Happy Pi Day 2022, and Happy Birthday Albert Einstein

WHAT IS PI DAY, AND WHEN IS IT CELEBRATED?

From here.

Today, March 14, is celebrated as Pi Day by some of the mathematically inclined in the US. Why? Well, in most of the world, the date is written as day/month/year. So in Israel, all of Europe, Australia, South America and China, today's date, March 14th, would be written as 14/3. 

But not here in the US. Here, we write the date as month/day/year; it's a uniquely American way of doing things. (Like apple pie. And guns.) So today's date is 3/14. Which just happen to be the first few digits of pi, the ratio of the circumference of a circle to its diameter.

And that's why each year, some (particularly geeky) Americans celebrate Pi Day on March 14 (3/14). The year 2015 was Pi'ish than all others, since the entire date (when written the way we do in the US, 3/14/15) reflects five digits of pi, and not just the first three: 31415. Actually we got even more geeky: This day in 2015 at 9:26 and 53 seconds in the morning, the date and time, when written out, represented the first ten digits of Pi: 3141592653.

So that's why Pi Day is celebrated here in the US -  and probably not anywhere else. (It has even be recognized as such by a US Congressional Resolution. Really. I'm not making this up. And who says Congress doesn't get anything done?) 

PI IN THE BIBLE

In the ּBook of Kings (מלאכים א׳ 7:23) we read the following description of  a circular pool that was built by King Solomon. Read it carefully, then answer this question: What is the value of pi that the verse describes?

מלכים א פרק ז פסוק כג 

ויעש את הים מוצק עשר באמה משפתו עד שפתו עגל סביב וחמש באמה קומתו וקוה שלשים באמה יסב אתו סביב 

And he made a molten sea, ten amot from one brim to the other: it was round, and its height was five amot, and a circumference of thirty amot circled it.

Answer: The circumference was 30 amot and the diameter was 10 amot. Since pi is the ratio of the circumference of a circle to its diameter, pi in the Book of Kings is 30/10=3. Three - no more and no less.

There are lots of papers on the value of pi in the the Bible. Many of them mention an observation that seems to have been incorrectly attributed to the Vilna Gaon.  The verse we cited from מלאכים א׳ spells the word for line as קוה, but it is pronounced as though it were written קו.  (In דברי הימים ב׳ (II Chronicles 4:2) the identical verse spells the word for line as קו.)  The ratio of the numerical value (gematria) of the written word (כתיב) to the pronounced word (קרי) is 111/106.  Let's have the French mathematician Shlomo Belga pick up the story - in his paper (first published in the 1991 Proceedings of the 17th Canadian Congress of History and Philosophy of Mathematics, and recently updated), he gets rather excited about the whole gematria thing:

A mathematician called Andrew Simoson also addresses this large tub that is described in מלאכים א׳ and is often called Solomon's Sea. He doesn't buy the gematria, and wrote about it in The College Mathematics Journal.

A natural question with respect to this method is, why add, divide, and multiply the letters of the words? Perhaps an even more basic question is, why all the mystery in the first place? Furthermore, H. W. Guggenheimer, in his Mathematical Reviews...seriously doubts that the use of letters as numerals predates Alexandrian times; or if such is the case, the chronicler did not know the key. Moreover, even if this remarkable approximation to pi is more than coincidence, this explanation does not resolve the obvious measurement discrepancy - the 30-cubit circumference and the 10-cubit diameter. Finally, Deakin points out that if the deity truly is at work in this phenomenon of scripture revealing an accurate approximation ofpi... God would most surely have selected 355/113...as representative of pi...

Still, what stuck Simoson was that "...the chroniclers somehow decided that the diameter and girth measurements of Solomon's Sea were sufficiently striking to include in their narrative." (If you'd like another paper to read on this subject, try this one, published in B'Or Ha'Torah - the journal of "Science, Art & Modern Life in the Light of the Torah." You're welcome.)

