In the Numbers 1 census, why do all the numbers from each tribe end in zero? Is this an estimate?
I will try to present both sides of the argument, and conclude based on what I know. As I noted above in the comments, a useful article analyzing these from a Jewish perspective can be found here.
In summary, there are a few approaches:
- Always rounded, some exceptions that need to be explained
- Always exact, likely through some miracle that numbers are almost always multiples of tens or hundreds
- There is some complex system to the countings/roundings, and they are sometimes exact, sometimes rounded in one way, sometimes the other. This is the approach taken by many modern Bible scholars, and is discussed here and at the above link.
In theory, reading the counting of the nation in the desert in it's simplest form would be to understand that these numbers are not rounded. This would appear to be the case based on a number of proofs:
- There are a number of additions/recounts of the tribes, both under their banners, and as the entire nation. In each of these additions/recounts, the final sum of the grouping is exactly the sum of it's parts. If these were to be estimates, we would expect that in one or two of the cases, the rounded numbers would cause the final sum to be rounded up or down; this is not the case. However, this can be refuted by saying that the rounded numbers were used as exact in the addition, and that no recounting to confirm ever took place.
- The number of firstborns counted in Numbers 3:43 (22,273) is an exact number, and why would that be counted exactly and not the other countings?
- Additionally, the number of Levites, counted in Numbers 3 and added there in verse 39, equaled exactly 22,000. Although it seems rounded, this must be an exact number, since the number of Levites was subtracted from the number of firstborns, giving a remainder of 273, which had to be redeemed. Again, if all the other numbers are rounded, how did these change?
- The tribe of Reuben in Numbers 26 is counted to be 43,730. This would indicate that the rounding, if it took place at all, would have been to the nearest 10, not to the nearest fifty or hundred.
On the other hand, the odds of each of the tribes having members numbering to exactly a hundred is astronomically unlikely. How can this be resolved?
The first resolution I will offer was presented by Professor Ely Merzbach, and is explained in the article linked above:
Thus far our assumption has been that in recording the censes, the Torah rounds figures to hundreds, as it would seem from the great majority of those that appear in chapters 1-4. But in section B. above, we note that there are a few figures in these chapters (and another one in parashat Pinchas) that end in tens, and not in hundreds. They are:
a. The tribe of Gad in our parasha 45,650 (1:25)
b. The tribe of Reuven in par. Pinchas 43,730 (26:7)
c. The family of Kehat aged 30-50 2,750 (4:36)
d. The family of Gershon aged 30-50 2,630 (4:40)
What is the reason for these exceptions to the system of rounding to hundreds?
Prof. Ely Merzbach, of the department of mathematics at Bar-Ilan University, addresses this question in his article, "The Censes of Bnei Yisrael in the Desert" (published inthe "Higgayon" - Studies in Rabbinical Thought, vol. 5, 5761). Here are some excerpts:
"It is always possible to attribute this phenomenon (of exactly rounded numbers) to a miracle, or coincidence, without any explanation (as some commentators have attempted to do). But explanations of this sort are rejected by the major commentators with the simple claim that a miracle must have some significance, or some benefit.
It seems to me that it is possible… [to explain the phenomenon] based on the following principles, which refer to fairly large numbers (and certainly to numbers greater than 5,000).
When the figure obtained is in whole tens (without units), the Torah records it as is, without rounding it. When the figure obtained is not in whole tens, then the Torah rounds it to the nearest hundred. The logic behind this system is simple: if a number ending in units already requires rounding, it is rounded to hundreds (with some small margin of inaccuracy). But if the figure ends in tens, it is left as is.
If we examine the data in the Torah, this becomes completely clear. In each of the two censes of Bnei Yisrael in the desert, 11 out of 12 figures are multiples of hundreds, while one (the tribe of Gad in the first census, and Reuven in the second one) is a multiple of tens. The probability of any number ending in zero but not being a multiple of 100 is 9/100. Therefore if any 12 numbers are chosen, we can expect the incidence of appearance of numbers with this characteristic to be 12 x 9/100 = 1.08. In other words: on average, out of 12 numbers, one will be a multiple of tens (and not a multiple of hundreds).
