I have some questions regarding gematria and zero:

  • which characters of alphabet means zero according to gematric rules?
  • since which time did Jews consider zero as a character equivalent to other alphabet characters=numbers?
  • if zero appears in gematria now, but didn't at gematria's beginning, how were adapted gematria perceptions?

Could somebody point me to further reading? Because of lack of knowledge, I ask Google the wrong questions, so I can't find something substantial to read historical studies about the subject.

  • 15
    Why would there be a 0 in Gematria? Gematria assigns a number to each letter, not a letter to each number.
    – rosends
    Aug 14 '14 at 16:44
  • 4
    would this link help ?math.stackexchange.com/questions/35894/…
    – eramm
    Aug 14 '14 at 16:49
  • 7
    There is no need for zero as a placeholder in gematria as it is a sign-value system.
    – Double AA
    Aug 14 '14 at 16:50
  • 1
    What about negatives, fractions, irrationals, or imaginaries? I’d like to see a word whose Gematria evaluates to (7/5)+πi.
    – DonielF
    Feb 24 '19 at 15:34

which characters of alphabet means zero according to gematric rules?

As pointed out in the comments, each letter matches a number. No letter has a zero value.

since which time did Jews consider zero as a character equivalent to other alphabet characters=numbers?

I don't know.

if zero appears in gematria now, but didn't at gematria's beginning, how were adapted gematria perceptions?

Irrelevant, as per above.

Could somebody point me to further reading -

A good source would be Gematria on Wikipedia.

  • 1
    From my link above "Gematria is not a base system, because the system is not positional, it is aggregate: the value of a letter/symbol does not depend on its position, the way it does in decimal, binary, etc., but only on the letter. So two words that are made up of exactly the same letters, though perhaps in different order, would correspond to the same value. This is not true of positional/base systems." So in this context a symbol that matched zero would have no value since it would be meaningless. Am I wrong ?
    – eramm
    Aug 18 '14 at 10:38
  • @eramm It would be useless as a placeholder, but it can still represent the number 0 (eg. as an additive identity).
    – Double AA
    Aug 18 '14 at 14:23
  • The Egyptians to Yosef after running out of money: "וְלָמָּה נָמוּת נֶגְדֶּךָ כִּי אָפֵס כָּסֶף"
    – Nic
    Aug 18 '14 at 18:56
  • @Nic - that means "...for the money is finished". Aug 19 '14 at 8:03
  • 1
    How about this one: "הוֹי מַגִּיעֵי בַיִת בְּבַיִת שָׂדֶה בְשָׂדֶה יַקְרִיבוּ עַד אֶפֶס מָקוֹם וְהוּשַׁבְתֶּם לְבַדְּכֶם בְּקֶרֶב הָאָרֶץ."
    – Nic
    Aug 19 '14 at 17:29

I too am very curious about the first use of zero as a concept in Judaism. Maybe Harel of Torah Thoughts would know.

Which characters of alphabet means zero according to gematric rules?

If we accept that Gematria is more than just comparing two words with the same numerical meaning, but that each number represents a Divine theme like we see most emphatically with seven and ten, then zero or nothing would be best represented by the characters אין rather than some glyph. Efes then as well as now carries a negative connotation.

since which time did Jews consider zero as a character equivalent to other alphabet characters=numbers?

Probably Egypt, but not in any script we’re used to.

if zero appears in gematria now, but didn’t at gematria’s beginning, how were adapted gematria perceptions?

The word kind of evolved with the times, eventually taking on more esoteric and mystical meanings.

Could somebody point me to further reading?