PI IN THE TALMUD

The Talmud echoes the biblical value of pi in many places. For example:

תלמוד בבלי מסכת עירובין דף יד עמוד א 

כל שיש בהיקפו שלשה טפחים יש בו רחב טפח. מנא הני מילי? - אמר רבי יוחנן, אמר קרא : ויעש את הים מוצק עשר באמה משפתו עד שפתו עגל סביב וחמש באמה קומתו וקו שלשים באמה יסב אתו סביב 

"Whatever circle has a circumference of three tefachim must have a diameter of one tefach."  The problem is that as we've already noted, this value of pi=3 is not accurate. It deviates from the true value of pi (3.1415...) by about 5%. Tosafot is bothered by this too.

תוספות, עירובין יד א

והאיכא משהו. משמע שהחשבון מצומצם וכן בפ"ק דב"ב (ד' יד:) גבי שני טפחים שנשתיירו בארון ששם ספר תורה מונח שהיא בהיקפה ששה טפחים ופריך כיון דלאמצעיתו נגלל נפיש ליה משני טפחים וכן בתר הכי דמשני בספר דעזרה לתחלתו נגלל ופריך אכתי תרי בתרי היכי יתיב משמע דמצומצם לגמרי וקשיא דאין החשבון מדוקדק לפי חכמי המדות

Tosafos can't find a good answer, and concludes "this is difficult, because the result [that pi=3] is not precise, as demonstrated by those who understand geometry." 

PI IN THE RAMBAM

In his commentary on the Mishnah (Eruvin 1:5) Maimonides makes the following observation:

פירוש המשנה לרמב"ם מסכת עירובין פרק א משנה ה 

צריך אתה לדעת שיחס קוטר העיגול להקפו בלתי ידוע, ואי אפשר לדבר עליו לעולם בדיוק, ואין זה חסרון ידיעה מצדנו כמו שחושבים הסכלים, אלא שדבר זה מצד טבעו בלתי נודע ואין במציאותו שיודע. אבל אפשר לשערו בקירוב, וכבר עשו מומחי המהנדסים בזה חבורים, כלומר לידיעת יחס הקוטר להקיפו בקירוב ואופני ההוכחה עליו. והקירוב שמשתמשים בו אנשי המדע הוא יחס אחד לשלשה ושביעית, שכל עיגול שקוטרו אמה אחת הרי יש בהקיפו שלש אמות ושביעית אמה בקירוב. וכיון שזה לא יושג לגמרי אלא בקירוב תפשו הם בחשבון גדול ואמרו כל שיש בהקיפו שלשה טפחים יש בו רוחב טפח, והסתפקו בזה בכל המדידות שהוצרכו להן בכל התורה.

...The ratio of the diameter to the circumference of a circle is not known and will never be known precisely. This is not due to a lack on our part (as some fools think), but this number [pi] cannot be known because of its nature, and it is not in our ability to ever know it precisely. But it may be approximated ...to three and one-seventh. So any circle with a diameter of one has a circumference of approximately three and one-seventh. But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations.

So what are we to make of all this? Did the rabbis of the Talmud get pi wrong, or were they just approximating pi for ease of use?  After considering evidence from elsewhere in the Mishnah (Ohalot 12:6 - I'll spare you the details), Judah Landa, in his book Torah and Science, has this to say:

We can only conclude that the rabbis of the Mishnah and Talmud, who lived about 2,000 years ago, believed that the value of pi was truly three. They did not use three merely for simplicity’s sake, nor did they think of three as an approximation for pi. On the other hand, rabbis who lived much later, such as the Rambam and Tosafot (who lived about 900 years ago), seem to be acutely aware of the gross innacuracies that results from using three for pi. Mathematicians have known that pi is greater than three for thousands of years. Archimedes, who lived about 2,200 years ago, narrowed the value of pi down to between 3 10/70 and 3 10/71 ! (Judah Landa. Torah and Science. Ktav Publishing House 1991. p.23.)

