# The Numbers Game

**Question #1: Pie r Squared**

Yanki is supposed to be watching his weight and therefore needs to figure out how many calories are in the pie he beholds. To figure out how big the pie is, he measures the diameter of the pie, and divides in it half to get the length of its radius. He then multiplies the length of the radius by itself to get “r squared,” and multiplies the result by three so that he knows the area of the pie’s surface. Is there anything wrong with his calculation?

**Question #2: Puzzled by the Pasuk**

“How can the *pesukim* tell us that the relationship between the circumference of a circle and its diameter is three to one, when simply taking a string and measuring around a circle demonstrates that the circumference is noticeably longer than three times the diameter?”

**Question #3: Performing Mitzvos Accurately**

“How accurate a calculation must I make when determining the size of an item to be used for a mitzvah?”

**Introduction**

In numerous places, both *Tanach* and *Chazal* approximate certain mathematical values, such as evaluating the ratio of the circumference of a circle to its diameter as three to one. The problem is that we can demonstrate mathematically that the ratio is greater than three and is almost 3 1/7. This leads to the following questions:

(1) Why would *Chazal* calculate using inaccurate approximations?

(2) When making *halachic* calculations, may we rely on these estimates, or do we need to be mathematically more accurate?

(3) A corollary question is: When providing an estimate, one must allow for a margin of error. Does *halachah* require a margin of error, and, if so, how much?

**The Slide Rule versus the Calculator**

Let me begin our discussion with a modern analogy, if something I remember can still be considered “modern.” When I first studied sophisticated mathematical estimates, I learned to use a slide rule, which today is as valuable to an engineer as his abacus. Relative to the calculator, a slide rule does not provide accurate measurements, and someone using a slide rule must allow for a fairly significant margin of error.

Today, complex computations are made with calculators, which provide far more accurate results that can be rounded off, as necessary, to the nearest tenth, millionth, quadrillionth or smaller. Of course, using a calculator still requires one to round upward or downward, but because it is much more precise, the margin of error is greatly reduced.

**How Irrational Are You?**

Numerous *halachic* questions require mathematical calculations that involve what we call “irrational numbers.” An irrational number means one that cannot be expressed in fractional notation. Another way of explaining an irrational number is that its value can never be calculated totally accurately, but can only be estimated. The two most common examples of irrational numbers that show up in *Chazal* are:

**Pi**

(1) The ratio of the circumference of a circle to its diameter, which we are used to calling by the Greek letter ∏ (pronounced like the word “pie,” and spelled in English “pi”). Since the 19^{th} century, the letter pi has been used to represent this number, because the Greek word for **periphery** is *peripherion, *which begins with the letter ∏. Hundreds of years earlier, the *Rambam* (*Commentary to the Mishnah,* *Eruvin* 1:5) noted that the ratio of the circumference of a circle to its diameter is an irrational number that can only be approximated, and that the scientists of his era used an estimate of 3 and 1/7, which is actually slightly greater than the value of ∏. The *Rambam* explains that since there is no accurate ratio, *Chazal* used a round number, three, for this calculation.

**The Diagonal of a Square**

(2) The length of a diagonal of a square, which is equal to the side of the square multiplied by the square root of two (√2). *Chazal* calculated the length of a diagonal of a square to be 1 and 2/5 times its side, which is slightly smaller than the value of the √2. (Another way of expressing this idea is that the ratio between the diagonal and the side is 7:5.) The fact that *Chazal*’s figuring is somewhat smaller than the mathematical reality is already proved by *Tosafos* (to *Sukkah *8a *s.v.* *kol*)*.*

Since both pi and the square root of two are irrational numbers, they can only be estimated but can never be calculated with absolute accuracy.

Based on the above-quoted statement of the *Rambam*, we can already address one of our earlier questions: “Why would *Chazal* have used inaccurate evaluations for calculation?” The answer is that any computation of the correlation of the circumference of a circle to its diameter will be an estimate. The only question is how accurate must this estimate be for the purpose at hand.

*Chazal* or *Tanach*?

Although the *Rambam* attributes the rounding of pi to *Chazal*, in actuality, there are sources in *Tanach* that calculate the ratio of the circumference of a circle to its diameter as three-to-one. Both in *Melachim *(I 7:23) and again in *Divrei Hayamim *(II 4:2), *Tanach* teaches that the *Yam shel Shelomoh, *the large, round pool or* mikveh *that was built in the first *Beis Hamikdash*, was thirty *amos *in circumference and ten *amos *in diameter, which provides a ratio of circumference to diameter of three-to-one. Thus, we can ask a question of the *Rambam*: Why does he attribute this ratio to *Chazal*, rather than the source for *Chazal*’s calculation, the *pesukim*?