While in modern Heb, zero is called efes, the Torah couches efes as Ain or Lo.1 Nevertheless, we see examples where efes precedes 1:

According to the Zohar II 162, where the subject of ‎ה' ‏אחד ושמו אחד‎, is discussed, [the problem being how there ‎can be more than one unique G’d, Ed.] the point is made ‎that the letter ‎א‎ in the word ‎אחד‎ is to be understood as the point ‎where absolute ‎אפס‎, absence of anything tangible, crosses over to ‎a world that progressively contains more tangible components.2 ‎This is understood as the reason why the Talmud absolutely ‎forbids us to draw out the letter ‎א‎ when we recite the word ‎אחד‎ ‎when reciting the k’riyat sh’ma. Seeing that this letter ‎represents a concept that is beyond our understanding, dwelling ‎on it by drawing it out would be close to blasphemous. ‎‎[My choice of word. Ed.] The position of that ‎letter ‎א‎ in the kabbalistic scheme of things is described as ‎מטי ולא ‏מטי‎, perhaps best translated as “in a state of inanimate ‎suspension, never at rest and never actually moving.”‎

The author compares this state of ‎מטי ולא מטי‎ to what is ‎known in Talmudic parlance as ‎שקלא וטריא‎, the discussion of, i.e. ‎weighing of pros and cons of different facets of a halachic ‎problem, before arriving at a conclusion.3

Rav Hirsch also appears to define efes as something other than zero mamash: being used up, a distant end, or valueless.4 I believe that this is, in part, due to everything in life being in a constant state of Flux. As Heraclitus quipped, “the only constant is change.” There is no absolute zero. When the ledger hits zero, the book is discarded. When we try to place a value on zero, we are insinuated that there is another beginning. Perhaps the aleph was not allowed to begin the Torah because it is used with efes. This seems to be the thrust from the Arizal. On sefer Eitz Chaim, Rabbi Shabtai Teicher comments on the notion of “above Chochma: the huili, Keter”:

Chochma is the first of the four. It follows, therefore, that chochma is called reishit. The Hebrew word for “the first” is reishit.

Indeed, reishit is the first word of the Torah that starts “In the beginning…” or “At first”. In Hebrew the phrase is “be-raysheet”. The prefix “be” simply means “in” or “with”. Also, the “official” Aramaic translation of the Bible, the Targum Yonatan, written by one of the great rabbis of the Talmudic period, R’ Yonatan ben Uziel, translates the Hebrew word “be-raysheet” as “be-chochma”, – with chochma, or in chochma.

You can understand why keter is a supernal aspect and yet not really part of the world to which it is the crown, similar to the crown of a king that is above his head and not part of his head. Accordingly, keter is not included as a sefira of the world, and in its place we count daat-knowledge, as stated in the Sefer Yetzira. Nevertheless, there are times when we do count keter as one of the ten sefirot. The entire matter may be understood from what has been explained beforehand in the name of the natural philosophers. There is an intermediary between each of the aspects.

The same idea was introduced by Ramban (R’ Moshe ben Nachman) in his commentary to the beginning of the verse, “And the earth was empty and void (tohu ve-bohu)…”5 He wrote, in the name of Sefer Habahir, that before the four elements were created there was created a primal matter called [in Greek] the “huili”. It is something that is prepared to take on the shapes of the four elements afterwards, but at first it is without any shape or distinction whatsoever. Since it is before the tohu (emptiness) it is called efes (zero), or nothingness, as it is written, “…They are accounted to him less than nothing and emptiness.”6

We see from the verse that “nothing”, the efes, comes before “emptiness”, the tohu.

Furthermore, the Hebrew word for “grasping” (nitfas) is always used for intellectual comprehension and perception. According to the Ari, the word efes – zero – nothing is derived from the same root because nothing can grasp it, as it says in the Zohar, “No thought can take hold whatsoever of You.”

The idea is as follows. The Infinite is also called Nothing (Efes-Zero) because nothing can grasp it. It has no substance and no shape whatsoever. After it there came into existence the tohu (emptiness), which is keter. After that there came into existence the bohu (void), which included the four elements, chochma and bina, tiferet and malchut.

However, the Bohu is not yet raysheet, Chochma. As we will soon see, it is the second aspect of Keter. It is not the four elements in actuality, but only in potential.