HAPPY BIRTHDAY, EINSTEIN

Today, March 14, is not only Pi Day. It is also the anniversary of the birthday of Albert Einstein, who was born on March 14, 1879. As I've noted elsewhere, Einstein was a prolific writer; one recent book (almost 600 pages long) claims to contain “roughly 1,600” Einstein quotes. So it's hard to chose one pithy quote of his on which to close.  So here are two.  Happy Pi Day, and happy birthday, Albert Einstein.

“As a human being, one has been endowed with just enough intelligence to be able to see clearly how utterly inadequate that intelligence is when confronted with what exists.”

— Letter to Queen Elisabeth of Belgium, September 1932

“One thing I have learned in a long life: That all our science, measured against reality, is primitive and childlike — and yet it is the most precious thing we have.”

— Banesh Hoffman. Albert Einstein: Creator and Rebel. Plume 1973

From the Talmudology Archives: ~ Gauss, Tosafot, and Counting the Sukkot Sacrifices

We are currently celebrating the seven-day festival (or eight outside Israel) of Sukkot (Tabernacles). When the Temple in Jerusalem stood, it was a time when a large number of animals were sacrificed. A very large number. For each of the seven days of the festival, in addition to two rams, fourteen lambs and one goat there were a number of bulls that were sacrificed. Thirteen on the first day, twelve on the second, eleven on the third, ten on the fourth, nine on the fifth, eight on the sixth and finally seven bulls on the last day.

במדבר 29:12-34

וּבַחֲמִשָּׁה עָשָׂר יוֹם לַחֹדֶשׁ הַשְּׁבִיעִי מִקְרָא־קֹדֶשׁ יִהְיֶה לָכֶם כָּל־מְלֶאכֶת עֲבֹדָה לֹא תַעֲשׂוּ וְחַגֹּתֶם חַג לַה׳ שִׁבְעַת יָמִים׃ 

וְהִקְרַבְתֶּם עֹלָה אִשֵּׁה רֵיחַ נִיחֹחַ לַה פָּרִים בְּנֵי־בָקָר שְׁלֹשָׁה עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם יִהְיוּ…׃ 

וּבַיּוֹם הַשֵּׁנִי פָּרִים בְּנֵי־בָקָר שְׁנֵים עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

וּבַיּוֹם הַשְּׁלִישִׁי פָּרִים עַשְׁתֵּי־עָשָׂר אֵילִם שְׁנָיִם כְּבָשִׂים בְּנֵי־שָׁנָה אַרְבָּעָה עָשָׂר תְּמִימִם׃… 

and so on…

Numbers 29:34

Pablo Picasso: “Bull,” 1945

On the fifteenth day of the seventh month, you shall observe a sacred occasion: you shall not work at your occupations.—Seven days you shall observe a festival of the LORD.— 

You shall present a burnt offering, an offering by fire of pleasing odor to the LORD: Thirteen bulls of the herd, two rams, fourteen yearling lambs; they shall be without blemish. 

The meal offerings with them—of choice flour with oil mixed in—shall be: three-tenths of a measure for each of the thirteen bulls, two-tenths for each of the two rams, 

and one-tenth for each of the fourteen lambs. 

And there shall be one goat for a sin offering—in addition to the regular burnt offering, its meal offering and libation. 

Second day: Twelve bulls of the herd, two rams, fourteen yearling lambs, without blemish… 

Third day: Eleven bulls, two rams, fourteen yearling lambs, without blemish…

and so on… 

So how many bulls would the manager of the Temple inventory have to make ready for the entire festival of Sukkot? Well, you could just add them up, which is not too hard (13+12+11+10+9+8+7=70), but there is another way, which is found in the medieval commentary known as Tosafot, on page 106 of the tractate Menachot. Let’s take a look. 