Moreover ... the greatest probability exists, once again, when there is exactly one number of this sort out of 12 numbers… Concerning the censes of the Levite families we could obtain similar results, but when the number of data is small (there are only three families), no statistical test may be applied."
Both of these Rabbis independently suggested that the census was not done by an individual counting each person that would pass in front of him/her. Rather, the counting was done by counting the leaders of already organized groupings (whether for battle or other organizational purposes), which are referenced repeatedly throughout the bible, such as in Exodus 18:21, 1 Samuel 8:12, Isaiah 3:83, etc.
The group sizes were thousands, hundreds, fifties, and tens. These Rabbis suggest that the groups were not all exactly comprised of these exact numbers, but rather, occasionally contained one more or one less. When the final counts were taken, however, each group was assumed to contain exactly the above amounts, thus, each group ended in a "round number", which did not come from rounding.
Rabbi Kalonymus suggested that the counts were done by having leaders of each group of 10 present themselves, and thus, every counting in the book is a multiple of 10. While this does help somewhat, it is still unlikely that so many tribes would happen to end in exactly 100.
Rabbi Kamenetzky suggested that the counts were done by counting the 50-person group leaders (which he considers battalion leaders, based on various bible sources). Therefore, numbers ending in fifties or hundreds are easier to explain. On the other hand, this makes it much harder to explain the numbers that end in other multiples of 10. The best way to deal with these questions is to take a case-by-case approach.
- Reuben's count in Numbers 26 ends in 30. Rabbi Kamenetzky explains that this stemmed from losses in the Korah rebellion, presumably from the families of Dathan and Abiram. Thus, one of the battalions was not 50 anymore, but rather only comprised of 30 people, and the counters left this in as a sign that they had been killed. That is also why their story is retold in the middle of the count in Numbers 26.
- Levites, who may not have been in such battalion groupings, were sometimes rounded or counted in groups of 10. Additionally, sometimes the Levites were counted from a very young age, which would also have required some other system of counting, and may have included rounding of some sort.
- The number of firstborn according to this, would have to have been exact, since they were likely gathered from their individual battalions.
It is also possible that they were counted in battalions of 100 (which has strong Biblical support as well), but this has it's own issues to address as well, and this is already far too long...
Rabbi Samet's approach can be found in the article linked above, and is also worthy of it's own post.
If population growth was random, a census could result in an ending number of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the chances one count would end in 0 is 1 in 10. The chances of 12 counts ending in 0 is 1 in 1012. When the 3 counts within the tribe of Levi are included, the chance of 15 counts ending in 0 is an infinitesimal 1 in 1015:
Rueben 46,500 Gershon 7,500 Simeon 59,300 Kohath 8,600 Gad 45,650 Merari 6,200 Judah 74,600 Issachar 54,400 Zebulun 57,400 Ephraim 40,500 Manasseh 32,200 Benjamin 35,400 Dan 62,700 Asher 41,500 Naphtali 53,400
There are three possible explanations:
- The census was an approximate count expressed in multiples of 10.
- An actual count was made then rounded off.
- The population growth was not random.
The census in Numbers was not the first time the Israelites were counted. When visiting with his father-in-law, Moses delegated his dispute resolution duties:
24 So Moses listened to the voice of his father-in-law and did all that he had said. 25 Moses chose able men out of all Israel and made them heads over the people, chiefs of thousands, of hundreds, of fifties, and of tens. (Exodus 18:24-25) [ESV]
The entire congregation was divided into groups of 10. There is no information whether this division was made along ancestral lines, or whether those placed into groups of 10 had the same selection criteria as those counted in the census. Nevertheless, this prior counting and division is consistent with an actual census of all tribes being a multiple of 10; it would also be a sign to the people there was something very unusual about the total number of people.