It has also been taught about the Bohu that the word is a combination of two other words: Bo, which means “in it,” and Hu, which means, “it is.” Thus, the word itself means “It is in it,” which is an indication that it is the first to assume the shape of the four elements, at least potentially, and within it, so to speak, is the Infinite Himself.7

The dilemma that follows (our intellectual Tohu v’Bohu) defines our bechira (sourced in Daat): do we side with Hashem or our own alien opinion? Without Hashem, efes takes on a negative meaning: destruction.8 In modern Hebrew, efes is also a derogatory term (i.e., dapar efes – lowest score of the low) meaning loser or idiot. Rabbi Dov Linzer notes:

The story of the spies returning with their evil report is well known, but the reason they were punished is not commonly understood. What did they do wrong? They reported what they saw accurately. Ramban suggests an answer. The key, he says, is in their use of the word efes, “however”9: “However, the people be strong that dwell in the land, and the cities are walled, and very great.”10 Ramban says that efes means “nothing” here (it later came to mean “zero”)11:

“Their wickedness was in their use of the word efes, which indicates that the matter is completely impossible.”12 To say that it was impossible demoralized the people and demonstrated, perhaps even propagated, a lack of faith in G-d. I would like to suggest that the key is a different word, one that they failed to use. That word is tova, “good,”13 a word introduced by Moshe.14

Years later, as Yael Eckstein observes, after Joshua had successfully led the people into Canaan, he composed a new prayer. This prayer, which is still recited every day, contains the phrase efes zulato, which means, “there is nothing but Him (G-d).”

It is not by accident that Joshua, who was one of only two spies to disagree with the others, used that same word which got his colleagues into so much trouble. His prayer teaches us that there is one time when we should use the word “but” — when we say, “There is nothing but G-d.”

This segues into the complete meaning of efes: endless15.16

If the Ein Sof B”H17 is One, then Ein is zero. Hashem is called the Endless One and not the Beginningless One (Ein Techila).18 If He were called the Beginningless One, it would be impossible to speak about Him at all. But we are able to conceive of Him to some small extent through His creation. This is a beginning, but it has no end19–“for indeed it does have a beginning, namely, the Luminary Who can freely choose to illuminate or not to illuminate. As Rambam states,20 ‘This is what the Prophet says: והוי-ה אלקים אמת – “And the L-rd G‑d is true,” meaning that He alone is the truth; nothing else is true as He is.’ For absolute truth is that which exists independently of everyone and everything else – a condition which can be ascribed solely to G‑d Himself, Whose existence derives from His very own essence and being. In contrast, all that which issues from Him exists only by His volition and sufferance, and not independently.”21 So too does our number system begin at zero, but it has no end. In this context, it is impossible to ascribe any negative numbers because there are no negative traits associated with Hashem.22

Another way of looking at Ein is through science. Michael J. Alter writes: “An atom is made up of a nucleus and its orbiting electrons. Physics has determined that despite its diminutive size, the nucleus (comprised of neutrons and protons) contains the predominate amount–more than 99.9 percent of the mass of the atom. The electrons are by comparison just clouds of moving fluff. Thus, atoms are mainly empty space. Hence, science and the kabbalah concur that matter is composed chiefly of nothing.”23 In a manner, Hashem is quite literally found everywhere and is everything.

Zero in relation to a base numeral system is nothing more than a place holder, but in Judaism the zero forces one to look elsewhere as I recall Greg Killian remark. Nobody concentrates on the zero in 10, but rather the ten emanating from zero. This is why zero in the Torah is nonexistent so-to-speak; Hashem is a given.

Arturo answers:

Hebrew proper does not have a base numeral system, much less an independent numbering system, because the system is not positional, it is aggregate: the value of a latter/symbol does not depend on its position, the way it does in decimal, binary, etc., but only on the letter. So two words that are made up of exactly the same letters, though perhaps in different order, would correspond to the same value. This is not true of positional/base systems.