Another offering, another math problem

We read there that the mincha offering is accompanied by a minimum of a one-issaron measure of flour. But a mincha can also be accompanied by a multiple of that number, up to a maximum of 60 issronot. What happens if a person vows to bring a specific number of issronot of flour to accompany a mincha offering but cannot recall how many he had in mind? What number of issronot of flour should he offer? Well it’s a bit tricky. The sages ruled that a single offering using the full sixty issronot of flour is all that needs to be brought. But the great editor of the Mishnah, Rabbi Yehudah Hanasi disagreed. In a spectacular way. Here is the discussion:

מנחות קו, א

תנו רבנן פירשתי מנחה וקבעתי בכלי אחד של עשרונים ואיני יודע מה פירשתי יביא מנחה של ששים עשרונים דברי חכמים רבי אומר יביא מנחות של עשרונים מאחד ועד ששים שהן אלף ושמונה מאות ושלשים

The Sages taught in a baraita: If someone says: I specified that I would bring a meal offering, and I declared that they must be brought in one vessel of tenths of an ephah, but I do not know what number of tenths I specified, he must bring one meal offering of sixty-tenths of an ephah. This is the statement of the Rabbis. Rabbi Yehuda HaNasi says: He must bring sixty meal offerings of tenths in sixty vessels, each containing an amount from one-tenth until sixty-tenths, which are in total 1,830 tenths of an ephah.

Since there is a doubt as to the true intentions of the vow, Rabbi Yehudah HaNasi covers all the bases and requires that every possible combination of a mincha offering be brought. So you start with one mincha offering accompanied with one issaron of flour, then you bring a second mincha offering accompanied with two issronot of flour, then you bring a third mincha together with three issronot, and so on until you reach the maximum number of issronot that can accompany the mincha - that is until you reach sixty. The total number of Rabbi Yehudah HaNasi’s mincha offerings is then calculated: 1,830.

How did the Talmud arrive at that number? We are not told, and presumably you simply add up the series of numbers 1+2+3+4….+59+60, which gives a total of 1,830. That certainly would work. But Tosafot offers a neat mathematical trick to figure out the sum of a mathematical sequence like this:

שהן אלף ושמונה מאות ושלשים. כיצד קח בידך מאחד ועד ששים וצרף תחילתן לסופן עד האמצע כגון אחד וששים הם ס"א שנים ונ"ט הם ס"א ושלש ונ"ח הם ס"א כן תמנה עד שלשים דשלשים ושלשים ואחד נמי הם ס"א ויעלה לך שלשים פעמים ס"א

How did we arrive at 1,830? Take the series from 1 to 60 and add the sum of the first to the last until you get to the middle. Like this: 1+60=61; 2+59=61; 3+58=61. Continue this sequence until you get to 30+31 which is also 61. You will have 30 sets of 61 (ie 1,830).

Tosafot continues with the math lesson, and lets us know the total number of bulls sacrificed over the seven day festival of Sukkot:

וכן נוכל למנות פרים דחג דעולין לשבעים כיצד ז' וי"ג הם עשרים וכן ח' וי"ב הם עשרים וכן ט' וי"א הם כ' וי' הרי שבעים

This method may also be used to count the number of sacrificial bulls on Sukkot, which are a total of 70. How so? [There are thirteen offered on the first day of sukkot, and one fewer bull is subtracted each day until the last day of sukkot, on which seven bulls are offered.] 13+7=20; 12+8=20; 11+9=20… [There are a total of 3 pairs of 20+ an unpairable 10]= 70.

In mathematical terms, the Tosafot formula for the sum (S) of the consecutive numbers in Rebbi’s series, where n is the number of terms in the series and P is the largest value, is S= n(P+1)/2. Which reminds us of…

Carl Friedrich Gauss

Carl Friedrich Gauss (1777-1855) was one of the world’s greatest mathematicians. He invented a way to calculate the date of Easter (which is a lot harder than you’d think), and made major contributions to the fields of number theory and probability theory. He gave us the Gaussian distribution (which you might know as the ”bell curve”) and used his skills as a mathematician to locate the dwarf planet Ceres. The British mathematician Henry John Smith wrote about him that other than Isaac Newton, “no mathematicians of any age or country have ever surpassed Gauss in the combination of an abundant fertility of invention with an absolute rigorousness in demonstration, which the ancient Greeks themselves might have envied.”

