# Why does the tribe of Gad have an unusual population count in Bamidbar 1?

When listing the census of the tribes in Bamidbar 1, only the tribe of Gad has the count 45,650 - I.e. - the count is not exactly on the hundreds as it is with all the other tribes. As the count was performed by a half shekel, I assume that all the numbers listed were accurate, or is it possible that somehow, there was an error, or the numbers were rounded? In either case, why is Gad the only tribe with this unusual count?

• Why would rounding be an error or a problem? On the contrary, it seems odd that everyone is exactly to the hundreds without rounding. My guess: Gad was exactly 50, so not subject to rounding. Commented May 22, 2014 at 18:55
• @Yishai - I'll admit that you could view it this way. I assume that "unusual" is that Gad is the exception to the rest.
– DanF
Commented May 22, 2014 at 19:04
• @Yishai I remember reading your "guess" somewhere. I'll try to find it, IY"H. Commented May 22, 2014 at 19:14
• israeldailypicture.com/2012/05/… israeldailypicture.com/2013/08/… somewhat relevant? Commented May 22, 2014 at 21:06
• Any new additional answers lately? Commented Jun 16, 2022 at 12:41

I actually saw a dvar torah this week that claimed that the tribe of Gad showed that the others were actually exact by a miracle in Mail Jewish (quoted below). See the quote from Rav Chaim Kanievsky below based on what his father the Steipler Rav told him.

Another explanation is that the counts were actually rounded to the nearest fifty or rounded up to the nearest fifty. In that case, the tribe of Gad is not different from the others in the rounding method. It is just that the rounding caused them to end in 50 rather than (1)00. Alternatively, since 50 is the precise middle, it was left as is because of uncertainty which way to round.

Aish.com has this explanation

The Shaarei Aharon quotes the Imrei Noam, who maintains that the Torah isn't particular about small numbers, and suggests that the census for each tribe was rounded to the nearest 100. Since the tribe of Gad had precisely 50 extra people, their count couldn't be rounded either way.

As proof that the Torah rounds numbers, the Imrei Noam cites the commandment to count 50 days of the Omer even though we count only 49, and the verse ordering 40 lashes to be given to certain transgressors even though we give only 39. This is also the position of the Meshech Chochmah (3:16).

However, Rabbi Chaim Kanievsky relates that he initially assumed that the census numbers were rounded, but when he mentioned this to his father, the Steipler responded that a number written in the Torah must be exact, and God must have had a reason why He miraculously caused each tribe to have such even numbers of people.

Mail Jewish also brings up the question and suggests an answer.

From: Sanford Lefkowitz
Date: Sun, May 18,2014 at 12:01 AM
Subject: Census counts

In Parshas Bamidbar, we see the first listing of census numbers by tribe. One rather anomalous feature of the counts is that 11 of the 12 counts are multiples of 100 and one is a multiple only of 10. One question this raises is "Are these exact numbers or round numbers?". If they are round numbers the rounding rule must be 'round to the nearest 10'. The probability that 11 out of 12 numbers, when rounded to the nearest 10, would also round to a multiple of 100 is on the order of one in 10 billion. The same anomaly, 11 out of 12 numbers being a multiple of 100, also occurs the second time the census counts are given in Parshas Pinchas. The probability that we would have two independent counts, rounded to the nearest 10, both producing results where 11 out of 12 counts round to a multiple of 100 if on the order of 10-20. This suggests there is something unusual going on here.

Shortly after the Bamidbar tribal count, we are given the count of the Levi'im, 22,000. That certainly looks like a round number. But shortly after that, we are given the count of the first born, 22,273 and told that each first born has to be redeemed by a Levi. The Torah then explicitly asks the question of what happens with the 273 remaining first born. Since 22,273 is clearly not a round number and the Torah explicitly mentions the number 273, it must be that 22,000 is an exact number. Given the unlikelihood of most of the tribal census counts being a multiple of 100 and the apparent fact that the Levi'im count is an exact number, it seems likely that all the tribal counts are exact numbers.