The commentaries there, however, already ask how the verse can make a calculation that we know is not accurate. The *Ralbag *suggests two options: either that the numbers used are intended to be a very broad estimate, or, alternatively, that the diameter is measured from the external dimensions of the *mikveh*, whereas the circumference is measured from its inside, which makes the estimate closer to mathematical reality. According to the second approach, we have no Biblical source that uses an estimate of three-to-one as a substitute for pi.

This will explain why the *Rambam* attributed the estimation of pi as three to *Chazal,* rather than to the *Tanach*. The *Rambam* was fully aware that one could interpret the verses according to the second approach of the* Ralbag*, in which case, there is no proof from the verse. He, therefore, attributed this estimate to *Chazal*.

*Gemara Eruvin*

However, the *Ralbag’s* approach seems to conflict with a passage of *Gemara*. The *Mishnah* in *Eruvin* states that if the circumference of a pole is three *tefachim*, its diameter is one *tefach*, which means that the *Mishnah* assumes a ratio of three-to-one.

The *Gemara* questions how the *Mishnah* knows that the ratio is three-to-one, and then draws proof from the above-quoted verse that the *Yam shel Shelomoh* was thirty *amos* around and ten *amos* across. The *Gemara* then debates whether the calculations of the *Yam shel Shelomoh* indeed result in a ratio of three-to-one, because one must also include the thickness of the poolitself, which offsets the computation. The *Gemara* eventually concludes that the verse was calculating from the inside of the pool, not its outside, and therefore the thickness of the pool’s containing wall is not included in the calculation (*Eruvin *14a).

However, this *Gemara*’s discussion leaves the mathematician dissatisfied, a question already noted by *Tosafos*. If the internal diameter of the *Yam shel Shelomoh* was ten *amos*, its circumference must have been greater than thirty *amos*, and if its circumference was thirty *amos*, then its internal diameter must have been less than ten *amos*.

**A Different Question**

The *Rosh*, in his responsa, is bothered by a different question, based on Talmudic logic rather than on mathematical calculation. He finds the *Gemara*’s question — requesting proof for the ratio between a circle’s circumference and its diameter — to be odd. The ratio between a circle’s circumference and its diameter is a value that one should calculate. By its nature, this is not a question that requires a Biblical proof or source.

In the literature that we have received from the *Rosh, *he asks this question in two different places. In his responsa (*Shu”t Harosh *2:19), we find a letter that he wrote to the *Rashba*, in which he asked a series of questions that the *Rosh *notes bother him tremendously and to whom he has no one else to turn for an answer. One of the questions the *Rosh *asks is: “Why does the *Gemara *ask for a Biblical source for a mathematical calculation?”

It is curious to note that a later commentary mentions that, in all the considerable literature that we have received from the *Rashba*, we have no recorded answer of the *Rashba* to this question of the *Rosh* (*Cheishek* *Shelomoh* to *Eruvin* 14a).

**Another Rosh**

As I mentioned above, there is another place where the *Rosh* asks why the *Gemara *wanted a Biblical source for a mathematical calculation, but in this second place the *Rosh *provides an answer to the question. In his* Tosafos Harosh* commentary on *Eruvin,* which was published for the first time fairly recently, the *Rosh* provides the following answer: Since the calculation of three-to-one is not accurate, the *Gemara* wanted a biblical source as proof that we are permitted to rely on this estimate.

(It is curious to note that the *Cheishek Shelomoh* whom I quoted above provided the same answer to this question as did the *Rosh* in his *Tosafos. *The *Cheishek Shelomoh *never saw the *Tosafos Harosh, *which had not yet been printed in his day.)

**Curiousity about the Tosafos Harosh**

There is an interesting historical point that can presumably be derived from this statement of the *Rosh. *As I mentioned, in the *Tosafos* *Harosh*, the *Rosh* does answer the question that he raised, and accredits this answer to himself. This should be able to prove which work the *Rosh *had written earlier, and also whether he ever received an answer to his question from the* Rashba*. This analysis is based on the following question: Why did the *Rosh *cite an answer in his *Tosafos¸*but not in his responsum, which was addressed as a question to the *Rashba. *There are three obvious possibilities:

(1) Although the *Rosh* wrote this answer in his *Tosafos*, he was dissatisfied with it, and therefore wrote a question to the *Rashba*. I would reject this possibility because, if it is true, then, in his correspondence to the *Rashba*, the *Rosh* would have mentioned this answer and his reason for rejecting it.