There is a delightful (though possibly apocryphal) story about Gauss as a bored ten-year old sitting in the class of Herr Buttner, his mathematics teacher. There are at least 111 slightly different versions of the story, but here is one, as told by Tord Hall in his biography of Gauss:

When Gauss was about ten years old and was attending the arithmetic class, Buttner asked the following twister of his pupils. “Write down all the whole numbers from 1 to 100 and add their sum…The problem is not difficult for a person familiar with arithmetic progressions, but the boys were still at the beginner’s level, and Buttner certainly thought that he would be able to take it easy for a good while. But he thought wrong. In a few seconds, Gauss laid his slate on the table, and at the same time he said in his Braunschweig dialect: “Ligget se” (there it lies). While the other pupils added until their brows began to sweat, Gauss sat calm and still, undisturbed by Buttner’s scornful or suspicious glances.

How had the child prodigy solved the puzzle so quickly? He had added the first number (1) to the last number (100), the second number (2) to the second from last number (99) and so on. Just like Tosafot suggested. The sum of each pair was 101 and there were 50 pairs. And so Gauss wrote the answer on his slate board and handed it to Herr Buttner. It is 5,050.

THE Number of bulls=70

Gauss was raised as a Lutheran in the Protestant Church, and so he did not learn of this method from reading Tosafot. But it is delightful to learn that the same mathematical method that launched Gauss into his career as a mathematician predated him by at least four-hundred years and can be found on page 106a of Menachot, where it also applies to the festival of Sukkot.

Happy Sukkot from Talmudology

Sukkah 8a ~ Rashi’s Mathematical Errors

In today’s page of Talmud, the discussion about the minimum size of a sukkah continues, and we get into some geometry. According to Rabbi Yochanan, the minimum circumference of a round sukkah is 24 amot. The Talmud tells us that this is based on the belief that when a person sits he occupies one square amah.

Next the talmud tells us that according to Rabbi Yehuda Hanasi the minimum size of a square sukkah is 16 square amot. But according to a “mathematical rule” on today’s page of Talmud, the perimeter of a square that surrounds a circle is greater than perimeter of the circle by one one quarter of the perimeter of the square.

סוכה ח,א

כַּמָּה מְרוּבָּע יוֹתֵר עַל הָעִיגּוּל — רְבִיעַ

Now, by how much is the perimeter of a square inscribing a circle greater than the circumference of that circle? It is greater by one quarter of the perimeter of the square.

If that is true, then the circle that surrounds a 16 amot square will be 3/4 or the perimeter, or 12 amot. Why does Rabbi Yochanan require a perimeter of 24 amot, which is twice as large?

The Talmud gives the following answer:

הָנֵי מִילֵּי בְּעִיגּוּל דְּנָפֵיק מִגּוֹ רִיבּוּעָא, אֲבָל רִיבּוּעָא דְּנָפֵיק מִגּוֹ עִגּוּלָא — בָּעֵינַן טְפֵי, מִשּׁוּם מוּרְשָׁא דְקַרְנָתָא

This statement with regard to the ratio of the perimeter of a square to the circumference of a circle applies to a circle inscribed in a square, but in the case of a square circumscribed by a circle, the circle requires a greater circumference due to the projection of the corners of the square. In order to ensure that a square whose sides are four cubits each fits neatly into a circle, the circumference of the circle must be greater than sixteen cubits.

