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Το ῎Ανευ Κλίσι Ελληνική
Greek Without Inflexions

το ἀριθμό

Before the revision of Autumn 2010, ΤΑΚΕ used some strange forms that had little to do with any ancient Greek words. For example, 'four' was τέτρα. It is true that τέτρα- occurred as a bound morpheme used a prefix in the language; but it is not the word for 'four' in any ancient dialect.

Using τέτρα for 'four' would be the same as if the 'Latino sine flexione' for 'four' was quadru. It is not; Giuseppe Peano had more sense and uses the regular Latin quattuor. Likewise ΤΑΚΕ will use the regular ancient word for 'four', namely: τέσσαρα. The same approach will be adopted with other numerals.

I cannot, however, adopt forms based on the ancient language for 'million', 'billion' and so on because they did not exist. Modern Greek has, of course, developed words for these but unfortunately, for reasons I explain in the final section below, ΤΑΚΕ cannot make use of them. I discuss some options and give examples of two possible systems before suggesting my own system.

1. Preliminary Considerations

1.1 zero
The ancient Greeks did not have a proper understanding of 'zero'. How can 'nothing' have any existence, was how they tended to think of it. The concept did not really become commonplace in our western world until the adoption of Arabic numerals. Our modern word is derived from French 'zéro' which in turn is derived from the Venetian 'zero' ['tsero], a contraction of the Italian 'zefiro', from the Arabic şifr (zero, nothing).

Modern Greek does not do this, but uses the ancient word μηδέν "nothing" to denote 'zero' (the modern Greek for 'nothing' is τίποτε). ΤΑΚΕ adopts the ancient word according the rules given on the 'Nouns, Adjectives & Definite Article' page; this gives the form μηδενό.

1.2 one
The ancient word was εἷς [m.], μία [f.], ἕν [n.] "one" which was a 1st & 3rd Declension adjective. We derive the ΤΑΚΕ word according the normal rules, namely: ἑνό.
1.3 two
The ancient word was δύο (or δύω according to dialect). In the earliest Greek it was declined with dual endings, but even from Homeric times there was a tendency towards making the word indeclinable. In Classical Attic and the Koine it was indeclinable δύο and this remains the ΤΑΚΕ word.
1.4 three & four
The ancient words for these two numerals were regularly declined 3rd declension adjectives, namely:
τρεῖς [m. & f.], τρία [n.] "three"; τέσσαρες [m. & f.], τέσσαρα [n.] "four."

There are no rules given on the 'Nouns, Adjectives & Definite Article' page for forming words from 3rd declension plurals; but where we don't use the genitive singular for forming the ΤΑΚΕ adjective, we use the neuter nominative. Thus we do the same here, namely: τρία (3) and τέσσαρα (4).

1.5 five ... ten
These were invariable (i.e. they did not decline) in the ancient language; they, therefore, remain the same in ΤΑΚΕ, namely: πέντε, ἕξ, ἑπτά, ὀκτώ, ἐννέα, δέκα.
1.6 eleven ... nineteen
The most common ancient word for 11 was ἕνδεκα, but occasionally we find δέκα εἷς. For 12 we find δώδεκα in Homer and Attic, but δυώδεκα in most other dialects and more rarely δυόδεκα. However, in the Delphian and Heraklean dialects and in late Attic we also find δέκα δύο.

For thirteen we find τρεῖς καὶ δέκα (three and ten), where τρεῖς changes to agree with the case and gender of the noun, τρεισκαίδεκα as a single, indeclinable word, as well as δέκα τρεῖς. Similar variations occur for 14...19. There are, therefore, clear ancient precedents for ΤΑΚΕ to adopt δέκα followed by the numbers one to nine. Thus 11 to 19 in ΤΑΚΕ are: δέκα ἑνό, δέκα δύο, δέκα τρία ... δέκα ἐννέα.

1.7 twenty ... nine hundred
In the ancient language 20 was εἴκοσι, 30 and 40 were formed with -άκοντα (τριάκοντα, τεσσαράκοντα) and 50 to 90 were formed with -ήκοντα. All these numerals were invariable. While πεντήκοντα and ἑξήκοντα are obvious enough, we then find that 70, 80 and 90 were ἑβδομήκοντα, ὀγδοήκοντα and ἐνενήκοντα where the first morpheme is somewhat different from the corresponding cardinal numbers.

