Friday, December 10, 2021
Can you Count to 70?
Monday, November 15, 2021
Goats and Amicable Numbers
In this week’s parasha, we find Yaakov preparing for his encounter with his twin brother Esav in several ways. Among other preparations, Yaakov sends him gifts consisting of various different kinds of animals. The Torah tells us (Bereishit 32:14–16) how many of each kind of animal Yaakov sent: 200 female goats and 20 male goats; 200 female sheep (ewes) and 20 male sheep (rams); 30 nursing camels with their young; 40 female cows and 10 bulls; 20 female donkeys and 10 male donkeys. What is the significance of these numbers?
In his ספר בעלי ברית אברם, R’ Avraham Azulai provides an explanation for the number of goats, which he attributes to R’ Nachshon Gaon of the 9th century. The total number of goats is 200 + 20 = 220. What significant property does the number 220 have?
Consider the factors of 220, that is, numbers that multiply together to give the product 220. We can factor the number 220 in the following ways:
220 = 1 × 220
220 = 2 × 110
220 = 4 × 55
220 = 5 × 44
220 = 10 × 22
220 = 11 × 20
Now consider only the “proper factors” of 220 – that is, all the factors in the above list, excluding the number 220 itself – and add them up:
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
So the proper factors of 220 add up to 284.
We now repeat the process, considering the factors of 284. We can factor the number 284 in the following ways:
284 = 1 × 284
284 = 2 × 142
284 = 4 × 71
Again, we consider only the proper factors of 284 – all the factors in the above list, excluding the number 284 itself – and add them up:
1 + 2 + 4 + 71 + 142 = 220
So the proper factors of 284 add up to 220. Does this number look familiar?
As we have just shown, the numbers 220 and 284 have the property that the proper factors of each number add up to the other number. A pair of numbers with this property is known as a pair of amicable numbers, or according to R’ Nachshon Gaon, מנין נאהב. Apparently it was known to the ancients that in order to gain the love of kings and princes, a person would give one of a pair of amicable numbers as a present, keeping the other number for himself. This is so that the factors of the number given add up to the number kept, and the factors of the number kept add up to the number given. So this is what Yaakov did. He sent Esav 220 goats, and kept 284 for himself.
Wait a minute: The Torah tells us that Yaakov gave Esav 220 goats, but where do we see in the Torah that he kept 284 for himself? Several pesukim later, as Yaakov gives instructions to the servants carrying the gifts, the Torah records (32:21), “כי־אמר אכפרה פניו במנחה ההולכת לפני” – “for he said: I will win him over with the gifts that are being sent ahead.” R’ Nachshon Gaon explains that this sentence contains a hint to the number 284, in the following way. The word אכפרה can be divided in two parts: אכ פרה. When the Torah uses the word אך, it is generally interpreted by the Rabbis to indicate exclusion or reduction. Calculating the numerical value of the second part of the word, פרה, we get: 80 (פ) + 200 (ר) + 5 (ה) = 285. Applying a reduction (indicated by אך) to the value 285 (given by פרה), we obtain a value of 284. This represents the number of goats that Yaakov kept for himself, according to R’ Nachshon Gaon.
Special thanks to Daniel Levenstein for bringing this insight to my attention.
Addendum: Yaakov sent a number of different types of animals. Why were only goats sent in amicable numbers? See an interesting thought from רבנו בחיי which may shed some light on this question.