Rashi explains that a circle of diameter 16 can inscribe a square whose perimeter is 12:

בעיגולא דנפיק מיגו ריבוע - אם היקפת בחוט של ט"ז אמה בקרקע בריבועא תמצא בתוכו ארבע מרובעות ואם היית צריך לעגלו מבפנים ולהוציא קרנות ריבועו אתה עוגלו בחוט י"ב ונמצא חיצון יתר על הפנימי רביע אף כאן אם היתה מרובעת סוכה זו דיה להיות כדי שישבו בהיקיפה ששה עשר בני אדם

But Rashi here is not precise, as you can see below:

Tosafot on our page of Talmud notes this inaccuracy, and points out another one. According to this page of Talmud,

כל אַמְּתָא בְּרִיבּוּעָא אַמְּתָא וּתְרֵי חוּמְשֵׁי בַּאֲלַכְסוֹנָא

Any square of side 1 will have a hypotenuse of 1.4

But as Tosafot points out, this is not correct (אין החשבון מכוון ולא דק דאיכא טפי פורתא). In fact the hypotenuse will be the square root of 2, which is an irrational number beginning with 1.141213….

Another Rashi with wrong math

In tractate Eruvin (78a) we read the following:

עירובין עח, א

אמר רב יהודה אמר שמואל כותל עשרה צריך סולם ארבעה עשר להתירו רב יוסף אמר אפילו שלשה עשר ומשהו

Rav Yehuda said that Shmuel said: If a wall is ten handbreadths high, it requires a ladder fourteen handbreadths high, so that one can place the ladder at a diagonal against the wall. The ladder then functions as a passageway and thereby renders the use of the wall permitted. Rav Yosef said: Even a ladder with a height of thirteen handbreadths and a bit is enough, [as it is sufficient if the ladder reaches within one handbreadth of the top of the wall].

Here is Rashi’s explanation:

עירובין עח, א רשי

סולם ארבעה עשר - שצריך למשוך רגלי הסולם ארבעה מן הכותל לפי שאין סולם זקוף נוח לעלות

A ladder that is 14 handbreadths high: The feet of the ladder need to be placed four handbreadths from the wall

Here is a diagram of the setup, and the problem with Rashi:

This error is noted by Tosafot (loc. cit.)

צריך סולם ארבעה עשר להתירו - פי' בקונטרס שצריך למשוך רגלי הסולם ארבעה מן הכותל ולא דק דכי משיך ליה י' טפחים נמי מן הכותל שהוא שיעור גובה הכותל יגיע ראש הסולם לראש הכותל דארבעה עשר הוא שיעור אלכסון של י' על י' דכל אמתא בריבועא אמתא ותרי חומשי באלכסונא

A ladder 14 handbreadths heigh is needed - Rashi explains that the feet of the ladder need to be placed 4 handbreadths from the wall. This is not accurate. For when the feet of the ladder are placed ten handbreadths from the wall, (a larger measure) the ladder will still reach the top of the wall. Because the length of the diagonal [i.e.the hypotenuse] is 14 handbreadths in a 10x10 [right angled] triangle. Because for every unit along the sides of square, the diagonal will be 1.4

What Tosafot is getting at is that in a 1x1 right angled triangle, the hypotenuse must be √2. But √2 is an irrational number, meaning the calculation never ends. However, an irrational number is not useful for our real-world measurements, and so Tosafot rounds √2 to 1.4, just as in the Talmud the value of π, another irrational number, is rounded to 3. (Shalom Kelman, a loyal Talmudology reader, sent us another explanation of the errors which you can read here.)

What to make of these Errors?

We have reviewed two mathematical errors made by Rashi, but how should we view them? As ignorance, mistaken calculations, or something else?

In his classic 1931 work Rabbinical Mathematics and Astronomy, (p58) W.M Feldman is firmly in the "Rashi was ignorant” camp.

It is however, most remarkable that although Rashi displayed great genius in mathematical calculation, he was quite ignorant of the most elementary mathematical facts. He was not aware that the sum of two sides of a triangle is greater than the third, for he says if a ladder is to be placed 4 spans from the foot of a wall 10 spans high so as to reach the top, then the ladder must be 14 spans high, (i.e. the sum of the two lengths), which of course is absurdly incorrect, the real minim length of the ladder must be only √(4x4)+(10x10)= √116=10.7 spans.