An Actual Count: 22,273 Firstborn
A consequence of the census was counting the number of the firstborn:
42 So Moses listed all the firstborn among the people of Israel, as the LORD commanded him. 43 And all the firstborn males, according to the number of names, from a month old and upward as listed were 22,273. (Numbers 3:42-43)
The total number of firstborn from among all tribes was exactly 22,273 and obviously does not end in zero and was not rounded off. The precision of the number is reinforced by the LORD who gives instructions on how to address the variance between the number of firstborn and Levites, whom the LORD exchanged for the firstborn:
44 And the LORD spoke to Moses, saying, 45 “Take the Levites instead of all the firstborn among the people of Israel, and the cattle of the Levites instead of their cattle. The Levites shall be mine: I am the LORD. 46 And as the redemption price for the 273 of the firstborn of the people of Israel, over and above the number of the male Levites, 47 you shall take five shekels per head; you shall take them according to the shekel of the sanctuary (the shekel of twenty gerahs), 48 and give the money to Aaron and his sons as the redemption price for those who are over.” 49 So Moses took the redemption money from those who were over and above those redeemed by the Levites. 50 From the firstborn of the people of Israel he took the money, 1,365 shekels, by the shekel of the sanctuary. 51 And Moses gave the redemption money to Aaron and his sons, according to the word of the LORD, as the LORD commanded Moses. (Numbers 3:44-51)
Given the actual counting of the firstborn from among all of the tribes, option #1 becomes less likely. If an actual count was made option #2 also becomes less likely as there would be no reason to change an actual count to record an approximate number. Therefore, however improbable, it is more likely the numbers are actual counts because the population growth was not random. That is, all counts ending in 0 reflects a purposeful work of God.
The lack of randomness in the growth of the tribes is reinforced by the number of firstborn: 22,273 is a prime number. Now it is true the chance the number of firstborn would be a prime number is not as small as 1 in 1015, yet this does reflect an anomalous condition. In other words, the number of firstborn was not random which supports seeing the growth of the tribes as other than random.
273 "Shortage" of Levites
Based on the census, the tribes can be ordered by size:
Birth Size Tribe Count 4 1 Judah 74,600 5 2 Dan 62,700 2 3 Simeon 59,300 10 4 Zebulun 57,400 9 5 Issachar 54,400 6 6 Naphtali 53,400 1 7 Rueben 46,500 7 8 Gad 45,650 8 9 Asher 41,500 14 10 Ephraim 40,500 12 11 Benjamin 35,400 13 12 Manasseh 32,200 3 13 Levi 22,300
Given the significance of the firstborn and the number 7, it is hardly coincidental that Reuben, the firstborn son, grew into the 7th largest tribe. Rather, the growth of the firstborn into the 7th has the appearance of a purposeful, divine, symbolic action. Similarly, it hardly be coincidental that the only count which does not end in "00" was the tribe of Gad who was the 7th son. So Gad "lost" the position of significance as the 7th, yet among all the tribes it still remains numerically unique.
In exchanging the firstborn for the Levites, it was determined there was a shortage of 273 Levites. This was compensated by paying Aaron and his sons 5 shekels per person. There is correspondence to the first 5 sons, the relative growth of each tribe, and the number 273:
Birth Census Tribe 1 7 Reuben 2 3 Simeon 3 13 Levi 4 1 Judah 5 2 Dan
1, 3, 7, and 13 are the factors of 273 (1x3x7x13=273). It is possible the population growth just happened to work this way, but it is also possible these conditions are a reflection of divine action. Obviously if God was responsible for the relative population growth among Jacob's sons, then He would also be responsible for the actual population growth and the actual number.
The relative population growth of Jacob's sons could be preserved by actual counts, or by actual counts which were rounded off. That is, the relative standing between the tribes would not necessarily be changed if the census was expressed as approximate numbers. However, the number 10 is important in bringing the people out of Egypt, the initial settlement, and future events in the land:
- 10 plagues in Egypt
- Passover lamb was chosen on the 10th day
- The Law given at Sinai began with the 10 Commandments
- 10 words written on the stone tablets
- 10 spies representing 10 tribes would give an evil report
- 10 tribes would reside west of the Jordan
- 10 tribes would break away to form the Northern Kingdom
Finally, tradition holds the Law was given at Sinai 50 days after the Passover. If accurate, this tradition corresponds to single count in the census which does not end in 00. The count for Gad ended in 50.
While 15 counts all ending in 0 is highly unlikely, it is also highly symbolic and a reflection of the specific works done by the LORD in bringing His people out of Egypt, into the land promised. It is a number which is significant for the initial and future divisions of the land.