References:
Leonard Eugene Dickson, History of the Theory of Numbers, Volume I: Divisibility and Primality, Carnegie Institute of Washington: Washington, 1919, p. 39, available at:
http://www.archive.org/stream/historyoftheoryo01dick#page/38/
ר' אברהם ב"ר מרדכי אזולאי, ספר בעלי ברית אברם, published 1873 but existed in manuscript for 300 years previously; pp. 48–49, available beginning at:
http://www.hebrewbooks.org/pdfpager.aspx?req=3997&pgnum=47
הנאהבים והנעימים - על רעות אצל מספרים, in Michlalah Jerusalem College's mathematical journal אלף אפס (ℵ₀):
http://alefefes.macam.ac.il/article/article.asp?n=15
(may not work in all browsers)
(Thanks to Yaaqov Loewinger for this link via Hebrew Wikipedia)
Friday, October 8, 2021
The Weight of the Teiva
Chazon Ish | R' Moshe Feinstein | GRA"CH Noeh | |
Length of amah | 57.66 cm. | 53.98 cm. | 48 cm. |
Volume of cubic amah (length/100)^{3} | 0.192 m^{3} | 0.157 m^{3} | 0.111 m^{3} |
Calculation | = 165000 x 0.192 ≈ 31630 | = 165000 x 0.157 ≈ 25950 | = 165000 x 0.111 ≈ 18250 |
Water Displacement | 31630 m^{3} | 25950 m^{3} | 18250 m^{3} |
Now that we have determined the amount of water displaced by the ark, all we have to do is calculate how much that water weighed. Then by Archimedes' Principle we can assume that the ark weighed the same amount. This, however, is not necessarily so simple. The density of sea water is slightly more than that of regular water at approximately 1025 kg/m^{3}. This figure usually remains about the same, without significant deviation, regardless of the exact temperature. The only drastic changes are observed when the water reaches extreme conditions such as freezing or boiling.
The first difficulty encountered is that during the initial 40 days of the flood, the waters were boiling hot (Rosh HaShanah 12a). This would change the density of the water substantially and consequently interfere with the calculation. However, it is important to note that Rashi's calculation is based on the rate at which the water receded after the 150 days which followed the 40 days of destruction. By that time, the waters had calmed down and most probably dropped to a more moderate temperature. Therefore, it can be assumed that the temperature of the water is a negligible factor in the calculation of the water density. However, what does seem problematic is that Rashi brings in the figure of 11 amos in 7:17 when the waters were at their highest intensity. It is almost certain that the density of the water at this point was much less than it was 190 days later. If the ark was calculated to have been submerged 11 amos by a calculation based on cooler waters, that figure should presumably be greater at the time of the actual flood.
Chazon Ish | R' Moshe Feinstein | GRA"CH Noeh |
31630 m^{3} | 25950 m^{3} | 18450 m^{3} |
x 1025 kg/m^{3} | ||
32420750 kg | 26598750 kg | ^{}18706250 kg |
Table of Metric Conversions | ||
57.66 cm | = | 22.7 in. |
53.98 cm | = | 21.25 in. |
48 cm. | = | 18.9 in. |
31630 m^{3} | = | 1117003 ft^{3} |
25950 m^{3} | = | 916416 ft^{3} |
18450 m^{3} | = | 651556 ft^{3} |
25^{o} C | = | 77^{o} F |
1025 kg/m^{3} | = | 2260 lb/61024 in^{3}(35.3 ft^{3}) |
32420750 kg | = | 71475519 lb = 35737.8 tons |
26598750 kg | = | 58640206 lb = 29320.1 tons |
18706250 kg | = | 41240222 lb = 20620.1 tons |
6372500 m | = | 20907152 ft |
53 cm | = | 20.87 in. |
The Constant Rate of Recession
Monday, September 20, 2021
How many בקשות in יעלה ויבוא
יעלה | ויבוא | ויגיע | ויראה | וירצה | וישמע | ויפקד | ויזכר | = 8 |
(Keep in mind that all the above verbs will apply to all of the following nouns:) | |
זכרוננו | ופקדוננו | וזיכרון אבותינו | וזיכרון משיח בן דוד עבדך | וזיכרון ירושלים עיר קדשך | וזיכרון כל עמך בית ישראל | x 6 |
= 48 | |
(And then the following modify all of the above:) | |
|לפניך לפליטה | לטובה | לחן | ולחסד | ולרחמים | לחיים ולשלום | x 7 |
So the entire first section | = 336 |
זכרנו ה' אלוקינו בו לטובה | ופקדנו בו לברכה | והושיענו בו לחיים טובים | + 3 |
= 339 | |
Now we add the final portion: | |
ובדבר ישועה | ורחמים | = 2 |
חוס | וחננו | ורחם עלינו | והושיענו | x 4 |
= 8 | |
So the final count is 339 + 8 | = 347 |
Wow, 347 בקשות packed into one small תפילה!