Judah Landa, in his book Torah and Science suggests that the Rabbis of the Talmud (and Rashi too, I suppose) had mistaken calculations. They did not give mathematics “the serious attention it deserved and that as a consequence their knowledge of it suffered.” Later commentators, like Tosafot and Maimonides knew that π was slightly greater than three, and Tosafot “demands to know why the Rabbis of Caesarea made statements without attempted verification by measurement and experiment.”

no error here

Rabbi Moshe Meiselman believes that the rabbis of the Talmud were incapable of making an error. Of any sort. In his book Torah, Chazal & Science he dedicates eight pages and copious footnotes to explain why, in fact, there were no errors in any of the math found on today’s page of Talmud. Among the sources he cites is a commentary on the Talmud written by the fifteenth-century Rabbi Simeon ben Tzemach Duran, better known as the Tashbetz. In his Sefer Hatashbetz, he addresses the parallel discussion in the tractate Sukkah:

ובתוס' תרצו כי התלמודיים טעו בדבריהם של רבנן דקסרי …וכל זה חיזוק וסמך שאין בכל דברי רז"ל דבר שיפול ממנו ארצה כי הם אמת ודבריהם אמת

Tosafot explains that the Talmud believed that the Rabbis (of Caesarea) were mistaken…but there is not a single error in all the words of the Sages, for they are true and their words are true…

Rabbi Meiselman concludes that not one of the commentaries on the Talmud “suggests that any of the Chachamim [Sages in the Talmud] made elementary errors in calculation or were ignorant of basic principles.”

Rabbi Meisleman aside, Rashi certainly seems to have made an error in his calculations. But why should this bother us? Of the hundreds of thousands of words written by Rashi, commenting on and explaining the entire Hebrew Bible and Babylonian Talmud, an error or two is hardly unexpected. (Just ask any author who has proof-read her manuscript a dozen times only to find a typo in the published book.) Did Rashi misunderstand, or was he ignorant of Pythagoras and his Theorem? In the end it doesn’t matter much. We all make mistakes. I only hope I make as few of them as Rashi did. Now that would be an accomplishment.

Yoma 55a ~ Yom Kippur, Counting, and Why the Chinese Are Good at Math

A Mishnah that we studied a couple of days ago on page 53a described how the Cohen Gadol (High Priest) would sprinkle the blood of his sacrifice in Temple on Yom Kippur.

נִכְנַס לַמָּקוֹם שֶׁנִּכְנַס, וְעָמַד בַּמָּקוֹם שֶׁעָמַד, וְהִזָּה מִמֶּנּוּ אַחַת לְמַעְלָה וְשֶׁבַע לְמַטָּה

He entered into the place that he had previously entered, the Holy of Holies, and stood at the place where he had previously stood to offer the incense, between the staves. And he sprinkled from the blood, one time upward and seven times downward.

And then he would count to avoid any error:

וְכָךְ הָיָה מוֹנֶה: אַחַת, אַחַת וְאַחַת, אַחַת וּשְׁתַּיִם, אַחַת וְשָׁלֹשׁ, אַחַת וְאַרְבַּע, אַחַת וְחָמֵשׁ, אַחַת וְשֵׁשׁ, אַחַת וָשֶׁבַע. יָצָא וְהִנִּיחוֹ עַל כַּן הַזָּהָב שֶׁבַּהֵיכָל.

And this is how he would count as he sprinkled, to avoid error: One; one and one; one and two; one and three; one and four; one and five; one and six; one and seven. The High Priest then emerged from there and placed the bowl with the remaining blood on the golden pedestal in the Sanctuary.