When we turn to the hundreds, we find that 100 was the invariable word ἑκατόν but that 200 ... 900 were formed with -ακόσιοι [m.], -ακόσιαι [f.], -ακόσια [n] which were declined like 1st & 2nd declension adjectives. Furthermore, although 300, 500, 600, 700 and 800 prefix the cardinal number, omitting the final vowel (except, of course, for ἕξ), 200, 400 and 900 use different first morphemes, giving us διακόσιοι, τετρακόσιοι and ἐνακόσιοι.

In the old version of ΤΑΚΕ, I had dispensed with εἴκοσι and all the numerals ending in -κοντα, forming compounds with -δέκα instead; this gave δυόδεκα, τριάδεκα, τετράδεκα and so on. There is no ancient precedent for this; on the contrary, as we have seen, δυόδεκα was a dialect form for 12. I have no doubt that an ancient would assume τριάδεκα and τετράδεκα were aberrant dialect forms for 13 and 14.

As for the hundreds, I had abstracted a word κόσιο = 100 and then formed compounds such as δυοκόσιο (200), τριακόσιο (300) etc. Although an ancient would probably have understood the compounds, I am quite sure that κόσιο would not have been understood. The abstraction of κόσιο reminds me of Glosa's nima "name" abstracted from the Doric Greek 'onyma' (ὄνυμα, gen. ὀνύματος; Attic ὄνομα, gen. ὀνόματος; ΤΑΚΕ ὀνόματο), i.e. patr-onym, and pseud-onym are falsely reanalyzed as patro-nym, and pseudo-nym. Indeed, before the present revision ΤΑΚΕ was tending towards a "Greek Esperanto" or Glosa written in Greek letters, rather than 'Graeco sine flexione.' So how shall I revise these? Let us first look at the next section.

1.8 thousands & more?

The ancient word for 1000 was χίλιοι[m.], χίλιαι [f.], χίλια [n]; this will give χίλια in ΤΑΚΕ. For multiples of 1000, the ancients said δὶς χίλιοι (twice 1000), τρὶς χίλιοι (thrice/ three times 1000), τετράκις χίλιοι (4 x 1000) etc. These are usually printed as single words, i.e. δισχίλιοι, τρισχίλιοι, τετρακισχίλιοι etc.; but occasionally we do find the words split as, e.g. τετράκις γὰρ χίλιοι [Thoukydides.6.31].

The ancients had no word for a million; but there was an adjective for 10 000, namely μύριοι [m.], μύριαι [f.], μύρια [n.]. Multiples of 10 000 were expressed in the same way as multiples of 1000, i.e. δισμύριοι (20 000), τρισμύριοι (30 000), τετρακισμύριοι (40 000) etc. This adjective was, according to ancient grammarians, a specialized use of the plural of μυρίος, μυρία, μυρίον "numberless, countless, infinite", with change of accent; cf. μύριος ὁ ὡρισμένος ἀριθμός, μυρίος ἀόριστος (mýrios the definite number, myríos indefinite) [Herodianos Grammatikos, 1.125].

I propose to extend this use of δίς, τρίς, τετράκις etc. to the hundreds and tens. The compound δισδέκα, which may, if preferred, be written δίς δέκα, would be perfectly well-formed in the ancient language and unambiguously means 'twice ten', i.e. twenty (whereas, as we have seen, pre-revision ΤΑΚΕ δυόδεκα actually occurred in some ancient dialects with the meaning '12'). I discuss these numeral adverbs further below.

In earlier ΤΑΚΕ I imagined a different timeline where counting of higher numbers was done in groups of four zeros, not three as in our own time line, i.e. there would special names for 104, 108, 1012 and so on. But ΤΑΚΕ is now firmly set in our world where, internationally at least, we have special names for 103, 106, 109 and so on. Modern Greek has words for these numerals; while the modern word for 106 would be an acceptable compound in the ancient language, it is not a well-formed compound in ΤΑΚΕ. Furthermore, the modern words for 109, 1012 are not acceptable with these meanings in either the ancient language or in ΤΑΚΕ. It will, however, be more convenient to discuss this later. For the moment, I shall merely observe that having a special word for 104 is unnecessary as δεκάκις χίλιοι means the same

Therefore, ΤΑΚΕ will not have a separate word for 10 000, but merely have the adjective μυρίο "numberless, countless, 'millions'" (cf. English: "myriad ... n. any immense number. - adj. numberless").