Friday, August 27, 2021
Balancing the Shevatim at Har Grizim and Har Eival
IMPORTANT UPDATE: Tradition magazine has been gracious enough to make their archives fully available to the public! So following the links above will now allow you to read the articles in their entirety.
Wednesday, July 21, 2021
Moshe's pleas
Gematrias off by 1
Sunday, July 4, 2021
Splitting up the Animals
The Probability of the Goral
2^{12} |
(24 x 23 x 22 x 21...x 2 x 1) |
1 |
12^{2} x 11^{2} x 10^{2} x 9^{2} x 8^{2} x 7^{2} x 6^{2} x 5^{2} x 4^{2} x 3^{2} x 2^{2} x 1^{2} ^{} |
תשע"ד: A similar drawing came up this week in דף יומי on :תענית כ"ז regarding the גורל used to determine the order of the משמרות in the בית שני. According to רש"י's understanding, there was a similar miracle at play. The probability associated with that drawing is discussed here.
Friday, June 25, 2021
Counting the Judges
Perhaps, it is possible that Rashi is not referring to the actual number given in Sanhedrin but to the calculation done there. Just as when B'nei Yisroel were 603,550 the number of dayanim was 78,600 the number of dayanim based on B'nei Yisroel's current population would be around 88,000. The next step, then, is to calculate what population would require 88,000 dayanim. Being that the dayanim included judges of groups of 1000, 100, 50 and 10 the following equation is given:
88,000 | = (x/1000) + (x/100) + (x/50) + (x/10) (where x = total population) |
= (100x + 20x + 10x + x)/1000 (multiplying by 1000/1000, i.e. 1) | |
= 131x/1000 (131 judges per 1000 citizens) | |
x | = 88,000,000/131 |
x | = 671,755 (rounded down) |
We are left, then, with only three incidents to consider. The first is the episode following the death of Aharon when B'nei Yisroel began to return toward Mitzrayim and B'nei Levi ran after them (see Rashi 26:13). The Yerushalmi records that only eight (or seven, see Rashi and the Yerushalmi inside) families were wiped out from Yisroel in that incident. This would seem to represent a significant loss but perhaps not 46,000.
The second is the episode with the snakes (Bemidbar 21) where, as recorded in the Mishna (Rosh HaShanah 29a), those who were bitten and did not have appropriate kavana when shown the copper snake perished. Here, too, there is no significant loss recorded but only that proper kavana was required for the snake to cure you. Before the cure, however, the pasuk states (pasuk 6) that a great multitude perished from Yisroel. We are not given any further information, though, on the number of casualties^{2}.
Finally, there is the plague of Ba'al Pe'or. That leads into a discussion as to what in fact transpired besides the loss of 24,000 in the plague. From the fact that the Torah refers to a magefa, it is doubtful that this refers to those killed by the shoftim. So how many people did the shoftim kill? The Ramban (on this pasuk) quotes a Yerushalmi, the simple reading of which implies that each shofet killed two men as commanded by Moshe. This would render well over 150,000 casualties. Ramban, however, concludes that the figure given by the Yerushalmi is just referring to how many would have been killed had Moshe's command been carried out but in the end the shoftim never had a chance to do so and they didn't kill a soul. Perhaps it is possible to take this idea of the Ramban that the shoftim were interrupted before having a chance to complete their mission, but to suggest that they had already begun to carry it out when they were interrupted. With or without such a supposition, one could suggest that in some way, these three incidents combined for a grand total of 46,000 fatalities. The other 24,000 died in the magefa. Now we have accounted for all 70,000 lives and all the figures work out.