Numbers and letters co-exist. While letters bring an item into existence, the numbers (carried through the letters) give them meaning such as Eliezer being 318.24 Therefore, each of the letters are important and cannot roll over to the next sequence.

To be sure, as mentioned by mi yodeya:

the Hebrew numerals constitute a decimal system, however it’s different from the Arabic decimal system in that the symbols do not recycle through all the orders of 10^n. So a symbol’s position in a string does not impact the symbol’s value. Because of this, the system, does not scale elegantly. It’s not really used for doing math (generally) nor representing very large numbers (though they could be if a system of infinite extension were devised). Let vector A = [1,2,3,4,5,6,7,8,9], then, left to right, the numerals B = [ט,ח,ז,ו,ה,ד,ג,ב,א] are (10^0)A respectively; and are the only symbols ever in the 10^0 place . Numerals C = [צ,פ,ע,ס,נ,מ,ל,כ,י] correspond to (10^1)A, and D = [ץ,ף,ן,ם,ך,ת,ש,ר,ק] alternatively, D = [תתק,תת,תש,תר,תק,ת,ש,ר,ק]) to (10^2)A. These basic symbols have been extended somewhat with apostrophes and dots, but is still quite limited in the range of represented values. Zeroes do not need a symbol in this system, because it is not dependent on position.25 If you were using Hebrew numerals as a tool for calculation, cognizance of the decimal exponent would need to be taken, as well as how the elements of B, C, and D scale from one vector to another through the various operations.

So, that was the symbols, but base 10 clearly shows up in the naming convention, too. Names for ones, tens, hundreds, and thousands. They cycle just as regularly as you would expect. The naming does get a little strange, though, with names like shesh me’ot elef usheloshet alafim vachamesh me’ot vachamishim, literally six hundreds of thousands and three thousands and five hundreds and fifty, for 603,550. They tended to split the number up, ordered in terms of hundreds, with higher orders said before lower orders, just as in English; but, within each order-group, hundreds will be said first, then ones, and then tens.

However, this is not a true base numeral system because each of the numerical segments are a complete unit in of themselves, insofar as they each retain an independent meaning (just as this census breaks down to its constituent parts per the shvatim and to that every Jew is a complete unit). Higher numbers come first to set the stage:

In terms of the Sefirot, single digits denote the emotive attributes (the Divine middot), double digits denote the intellective attributes (the Divine mochin), hundreds designate the level of Divinity that transcends Divine intellect, while thousands and tens of thousands respectively denote the levels of Divinity known as Ratzon (“the Divine Will”) and Taanug (“Delight”). In terms of the levels of the soul within an individual Jew, the five classes of numbers correspond to the five soul-levels called (in ascending order) Nefesh, Ruach, Neshamah, Chayah and Yechidah.26

In Kabbalah, a homonymous word is deployed to describe nothingness: ayin. On the pasuk “and he [Jacob] took stones from that place and put them under his head,”27 the Maggid of Mezritch elaborates:

It is known that the “stones are [Heb] letters. When the tzaddik prays with these letters and binds (me-kasher) himself to the supernal wisdom (hokhmah elyonah), as is known, he has already entered the gate of eternity/nothingness (shaar ha-ayin). He will elevate his heart until it is as if G-d’s power is in it. At that moment he achieves complete nullity (efes mukhlat). As such, everything is divine power (koach) and his [the tzaddik’s] speech is from the speech of G-d that created the world. The world of speech (olam ha-dibbur) is drawn from the supernal wisdom (chokhmah) which is the pleasure (ta’anug) and playfulness (sha’ashuim) that G-d gets from the world. And now he [the tzaddik] speaks only for the playfulness (sha’ashua) of G-d. And through this the letters return to their original source, which is the wisdom (chokhmah) from which they were drawn.28

1-Ibn Ezra on Isaiah 34:12 (Ain); Ibn Ezra on id. 41:12 (Lo).