1.9 Cardinals, ordinals & numeral adverbs
It will be seen from the above that in the revised ΤΑΚΕ I shall be using numeral adverbs. Also I now dislike the use of θετό to form ordinals. Certainly the ancient adjective θετός was connected with the verb τιθέναι "to place", but it usually had the meaning "placed as one's son/daughter" i.e. "adopted". There simply is no ancient precedent for using it to denote ordinal position.

This peculiar pre-revision ΤΑΚΕ use of θετό was a move towards pidginizing the language. But what ΤΑΚΕ is really about is dropping all inflexions, not pidginization. 'Latino sine flexione' does not use any word, particle or suffix to denote ordinal number; it is quite happy for 4 and 5 to be 'quattuor' and 'quinque', while 4th and 5th are 'quarto' and 'quinto'. Likewise, ΤΑΚΕ ordinals 1st to 10th will be formed from ancient Greek ones according to the normal rules, as will the ordinals for 100th and 1000th. I shall also use the ancient forms for the corresponding numeral adverbs. These 36 words will constitute the 'basic numerals' and all others will be formed by compounds of them, as explained below.


2. The Basic Numerals

three times
four times
five times
six times
seven times
eight times
nine times
ten times
hundred times
thousand times
2.1 Forms used as a prefix ("compositional base")
four ..1000: as the adverb without the final -κις. i.e. τετρα-, πεντα- ... χιλια-.
two & three: as the adverb without the final -ς. i.e. δι-, τρι-.
one: we use instead μον(ο)- ← μόνο (ancient: μόνος) "solitary, alone".
For example:
μονόποδο = one-footed, monopod (cf. ancient Greek: μονόπους, [gen.] μονόποδος)
δίποδο = two-footed, biped (cf. ancient Greek: δίπους, [gen.] δίποδος)
τρίποδο = three-footed, tripod (cf. ancient Greek: τρίπους, [gen.] τρίποδος)
τετράποδο = four-footed, quadruped (cf. ancient Greek: τετράπους, [gen.] τετράποδος)
ὀκτάποδο = eight-footed, octopus (cf. ancient Greek: ὀκτάπους, [gen.] ὀκτάποδος)
(ποδό = foot ← ancient Greek: πούς, [gen.] ποδός
- ancient Greek also had an adjective ἐννεάπους, [gen.] ἐννεάποδος = nine feet long)
  1. 0 (zero) = μηδενό. On those occasions when 'zero(e)th' is required modern Greek has coined the word μηδενικός; ΤΑΚΕ likewise has μηδενικό and for 'zero times' we may use μηδενάκις.
  2. Ancient ὄγδοος (eighth) is for earlier ὄγδοϝος (actually found in the Aitolian dialect). It appears that in this word the oo was never contracted, not even in the Attic dialect. ΤΑΚΕ, therefore, does not contract the vowels either.
  3. Text books and grammars give ἐνάκις for 'nine times'; the word, however, is not over-common and is more generally written as ἐννάκις in codices. Also the forms ἐννεάκις ἐννεακισχίλιοι (9000) are attested as well. The numeric prefix is always ἐννεα- in the ancient language and the collective, which ΤΑΚΕ forms from the adverbial numerals for 'four' and above (see below) was always ἐννεάς (gen. ἐννεάδος) in the ancient language. Therefore, I have adopted the form ἐννεάκις since (a) it does have ancient precedent, and (b) we avoid unnecessary exceptions since the ΤΑΚΕ numeric prefix and collective noun are formed regularly.
  4. ΤΑΚΕ also has μυριάκις "innumerable time, countless times, 'millions of times'" and thus μυρια- used as a prefix (but the ordinal μυριοστό will be about as rare as "myriadth" in English).
  5. At least two different systems of using alphabetic symbols to represent numbers were found among the ancients: the earlier Attic numerals and, from the 4th century BCE, by a system known Ionian, Milesian or Alexandrian numerals. As an example 2011 is written ΜΜΔΙ in the former system and ͵βιαʹ in the latter. Both had disadvantages, especially the lack of a symbol for 0 (zero). ΤΑΚΕ will, therefore, not attempt to resurrect either of these systems; it will use the 'Western Arabic' numerals as in the table above.
  6. The ancients had no ordinal indicator; but in ΤΑΚΕ we will use the old numeral marker ʹ (see v above) as an ordinal marker, e.g. 2ʹ (δεύτερο) = 2nd, 100ʹ (ἑκατοστό) = 100th. This symbol is technically known as "dexia keraia" (δεξία κεραία) has its own Unicode character (U+0374), but for convenience in typing the acute accent may be used, e.g. 2´ 100´