Today’s page of Talmud comments on this interesting way of keeping track of the sprinklings:

יומא נה, א

תָּנוּ רַבָּנַן: אַחַת, אַחַת וְאַחַת, אַחַת וּשְׁתַּיִם, אַחַת וְשָׁלֹשׁ, אַחַת וְאַרְבַּע, אַחַת וְחָמֵשׁ, אַחַת וָשֵׁשׁ, אַחַת וָשֶׁבַע, דִּבְרֵי רַבִּי מֵאִיר. רַבִּי יְהוּדָה אוֹמֵר: אַחַת, אַחַת וְאַחַת, שְׁתַּיִם וְאַחַת, שָׁלֹשׁ וְאַחַת, אַרְבַּע וְאַחַת, חָמֵשׁ וְאַחַת, שֵׁשׁ וְאַחַת, שֶׁבַע וְאַחַת.

וְלָא פְּלִיגִי: מָר כִּי אַתְרֵיהּ וּמָר כִּי אַתְרֵיהּ

The Sages taught in a baraita that when sprinkling, the High Priest counted: One; one and one; one and two; one and three; one and four; one and five; one and six; one and seven. This is the statement of Rabbi Meir. Rabbi Yehuda says that he counted: One; one and one; two and one; three and one; four and one; five and one; six and one; seven and one.

The Gemara comments: They do not disagree about the matter itself that the High Priest sprinkles once upward and seven times downward. Rather, this Sage rules in accordance with the norm in his place, and this Sage rules in accordance with the norm in his place. In one place they counted the smaller number first, while in the other place they would count the larger number first.

WHo excels at math?

In a 1994 study of forty second-generation Chinese-American and 40 Caucasian-American preschoolers and kindergartners, the Chinese-American children outperformed Caucasian-American children on measures of mathematics, spatial relations, visual discrimination, numeral formation, and name writing. A 2011 study that explored cultural differences in young children’s early math competency prior to their school showed that Taiwanese children performed better than U.S., Peruvian, and Dutch children. More Taiwanese four-year-olds were able to count up to at least 21 when compared with children from the other three countries. There are more studies like these, but you get the idea. But why should some cultures be especially good at math? The answer, it appears, is in the language.

The Best language to learn math…is not english

From here.

It turns out that the way numbers are counted in different languages may make arithmetic easier - or harder. In a fascinating article in The Wall Street Journal, Sue Shellenbarger noted that Chinese, Japanese, Korean and Turkish use simpler number words and express math concepts more clearly than English. And this makes it easier for small children to learn counting and arithmetic. “The trouble starts at "11” she wrote:

English has a unique word for the number, while Chinese (as well as Japanese and Korean, among other languages) have words that can be translated as "ten-one"—spoken with the "ten" first. That makes it easier to understand the place value—the value of the position of each digit in a number—as well as making it clear that the number system is based on units of 10.

English number names over 10 don't as clearly label place value, and number words for the teens, such as 17, reverse the order of the ones and "teens," making it easy for children to confuse, say, 17 with 71, the research shows. When doing multi-digit addition and subtraction, children working with English number names have a harder time understanding that two-digit numbers are made up of tens and ones, making it more difficult to avoid errors.

These may seem like small issues, but the additional mental steps needed to solve problems cause more errors and drain working memory capacity…

This suggestion was supported by a more recent study that showed that among Chinese children language abilities were able to significantly predict both informal and formal math skills.

Language	17	27
English	'seventeen'	'twenty-seven'
Chinese	'ten-seven'	'two-ten-seven'
Japanese	'ten-seven'	'two-ten-seven'
Turkish	'ten-seven'	'two-ten-seven'
Hebrew	'seven-ten'	'twenty-seven'

Today’s discussion in the Talmud notes that there were different ways of counting the one “upward sprinkling” and the seven “downward” ones. Of course neither effected the way that numbers higher than eleven are counted in Hebrew, but it is a reminder that the order in which we count things plays a very significant role in how we might see the world. And how good we are at getting our sums right.