2-See Duties of the Heart, First Tr. On Unity 5.

3-Kedushat Levi, Exodus, Ki Tisa.

4-Etymological Dictionary of Biblical Hebrew by Matityahu Clark, p. 15.

5-Genesis 1:2.

6-Isaiah 40:17.

7-From Tohu to Bohu appears to mirror the Yiddish word eppes which means something or a little bit of anything. Curiously, the aleph of efes assumes an ayin in eppes which is more of a guttural sound. This is probably what led Isaac Mozeson to connect the reverse (pei samach) to the hapax legomenon פסה in Tehillim 72:16. See idem, Origin of Speeches; see also Klein Dictionary, פִּסָּה. Without this opening in the throat, we are left with a closed dental lisp of samech. See also Aryeh Kaplan, cmt, Sefer Yetzirah, p.102 (explaining interchangeable letters); see also Rashi, Vayikra 19:16. Samech is a closed loop or a chain like a Syrian semkath ܣ. See Etymological Dictionary of Biblical Hebrew, p. 203, s.v., פסס. Psst! Is not Edenics fun?

8-Rashi, Is 34:12.

9-See Ibn Ezra on Numbers 22:35:2 (defining efes as howbeit); see also Ber Rab 40:4 (attributing efes to the Gk word hafes). I believe that this is referring to the word ἄφεσις or aphesis. Ironically, aphesis or Apheresis linguistically speaking is when the initial vowel is let go to produce a new word. In Greek (and Russian) there’s no initial ‘h’ even though the apostrophe (in classical Gk) denotes such. Therefore, hafes becomes efes.

The weak (rapha) hei in Hebrew is also quiescent. There is an example of this in the word ‘hevel’ found in Ecc 1:2. “Vanity is something empty of substance, it is zero. However, when combined with something of substance, such as Torah study, it adds value to it, in the same manner that a zero that is placed next to a number adds value to it.” Mashal Umlitzah.

This seems to align with academic notions connecting the word ‘zero’ to Zera and Zeroa (a sowing seed), written as “–” similar to a hyphen.

The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. The system was adapted from that of the Greek numerals in the late 2nd c BCE.

The current numeral system is also known as the Hebrew alphabetic numerals to contrast with earlier systems of writing numerals used in classical antiquity. These systems were inherited from usage in the Aramaic and Phoenician scripts, attested from c. 800 BCE in the so-called Samaria ostraca and sometimes known as Hebrew-Aramaic numerals, ultimately derived from the Egyptian Hieratic numerals.

The Greek system was adopted in Hellenistic Judaism and had been in use in Greece since about the 5th c BCE.

10-Nm 13:28.

11-“He writes subsequently that their sin consisted of the word אפס, that though they had spoken truthfully, and had reported on what they had been asked to report, that word means ‘zero,’ in the sense that the chances of accomplishing one’s objective were ‘zero,’ non existent.” Shenei Luchot HaBerit, Torah Shebikhtav, Sh’lach, Torah Ohr.

12-Ramban, v 27.

13-Radak, Gen 1:31:

Even though, as a result of these six “days” of G’d’s creative endeavour, something had been lost forever, i.e. the אפס, the absolute nothingness which had preceded creation of the universe, silence this very “nothingness” had now become inextricably interwoven with the “something,” all the phenomena which G’d had called into existence, G’d described the sum total as good.

Maimonides, in commenting on the words והנה טוב מאד,, (Moreh 3,10) writes: “even death, which appears to constitute a return to אפס to nothingness, G’d considered as something positive, constructive, seeing it is only a prelude to rebirth, albeit sometimes in a different guise than that the previous incarnation. Death is perceived as the result of the ‘nothingness’ which had preceded the universe having become an integral part of this universe. Hence it had become a necessary phenomenon. [these words are mine, I am not sure that I could literally translate the words of Maimonides, themselves a translation, rendered differently by different super-commentaries, such as Crescas, Shem Tov, and Afudi, as well as Abravanel. Ed.]