3. Compound numerals

As we saw in eleven ... nineteen above, "There are, therefore, clear ancient precedents for ΤΑΚΕ to adopt δέκα followed by the numbers one to nine. Thus 11 to 19 in ΤΑΚΕ are: δέκα ἑνό, δέκα δύο, δέκα τρία ... δέκα ἐννέα." In ΤΑΚΕ the larger number always precedes the smaller one.

Also we have seen that compounds of tens, hundreds and thousands are expressed in ΤΑΚΕ by prefixing the numeral adverbs to δέκα, ἑκατόν and χίλια respectively. I had considered allowing the adverb to be written separately or compounded according to preference; but that would make expressions such as δίς δέκα πέντε ambiguous. Does it mean 2 × 10 + 5 = 25 or 2 × (10 + 5) = 30? However δίσδεκα πέντε is clearly 2 × 10 + 5; and if we insist upon compounding, then δίς δέκα πέντε becomes unambiguously 2 × (10 + 5). Therefore in ΤΑΚΕ multiples of 10s, 100s and 1000s will be compound words.

In the ancient language compounds were given a 'recessive' accent, i.e. they were accented on the third syllable from last if the last syllable contained a short vowel. Thus compounds of ἑκατόν will be accented -έκατον, e.g. δισέκατον (= δίς ἑκατόν 'twice a hundred') = 200. The final -α of δέκα and χίλια was also short, thus, e.g. τετρακίσδεκα = 40 and τετρακισχίλια = 4000. Thus we will find in ΤΑΚΕ:

δίσδεκα ἑνό = 21τετρακίσδεκα τέσσαρα= 44
ἑπτακίσδεκα πέντε = 75ἐννεακίσδεκα ἕξ = 96
ἑκατόν ἑνό = 101τρισέκατον πέντε = 305
πεντακισέκατον τρίσδεκα ὀκτώ = 538ἐννεακισέκατον ὀκτακίσδεκα ἑπτά = 987
χίλια τρία = 1003δισχίλια δέκα =2010
πεντακισχίλια ἑπτακίσδεκα δύο = 5072ἐννεακισχίλια ἐννεακισέκατον ἐννεακίσδεκα ἐννέα = 9999
3.1 Compound Ordinal Numbers
In both ancient and modern Greek compound ordinals use ordinal forms for each part; for example, 14th is τέταρτος καὶ δέκατος in ancient Greek, and δέκατος τέταρτος in modern Greek. Therefore, in ΤΑΚΕ 14th is δέκατο τέταρτο. Other examples are:
ἑπτακισδέκατο πέμπτο (75ο) = 75th
πεντακισχιλιοστό ἑπτακισδέκατο δεύτερο (507ο) = 5072th.
  • The suffix -οστός, -οστή, -οστόν took the acute on the last vowel even in compound, e.g. 3000th was τρισχιλιοστός. So in ΤΑΚΕ 3000th is τρισχιλιοστό (3000ο).
3.2 Compound Numeral Adverbs
Similarly, compound numeral adverbs also use the adverbial form for each element, e.g.
ἐννεακισδεκάκις ἑξάκις = 96 times
δισχιλιάκις δεκάκις τρίς = 2013 times.
  • In both ancient Greek and ΤΑΚΕ the suffix -άκις at the end of a single word or compound word bore the accent on the next to last syllable.