Nevertheless, it is doubtful that Rashi actually wrote 88,000 and had this convoluted calculation in mind. Rather, it is more reasonable to assume that this version of Rashi is a mistake and that he originally wrote 78,000, particularly because the Ramban had such a reading of Rashi. However, I am not the first to try and justify the figure of 88,000. The Margaliyos HaYam on the gemara in Sanhedrin cites the sefer Techeles Mordechai who offers a calculation based on a population of 603,550:
603 | shoftim over a thousand |
6,035 | shoftim over a hundred |
12,071 | shoftim over fifty |
60,355 | shoftim over ten |
70 | zekeinim |
276 | (12x23 small Sanhedrin for each tribe) |
72 | (12x6 Nesi'im for each tribe) |
8,580 | Levi'im |
=88,002 |
Friday, May 28, 2021
Piles of Quail
Thursday, May 13, 2021
No Population Increase
Tens and Ones
The first question is, why is this the case.
Furthermore, I noticed something this week that I don't recall ever noticing before: In one instance in the parasha, this style is violated. Pasuk 2:9 gives the total of the Eastern Camp, including the tribes of Yehuda, Yissachar, and Zevulun. The number is 186,400, written as follows:
כָּל-הַפְּקֻדִים לְמַחֲנֵה יְהוּדָה, מְאַת אֶלֶף וּשְׁמֹנִים אֶלֶף
וְשֵׁשֶׁת-אֲלָפִים וְאַרְבַּע-מֵאוֹת--לְצִבְאֹתָם; רִאשֹׁנָה, יִסָּעוּ.
"... a hundred thousand and eighty thousand and six thousand and four hundred ..."
This is a clear departure from the usual style, which would have been "ששה ושמנים אלף", "six and eighty thousand". I am not aware of any other such departure from the usual style. Any ideas why this is?
Rounded numbers
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):
- Scriptures had to say what the total was after giving the detailed figures because Moses and Aaron were commanded to know the census of the entire people and the census of each tribe, for that is the way of kings when counting the people. But the reason underlying this commandment--why the Lord commanded it--escapes me. I do not know why they had to know the number; why were they commanded to know it?
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):
- Perhaps Scripture says regarding the number of Israelites, "You ... shall record them by their groups, from the age of twenty years up, all those in Israel who are able to bear arms" (1:3), because they were counted not in units, but only by tens. Therefore, none of the numbers has units, because each head of Israelites gave his number of men, and they were heads of tens. Every small number was rounded, and fewer than ten did not have a head over them alone. Therefore it says "by their groups," for no camp has fewer than ten, as explained in the Jerusalem Talmud, at the end of the first chapter of Eruvin.
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.
- Above, in Parshat Bemidbar (1:25) we cited Imrei Noam to the effect that the reason "units and tens" were not mentioned in the counting, and each tribe was reckoned in hundreds except for the tribe of Gad (45,650), is that Scripture is not strict about the few. There we explained that the reason must be that the Torah completed the counting to the nearest hundred, and that one should not interpret that it rounded to the nearest ten, for if so how does one explain the improbable fact that no tribe had a number ending in tens save for the tribe of Gad? As for the tribe of Gad not being rounded to a hundred, that is because their number came exactly to fifty, and it was not possible to round it to a hundred since it falls just in between.
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 humble opinion, the counting was done by the chieftains of fifties, since we see in Parshat Jethro that the leaders were divided into heads of tens, heads of fifties, heads of hundreds, and heads of thousands, and apparently the army was divided into heads of fifties. Likewise, we see in the beginning of II Kings (1:9-10) that there were captains of fifty with their fifty men. It was by these captains that the Israelites were counted, and hence there were either complete hundreds or fifties. Except that this theory encounters a problem in Parshat Pinehas (26:7), where the tribe of Reuben totals forty-three thousand, seven hundred and thirty. Possibly the Torah subtracted those of Korah's followers who were swallowed in the earth from the rounded-out fifty, and since their number was twenty that left exactly thirty, and this accounts for the exact number in 26:7, but nevertheless the matter needs more thought.
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.
Prepared for Internet Publication by the Center for IT & IS Staff at Bar-Ilan University.