14-“It’s Good Because I Say So” on parashat Shelach; see “One Wrong Word” by Rav Yaakov Beasley.

15-The word used here is keitz, which is understood to mean the ends of physicality as we see in the writings of Rebbe Nachman:

“‘All ÆFSei the earth have seen’—In other words, the physical matter of the thing becomes EFeS (worthless) <and emptiness, as explained above>.” Likutei Moharan 94:3 (cit. Teh 98:3).

16-Radak, Ps 2:8.

17-There is a basic difference between the way man looks at things and in the way G-d looks at things. Man begins with the phenomena that he sees and hears and proceeds by means of those phenomena to comprehend or assume the existence of other less familiar phenomena. Such unfamiliar phenomena may exist only in the abstract, never to be perceived visually or aurally. G-d, on the other hand, proceeds in the opposite fashion. Sof ma’asseh bamachashavah techillah, the final product had been envisaged at the very outset. This is the way He operates. Every manifestation of any phenomenon represents merely a step planned and necessary to lead to the ultimate total structure. Having planned the final product, He comprehends the significance of every step, every constituent part, even if it is apparently unrelated to the whole. He is aware that such a phenomenon is indispensable, an integral part of the whole to be revealed later. Akeidat Yitzchak 49:1:4.

18-If we think of this eternal return in terms of sefirot, then the beginning is keter. “Similarly, keter contains within it the ultimate goal which it seeks to attain. The aim of every plane of reality is to reveal the Infinite Light appropriate to that plane of reality. Thus, keter contains within itself the endpoint which it desires to achieve. And when it has achieved that desired endpoint, it ceases to drive further. This is the meaning of the statement of the Sages in Sefer Yetzira (1:7): “the end is wedged in the beginning, and the beginning is wedged in the end.” The purpose of the emanation of the sefirot which is expressed in keter is to unfold the level of Infinite Light appropriate for that plane of reality, all the way through the various stages of development (i.e., through the various sefirot) to the final revelation and manifestation of that light in the sefira of malchut. Thus, keter is “wedged” in the lowest sefira, malchut. And malchut is wedged in the highest sefira, keter, for it is the end-point which fulfills the purpose of the entire emanation.” Rabbi Moshe Miller on Keter.

19-Mezritcher Magid in Imrei Tzadikim, quot. In Torat HaMagid (Tel Aviv, 5729), vol. II, p. 162, trans. In Meditation and Kabbalah by Aryeh Kaplan, p. 303.

20-Hil. Yesodei HaTorah 1:4.

21-Reishit Goyim Amalek, ch. 1.

22-Cf., Tanya, Shaar HaYichud VeHaEmunah 9 (86b).

23-Why the Torah Begins with the Letter Beit, p. 5, by Michael Alter.

24-See Breshis 14:14.

25-This was also the case with Hebraic hieratic numerals as they did not need an attendant zero. See “The Use of Hieratic Numerals in Hebrew Ostraca and the Shekel Weights” by Yohanan Aharoni; “The Hieratic Scribal Tradition in Pre-exilic Judah” by David Calabro.

While I employ the notion of a sexagesimal system in place during the Talmudic Era, that was solely in relation to the times and dealings with the people of that Era which is why that system was used for tort claims, business, and the like.

What I will say here is that all things considered, it appears that the original weight, what was also used in Cana’an and throughout Mesopotamia, was sixty. It makes sense as 60 is a unitary perfect number. A sexagesimal base numeral system was quite ubiquitous, used by Sumarians and Akkadians alike.