Why are 11 out of 12 tribal counts multiple of 100 each of the times the count is given? Here is a speculation. Perhaps the Torah is trying to call our attention to the anomaly. If all the counts had been a multiple of 100, that would have been even more unlikely than 11 out of 12 counts being a multiple of 100. But if that had been the case, we might have just assumed they were all being rounded to the nearest 100 and not considered it very interesting. If the counts had been numbers like 21,906, we might just say, "OK, that's what the number turned out to be. No big deal". But by having exactly 11 out of 12 counts be multiples of 100 on two occasions, the Torah is telling us to take notice. The only way such an unlikely event could occur is if Hashem is in control. He is taking care of everything, even down to the population counts.

Sanford Lefkowitz

However, I did find a different explanation at Rounding of Numbers in the Censes of Bnei Yisrael by Rabbi Elchanan Samet which discusses the subject at length and attempts to account for the other countings as well

D. ROUNDING TO HUNDREDS OR TO TENS - ARTICLE BY A. MERZBACH

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."

E. ALTERNATIVE EXPLANATION FOR NUMBERS ENDING IN TENS IN CHAPTERS 1-4

Merzbach's two rules for rounding figures explain all four exceptional figures listed at the beginning of section D. above, and even match the statistical probability of the phenomenon of the two numbers - the tribe of Gad and the tribe of Reuven - in the two censes held in the desert. But the "simple logic" that he employs is actually not so simple. He writes, "If a number ending in units already requires rounding, it is rounded to hundreds. But if the figure ends in tens, it is left as is." We may ask: if a number ending in tens is considered a round number, then why are numbers ending in units not rounded to the nearest ten, thereby diminishing the maximal inaccuracy from 49 to 4? The proposition that "If a number… already requires rounding, it is rounded to hundreds" is not a mathematical one; it is a matter of personal taste.

Setting aside for a moment the exception of the tribe of Reuven in parashat Pinchas, focusing instead only on the numbers in chapters 1-4, we may solve the difficulty of the three exceptional figures without reliance on Merzbach's two rules.

Now we must deal with the number of the tribe of Gad: 45,650. Why is this number not rounded to the nearest hundred? Perhaps because it ends precisely with 50, and therefore cannot be rounded either upwards or downwards.

But we may suggest a slightly different idea: since the exact number 50 cannot be rounded, it may itself be considered a rounded number, in a sense, even within a system of rounding to hundreds. Therefore it is possible that where the real number is close to fifty, the number is rounded to 50 rather than to 100 (thereby diminishing the inaccuracy that would result from rounding to 100).

• This answer assumes that the numbers were rounded. however this is very problematic, since the actual sum of all the tribes would exceed the number given in the Torah (600,3550)?
– Bach
Commented May 25, 2017 at 14:09
• @Bach Part of the answer says that the numbers were exact by a miracle. Other parts cited deal with the rounding that may have occurred and whatever problems may have arisen. Commented May 25, 2017 at 19:58
• @Bach You had a typo in the number it should have been 603,550 which could be the rounded total. Or indeed it could be an exact total. Commented May 26, 2017 at 1:49
• Your right, my mistake. Anyways, you don't understand the issue here. You suggest that the remainders were dropped and the numbers were rounded. However, if you would add these remainders together, you would surely get a sum of a few hundred. Now add them to the sum of the rounded numbers which is recorded in the Torah as 603,550, the entire sum (remainder+rounded) should be much more than those recorded in bamidbar! My question is, according to your interpretation, why is the Torah ignoring the remainders (which add up to another few hundred)?
– Bach
Commented May 26, 2017 at 2:15
• @larry909 No new answers Commented Jun 17, 2022 at 21:15

See Shiras Dovid (to Bamidbar 26) by R. Aharon Dovid Goldberg. He also explains that the unusual Gad's counting in 1:25 came exactly to 50 because it was not possible to round it to a (1)00 since it falls just in between.