of the thread, the series of ordinals has a "centreless or 'even' symmetry" with respect to sign, with no central element at the axis of symmetry equivalent to cardinal unsigned zero. This is a fundamental difference between cardinals and ordinals, and precisely the one why it is not possible to represent ordinals in a coherent way using a cardinal way of thinking.

Let's see how we can define ordinals in their own terms, before moving on to explaining how positional ordinals work.

Ordinal numbers are meant to represent the concept of order. So let's think of an ordered set. What do we need to define it as ordered? Well, we need to have a binary relation of precedence. That is, a relation that given two elements of the set will tell us which one goes "before" or "after" which (or in cardinal terms, which one is "less than" or "greater than" which). Next, we have to be careful to distinguish between two kinds of order: total order and partial order. An order is total if all the elements in the set are comparable under the above binary relation. An example of a set with a total order is the set of real numbers, and an example of a set with a partial order but without a total order (or at least not one that is compatible with its arithmetic) is the set of complex numbers. Total order is also known as linear order (because the set can be arranged in a one-dimensional, linear fashion under the order relation), and in mathematical terms a set with a linear order can be referred to as a chain. This is the kind of order that interests us for the purpose of defining ordinal numbers.

So we have a binary relation of precedence between ordinal numbers, and thus we can say that some ordinal goes before or after some other. But ordinals tell us more than just this. Two ordinals can be immediately adjacent to one another, meaning that they have a "point" in common (see below) and there is no other ordinal "between" them (in mathematical terms, given an ordinal A that precedes an ordinal B, ordinal A is said to be immediately adjacent to ordinal B iff there exists no ordinal C such that C is after A and before B). So, in addition to the binary relation of precedence, we can define a pair of functions for immediate precedence: a function "next" that tells us which ordinal goes immediately after another, and a function "previous" that tells us which one goes immediately before. So we can say that "ordinal C goes before ordinal E and after ordinal A", but also that "the next ordinal to ordinal C is ordinal D and the previous one to it is ordinal B". Thus, we can conceptualize the chain of ordinals as a chain of immediately adjacent segments on a line. Each of those segments has a certain size, and is defined by two points: an initial point and an end point. Its initial point being the one that the given segment has in common with its previous segment (previous ordinal), and its end point the one in common with its next segment (next ordinal). So we can say that ordinals have a "size" (I'll elaborate on this in a later post) and two "sides" (the side "facing" its previous ordinal and the one "facing" its next).

What's more, an ordinal also tells us its exact relative position in the whole chain of ordinals. And for this we need to define a point of reference: the starting point of the ordinals. Which leads us to the concept of first. Using the above conceptualization in terms of segments, we can define first as "the ordinal whose initial point (or 'previous' side) is at the starting point". Which, for the set of unsigned ordinals, can be rephrased as "the ordinal with no previous ordinal".

Once we have the concept of first and the concept of immediate precedence, we can already define all other ordinals: the second is "the ordinal whose immediate previous is the first", the third "the one whose immediate previous is the second", etc. When we introduce the concept of sign and the chain of positive ordinals is mirrored with the starting point as the centre of symmetry, we can then define negative first as "the one immediately previous to positive first", negative second as "the one immediately previous to negative first", etc. That is, negative first is "the ordinal whose end point (instead of initial point) is at the starting point", and the change of sign can be thought of as kind of "flipping" or "reversing" the "sides" of the ordinal (the one "facing" its next becomes the one "facing" its previous and viceversa). Note that, as a different kind of "mirror image" of the concept of first, we can also define the concept of last, if instead of (or in addition to) a starting point, our chain has a defined final point. The last is thus "the ordinal whose end point is at the final point" (or "the [unsigned] ordinal with no next ordinal"), and similarly we can define all other ordinals in terms of the last instead of in terms of the first: second to last, third to last, etc. (and here we can also introduce the concept of sign and define negative last as "the ordinal whose initial point is at the final point", and then define the subsequent negative second to last, etc.).

As a side comment, note the etymology of the English word first, which is (quite obviously) not in any way related to the word for cardinal one. Instead, it (and its Latinate cognates primary and related words such as prime) etymologically means "foremost" (PIE root *per- + a superlative sense through derivation). First is in fact a cognate of fore; not surprising since the first is the one at the fore or before all others (in the unsigned or positive series). The (Latinate, through French) word second is also not in any way related to the word for cardinal two, and instead etymologically means "the one about to follow [the first]" (Latin secundus, the future participle of deponent verb sequor "to follow", from PIE root *sekʷ-). As for the word last, like first it is etymologically a superlative (the superlative of comparative latter, and a cognate of late, since the last is "the one coming latest").

However, even though ordinals and cardinals are distinct concepts and, as we have just seen, ordinals can be defined entirely in their own terms completely independent of the cardinals, there seems to be an apparently innate tendency to associate ordinal number first with cardinal number one, ordinal second with cardinal two, and in fact all other ordinals in English are named after their associated cardinal in this way (third, fourth, etc.). This mental association probably stems from the fact that in order to have a first you need to have at least one, in order to have a second you need to have at least two, etc. Note that there is no element to be named ordinally if you have zero elements, so languages have historically never felt any need to come up with a word for a supposed ordinal zeroth, even centuries after a specific word for zero entered the language.

Nevertheless, it is precisely because of this close mental (and linguistic and representational) association between cardinals and ordinals that, once people realized of the nature and importance of number zero for the cardinals, and in fact turned this discovery into a kind of badge of honour of how evolved and mature modern mathematics are in comparison with the "primitive" mathematics of the ancients that lacked the use and concept of 0 as a number, that in recent times certain people have tried once and again to artificially graft this purely cardinal concept (the concept of no quantity) onto the ordinals, absurdly trying to create the (unnecessary and impossible) concept of an ordinal zeroth. Curiously, people have not been so keen to, viceversa, try to graft the purely ordinal concept of last onto the cardinals. :rolleyes:

As should be (hopefully) clear by now —once that ordinals have been defined in their own terms, without the need to use, nor room for, an ordinal concept analogous to the cardinal concept of zero—, and as I previously tried to explain with illustrative examples (such as the one of a person walking successive steps from an initial point of rest), the word zeroth only makes sense in an indicial sense. That is, meaning not a true ordinal "zeroth" (an absurdity), but merely acting as a tag, an index that is placed on some element to mean that the tagged element is "in some way related to cardinal number zero". For example, because a label with a printed "0" has been physically placed on it, such as on a door or floor (e.g., to indicate how many floors there are from this floor to the ground—i.e., to indicate a quantity, not really an order, although an order can be derived in such cases from the fact that the cardinal numbers are themselves ordered). Or because cardinal number 0 is used as a sequential tag to identify the element in an indexed series because of some practical reason, such as in the case of the indices commonly used to identify array elements in modern computer science. A practice originating from the fact that it eases pointer arithmetic when accessing the memory location of the element, since this way to access element i you just have to calculate the pointer m + i, where m is the pointer to the memory address of the first element of the array. Another reason is the fact that indices starting at zero may use the full range of unsigned integers representable on computers, whereas indices starting at one would "waste" the available index zero. Note, however, that the initial practice in programming languages was to use an "intuitive" ordinal-inspired approach of one-based indexing, and it was only later because of said practical reasons that the usage was majoritarily changed to the current dominant one of zero-based indexing; and then it was not long before too-self-assured computer geeks—which do not seem to realize of the difference between ordinals and indicials—started to believe that "zeroth" as an ordinal was a sensible concept, since they think they are using it when they refer indicially to the first element of an array as its "0-th" because of its index label. :rolleyes: