Lectures on the weekly Torah reading by the faculty of Bar-Ilan University in Ramat Gan, Israel. A project of the Faculty of Jewish Studies, Paul and Helene Shulman Basic Jewish Studies Center, and the Office of the Campus Rabbi. Published on the Internet under the sponsorship of Bar-Ilan University's International Center for Jewish Identity.
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Parashat Bemidbar 5759/1999
The Census of the Israelites in the Wilderness
Prof. Eli Merzbach
Department of Mathematics and Computer Science
Several censuses of the Israelites are mentioned in the Torah. It is interesting that almost all the numbers listed in these censuses appear to be round numbers, i.e., without units and even mostly without tens. Of course this can be ascribed to miracle or viewed as an inexplicable random occurrence (as some have tried to do). The great commentators have rejected interpretations of this kind on the simple grounds that there are no miracles that do not have significance or purpose.
Another question arises when we consider all the censuses that appear in Numbers: why does the Torah have to relay the subtotals both for the various tribes and also according to their banners? Nine sums appear in Numbers, each and every one of them accurate; but what did the Torah wish to say by relaying these sums? On this question Nahmanides wrote (Num. 1:45):
Here are the detailed figures and the totals of the censuses recorded in the Torah:
Numbers 1 Numbers 2 Numbers 26
(on the Plains of Moab)
Reuben 46,500 43,730
Simeon 59,300 151,450 22,000
Gad 45,650 40,500
Judah 74,600 76,500
Issachar 54,400 186,400 64,300
Zebulun 57,400 60,500
Ephraim 40,500 32,500
Manasseh 32,200 108,100 52,700
Benjamin 35,400 45,600
Dan 62,700 64,400
Asher 41,500 157,600 53,400
Naphtali 53,400 45,400
Total 603,550 603,550 601,730
Census of the Levites Numbers 3 Numbers 4
(1 month and up) (age 30 and up)
Gershon 7,500 2,630
Kohath 8,600 2,750
Merari 6,200 3,200
Total Levites 22,300 8,580
These are all the census figures given in the Torah (with the exception of the enumeration of first-borns, whose number is very small in comparison to these figures).
In the literature of the rishonim, or early commentators, I did not find any direct attempt to deal with the two questions which I raised (why the figures are rounded, and what use there is in the totals or sums). Aharonim, or later commentators, however, struggled with these questions. In Meshekh Hokhmah Rabbi Meir Simha ha-Cohen of Dvinsk addressed the first issue as follows (Num. 3:16):
Regarding the number of Levites, from age one month up--Moses entered the Tent and a divine voice called out and said, "Thus and so many babes," also not counting by individuals, for the count was "as he was bidden" regarding the number of Israelites [no less than ten]. But this was not the case with the list of first-borns, where each individual was reckoned.
But our explanation is problematic, since here in chapter 26, all the numbers are rounded out to the nearest hundred save for Reuben (43,730) ending with thirty. Clearly in the reckoning here the Torah did not complete to the nearest hundred. If so, it is wondrous how they all ended with hundreds save for one; so this must be studied further.
At least a partial resolution of the problem is provided in Emet le-Ya'akov, written by Rabbi Jacob Kaminetsky:
In my opinion both questions can be answered by relying on the following general rules that pertain to fairly large numbers (certainly to numbers greater than 5,000).
1) When the number obtained was in tens (with no units), then it was registered as is and the Torah did not round it.
2) When the number obtained was not in complete tens, it was rounded to the nearest hundred.
There is a simple logic to these rules: if you round a number that ends in units, then it is rounded to hundreds (the error being less than a hundredth), but a number that ends in tens is left as is. It should be noted that the simple notion which we understand of rounding numbers to the nearest hundred was totally foreign to science until the end of the Middle Ages. Otto Neugebauer, in "The Astronomy of Maimonides and its source," HUCA 22 [1949], p. 340, notes that also ancient astronomers who were expert in complicated computations and who regularly used rounding did not generally round to the nearest whole number. Rounding was generally done downwards, unless the number was very close to the larger number (e.g., greater than 0.75). Neugebauer stresses that Maimonides in his astronomic computations to determine when the new moon occurs rounded to the closest integer and that this was a major innovation in comparison with his predecessors such as Ptolemy or even Al-Battani.
Now let us return to the census in the Torah. As we have said, the numbers were rounded according to the two rules I mentioned above. If we look at the figures in the Torah, this is patently clear. In each of the two censuses of the Israelites in the wilderness, 11 out of 12 figures are multiples of hundreds, whereas one (in the first census the tribe of Gad, and in the second census the tribe of Reuben), is a multiple of ten. The probability of any number ending in zero but not being a multiple of one hundred is 9/100, therefore if one takes any 12 numbers, the expectancy of such a number appearing is equal to 12 x 9/100 = 1.08. In other words, on the average, out of 12 nu, one will be a multiple of ten (but not of a hundred). Moreover, if we compute the different probabilities (according to binomial distribution), it turns out that the greatest probability is obtained when exactly one out of twelve numbers has this form. The probability of this equals 12 x (1-9/100)11 x 9/100, and all the other probabilities are smaller.
Examining both censuses together also yields the same results: out of 24 figures, the average number of occurrences of the specifically desired form is close to 2, and the maximal probability is obtained when k=2, which is indeed what happened.
As for the censuses of the Levites, similar results can be obtained, but with a small number of figures (there being only three families) no statistical analysis can be made.
The rules that we used enable us to answer the question about the sums. Now it is clear why the Torah had to write down the total sums of Israelites in both censuses. Since all the numbers were rounded, one could have had a situation where the grand total obtained would be far off from the actual number in the census. In theory, for the census of the Israelites the deviation could be as great as 588 people. For example, if the number of people counted in each tribe ended in 49, then the numbers would be rounded down to the nearest hundred, so that after totalling all twelve tribes one would have a figure smaller by 588 (actually by 600, after rounding) than the actual census count. Of course this is a rather extreme example, and actually there is a mathematical theorem stating that as the number of figures being summed increases, the deviations resulting from rounding are more likely to offset one another. Actually that is precisely what happened with the census of the Israelites. All the deviations, both upwards and downwards, counterbalanced so that the sum matched the total census taken (of course, to the nearest 50), and therefore it was very important that all these figures and sums be reported in the Torah.
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