4. Fractions, N-tuples & Collectives

4.1 Fractions
As in the ancient language, so in ΤΑΚΕ:
  • ἡμι- is a prefix meaning "half"; e.g. κύκλο = "circle", ἡμίκυκλο "semicircle"; ἄνθρωπο "human [being]", ἡμιάνθρωπο "half-human".
  • ἥμισυ (← ἥμισυς [m.], ἡμίσεια [f.], ἥμισυ n.]) "half" may, just as in the ancient language, either be used an adjective, e.g.ἥμισυ ὥρα "half an hour" or, more commonly, used with partitive expressions, e.g. ἥμισυ ἐκ το ἡμέρα "half [of ]the day(s)" When used with other numerals the word order is τρία ὥρα και ἥμισυ "three and a half hours".
All other fractions
  • the denominator is the ordinal with μέρο (part) 'understood';
  • for numerator is the normal cardinal number.
For example:
ἑνό τέταρτο " one quarter, a quarter"
τρία τέταρτο "three quarters"
ἑπτά δέκατο "seven tenth"
δίσδεκα τρία τρισδέκατο δεύτερο (23/32)
τρία τέταρτο ἐξ ὥρα "three quarters of an hour"
δύο ὥρα και τρία τέταρτο "two and three quarter hours"
4.2 N-tuples (single, double, etc.)

These adjectives are formed by adding the bound morpheme -πλάσιο (-fold) to the forms used as a prefix ("compositional base"), e.g. διπλάσιο = double; τριπλάσιο = triple, treble, threefold; δεκαπλάσιο = tenfold; ἑκατονταπλάσιο = a hundredfold (cf. ancient Greek: διπλάσιος, τριπλάσιος, δεκαπλάσιος, ἑκατονταπλάσιος).

Also in ΤΑΚΕ: δισδεκαπλάσιο = twentyfold, ἑπτακισχιλιαπλάσιο = seven-thousandfold etc. But not *μονοπλάσιο which makes no more sense than 'onefold' in English. "Single" in ΤΑΚΕ is just plain μόνο.ΤΑΚΕ does, however, have μυριαπλάσιο "countless times as many as, myriadfold".

4.3 Collectives
These are nouns denoting a group of so many units and end in the formative suffix -άδο (from ancient -άς, [gen.] -άδος). We suffix this to:
  • μον(ο)- for 'one', thus: μονάδο = unit, monad, ace.
  • the cardinal number without final vowel for 'two' and 'three', thus: δυάδο = group of two, a couple, pair, dyad; τριάδο = a threesome, trio.
  • the adverbial numeral without -άκις for the rest, e.g. τετράδο = a foursome, quartet; ὀκτάδο = a group of eight, an octad, an octet(te) ; ἑκατοντάδο = a group of 100; χιλιάδο = a group of 1000, a chiliad; μυριάδο "a countless number, 'millions [of]', myriad."

5. Millions, Billions, Trillions etc.

The highest number with its own name in Classical Latin was mille 1000, and in Classical Greek was μύριοι 10 000. Clearly this state affairs would not continue as mathematics developed. In early Italian the word millione (modern Italian: milione), was coined to denote 1 000 000; it was derived from mille 1000 with the augmentative suffix -one, i.e a million was a 'big thousand'.

Names for larger numbers are coined by prefixing to -illion (suggested by the word 'million') Latin numeric prefixes, e.g. billion and trillion with the Latin prefixes bi- (2) and tri- (3), respectively. This system was evolved from systems used by the 15th century French mathematicians Jehan Adam and Nicolas Chuquet. Unfortunately, it has become implemented in two different ways:

  • Long scale (French: échelle longue) in which every new term greater than a million is 1 000 000 times the previous term, i.e. 106n where n is the value of the Latin prefix; e.g. 1 billion = million millions (1012), 1 trillion = million billions (1018), and so on.
  • Short scale (French: échelle courte) in which every new term greater than a million is 1 000 times the previous term. i.e. 103n+3 where n is the value of the Latin prefix; e.g. 1 billion = thousand millions (109), 1 trillion = thousand billions (1012), and so on.

The short scale has been traditional in the USA while the long scale has been traditional in much of Europe and Latin America (but not Brazil which uses the short scale). When I was at school in the 1950s Britain universally used the long scale; however, in the mid 1970s the UK Government switched to the short scale as did the mass media. Now for all practical purposes the UK can be considered as in the short scale camp. Also there are some other European countries that use the short scale, including Greece.