“Six represents the material, physical, secular. Ancient Mesopotamia, the birthplace of Abraham, originally used a numerical system based on the number six. Western civilization still bears traces of this in the twenty-four-hour day (2 x 6 hours of light, plus 2 x 6 of darkness); the sixty (10 x 6) minutes in an hour, and seconds in a minute; and the 360 degrees in a circle (6 x 6 x 10). All of these originated in astronomy, at which the ancient Mesopotamians excelled. Judaism acknowledges the six-part structure of time and space but adds that G-d exists beyond time and space. Hence seven—the one beyond six—became the symbol of the holy.” Rabbi Jonathan Sacks, Koren Siddur, p. xxviii.

However, according to Otto Neugebauer, the origins of the sexagesimal system are not as simple, consistent, or singular in time as they are often portrayed. See Neugebauer, O. (1969), “The Exact Sciences in Antiquity”, Acta Historica Scientiarum Naturalium et Medicinalium, Dover, 9: 17–19. This is why we find fractions. It appears that with the dumbing down of civilization, society moved to a base 50 system briefly, as we find with the different types of shekels, see “Money, Prices and Market in the Ancient Near East” by Bert van der Spek (of VU Univ. Amsterdam), p. 30, Yale Univ. New Haven, Economics Dep’t, Economic History Sem., March 30th, 2015, before settling for base 10 (and now with binary basically ruling the world). This system, though, will return in the future (as is needed to weigh a few things in the third Temple). However, base 60 also appears to be eternally incomplete as there has never been a true sexagisimal system with 60 different symbols.

Krieger delves into this further (cit. Mathematics in the time of the pharaohs by Richard J. Gillings, p. 257), writing:

Hebrew uses a decimal system, with hieratic tokens.

The decimal nature corresponds to that there are symbols for 1, 10, 100, etc.

The hieratic nature corresponds to using separate symbols for 1, 2, 3, 4, 5, …, 9 in each place. (This was the style of the hieratic system in Egypt, and used in alphabetic systems throughout the ancient world. The late egyptian style was to use a symbol meaning e.g. (70). In any case, the argument presented here is that there was a special symbol for ‘70’ that did not need an attendant 0.)

The modern form might be seen in a series of stamps for 1c, 2c, … 9c, then 10c, 20c, 30c, .. 90c, and then £1, £2, …, £9, … etc., as formerly used on railway small freight. One makes up £1.63 with a £1 stamp, a 60c and a 3c stamp.

Of course, one notes that the choice of letter implies a notion of place, even if this is not so presented. Consider the abacus. This was the mainstay of calculation in ancient times, and ultimately the reference of numbers.

The western abacus consists of columns and rows, where the column value is the product of the row values. Carry depends on if there were a change of column.

The notation of the chinese and mayan is to attach a column weight and position, like saying 3c 2x 4i, meaning 3 in the hundreds, 2 in the tens, and 4 in the units. Other notations use different ways of showing this. The tokens of the romans and kindred folk [such as the etruscans], use the same scale we use of coins: each token means 1 or 5 in a specific column. So 2x is written as XX.

The alphabetic codes like the Hebrew, greek and gothic, use a large number of symbols for “n in column C”, the order of the alphabet provides a ready-made collection.


Since the notion of base and position rests in the abacus, and not the notation, it is fair to say that the base is decimal, and the selection from 7i vs 7x or 7c suggests that position was perfectly understood.

The modern notation derives from the greek alphabetic one, with 1x drafted to mean 0i. This is why, for example, 0 is after 9, rather than before 1. The use of the symbols for the I column, and a later substitution of say 3i 2i as 3x 2i is an unfolding of the results. Place was understood: it is what makes a base a base. Position is a function of notation, and does not destroy the place values.

26-Tzemach Tzedek on Iggeret Hakodesh, Epistle 21.

27-Gen. 28:1.

28-The Maggid of Mezritch, Torat ha-Maggid, vol. 1, 73a/b. Cf., 1, 76a.

  • 1
    Ha, exactly 30000 characters!
    – Kfir
    Jun 30 at 7:34
  • 4
    That's a problem as I'd recommend you summarize your main points at the top as this is too long for anyone to bother reading.
    – robev
    Jun 30 at 7:55

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