5.1 Why ΤΑΚΕ cannot use the modern Greek terms
(Note: modern Greek words are given in the pre-1980 polytonic spelling)
The modern word for a 'million' is ἑκατομμύριο(ν) ← ἑκατόν+μύριον, i.e. 100 × 10 000. This would have been an acceptable compound in the ancient language since, besides using ἑκατοντα- as a prefix just as ΤΑΚΕ does, it also allowed ἑκατογ-/ἑκατομ-/ἑκατον- as prefixes, the final nasal consonant being determined by what followed (but arguably ἑκατοντακισμύριον would have been better formed). Although ΤΑΚΕ uses only ἑκατοντα- and does not use μύριο = 10 000 (but only μυρίο "countless"), we could perhaps have accepted ἑκατομμύριο. However, it is what happens with 'billion' and 'trillion' etc. that is the real problem.

In modern Greek we find δισεκατομμύριο(ν) = (109), τρισεκατομμύριο(ν) (1012), and so on. This just will not do in ΤΑΚΕ where δίς means "twice" and τρίς means "three times". There is no doubt that to an ancient Greek δισεκατομμύριον would mean 2 × 100 × 10 000, i.e. two million, and τρισεκατομμύριον would mean 3 × 100 × 10 000, i.e. three million. Exactly the same would apply in ΤΑΚΕ (assuming we had adopted ἑκατομμύριο). Clearly ΤΑΚΕ has to adopt a different method.

5.2 Some possibilities
One seemingly simple solution would be simply to adopt Hellenized forms of 'million', 'billion' etc., e.g. μιλλιῶνο, βιλλιῶνο (or διλλιῶνο), τριλλιῶνο. But that would merely perpetuate the confusion of having short scale and long scale conventions (which has caused problems in would-be IALs).

Indeed, the ending -illion (-ιλλιῶνο) implies a system based on a million, i.e. billion = 1000 0002, which is the way it works in the long scale. However, the short scale, which is, despite the names given to numbers, based on powers of 1000, has become more internationally widespread because it has proved convenient to have names for powers of a thousand rather than just of a million (cf. the SI prefixes). Indeed, some users of the long scale have words ending in -iard for the "missing" thousand names, as we see in the table below.

Russ Rowlett of the University of North Carolina has observed that the present systems cause widespread confusion and has suggested we dump the Latin-based numbers of the short and long scales and adopt a Greek-based system, thus:

n103n Short scaleLong scale SI prefix Russ Rowlett's proposed
Greek-based name
3109 billionthousand million/
giga- gillion
41012 trillionbilliontera- tetrillion
51015 quadrillionthousand billion/
peta- pentillion
61018 quintilliontrillionexa- hexillion
71021 sextillionthousand trillion/
zetta- heptillion
8 1024 septillionquadrillionyotta- oktillion

The trouble, in my opinion, is that the proposed system is only half Greek. The Latin-derived termination -illion is still there; and 'tetr-illion' is surely going to suggest 'quadr-illion', 'pent-illion' to suggest 'quint-illion' and so forth, which could cause further confusion. The only thing that users of both the long and short scales will understand unambiguously are the SI prefixes themselves. It would, I think, have been better if 10009 had been named 'gigillion', and the rest 'terillion', 'petillion' and so on.

Modern Greek uses similar terms for the SI prefixes and so shall I (except that ΤΑΚΕ will spell 'yotta-' as ἰοττα- and not γιοττα- as in modern Greek); and therein, it seems to me, lies the obvious solution for ΤΑΚΕ.

5.3 "Large numbers" in ΤΑΚΕ
We just use the SI prefixes as words in their own right, thus:
n103n Short scaleLong scaleΤΑΚΕ
2106millionmillion μέγαμεγοστόμεγάκις
3109billionthousand million/
41012trillionbillion τέρατεροστότεράκις
51015quadrillionthousand billion/
61018quintilliontrillion ἔξαἐξοστόἐξάκις
71021sextillionthousand trillion/
81024septillionquadrillion ἰότταἰοττοστόἰοττάκις

It may be objected that these numerals are 'artificial', but ΤΑΚΕ itself is artificial. There just are no ancient words for these numerals and the modern Greek forms are, for reasons explained above, not acceptable in ΤΑΚΕ. So why not just adapt the SI prefixes?

The other objection that may be made is that no number above 1024 has a name. It may seem inelegant and "incomplete", but how often do we really have to talk about very large numbers? Anything more than a trillion (short scale)/ billion (long scale) or so is rarely encountered. Most people do not care whether the universe contains a septillion, a vigintillion, or a bazillion atoms! In practice numbers of these magnitudes are encountered only in mathematics and allied sciences where scientific notation will be used.