(2) The *Rosh* indeed received an answer, either this one or a different answer, from the *Rashba*. I reject this approach also, because, were it true, the *Rosh* would have quoted the *Rashba*’s answer in his *Tosafos* and, if need be, discussed it.

(3) Therefore, I conclude that the *Rosh*, indeed, never received an answer to the question he asked of the *Rashba* and subsequently reached his own conclusion as to how to answer the question, which he then recorded in the *Tosafos* *Harosh*. This would lead us to conclude that the *Tosafos* *Harosh* were written later in his life than his responsa, or, at least, than this responsum.

**Mathematical Accuracy**

At this point, we can address one of earlier questions.** “**When making *halachic* calculations, may we rely on these estimates, or do we need to be mathematically more accurate?” We might be able to prove this point by noting something in the *Mishnah* in *Eruvin*. The *Mishnah* there ruled that, under certain circumstances, an area that is fully enclosed on three of its sides and has a beam, a *tefach* wide, above the fourth side, is considered *halachically* fully enclosed, and one may carry inside it. The *Mishnah* then proceeds to explain that if the beam is round and has a circumference of three *tefachim*, one may carry inside the area because, based on the calculation that the relationship of its circumference to its diameter is three-to-one, the beam is considered to be a *tefach* wide. However, as the *Rambam* notes, the beam is actually less than a *tefach* in diameter, and therefore, one should not be permitted to carry in this area!

The* Aruch Hashulchan* (*Orach Chayim *363:22; *Yoreh Deah* 30:13) notes this problem and concludes that one may carry in this area. He contends that this is exactly what the *Gemara* was asking when it requested Scriptural proof for a mathematical calculation. “Upon what *halachic* basis may we be lenient in using this estimate of three-to-one, when this will permit carrying in an area in which the beam is less than a *tefach* wide? The answer is that this is a *halachah* that we derive from the verse.”

To clarify this concept, the *Chazon Ish* notes that the purpose of mitzvos is to draw us nearer to *Hashem*, to accept His reign, and to be meticulously careful in observing His laws. However, none of this is conflicted when the Torah teaches that we may use certain calculations, even if they are not completely mathematically accurate. In this instance, relying on these estimates is exactly what the Torah requires (*Chazon Ish, Orach Chayim *138:4). As expressed by a different author, the *Gemara* (*Eruvin* 4a; *Sukkah *5b) teaches that the measurements, the *shiurim*, required to fulfill mitzvos are all *halachah leMoshe miSinai*, laws that *Moshe Rabbeinu* received as a *mesorah* in *Har Sinai*. Similarly, these estimates of irrational numbers mentioned above are all *halachah leMoshe miSinai* that one may rely upon to fulfill mitzvos, whether or not they are mathematically accurate. The same Torah takes these calculations into consideration when instructing us which dimensions are required in order to fulfill these specific mitzvos (*Shu”t Tashbeitz* 1:165).

In the context of a different *halachah* in the laws of *Eruvin*, the *Mishnah Berurah* makes a similar statement, contending that we can rely on *Chazal*’s estimates, even when the result is lenient. However, the *Mishnah Berurah* there vacillates a bit in his conclusion, ruling that one can certainly rely on this when the issue is a rabbinic concern (*Shaar Hatziyun* 372:18). In a responsum, Rav Moshe Feinstein questions why the *Mishnah Berurah* limits relying on this approach, and Rav Moshe rules unequivocally that the rule permitting one to rely on these estimates holds true even germane to *de’oraysa *laws and even leniently (*Shu”t Igros Moshe, Yoreh Deah *Volume3 #120:5).

**How Straight Are My Tefillin?**

Personally, I find the context of Rav Moshe’s *teshuvah* very interesting. There is a *halachah leMoshe miSinai* that requires that the boxes of the *tefillin*, the *batim*, must be perfectly square. In a responsum dated 21 Adar II, 5736, Rav Moshe was asked whether there is a *halachic* preference to use scientific measuring equipment to determine that one’s *tefillin* are perfectly square. Rav Moshe rules that there is neither a reason nor a *hiddur* to measure the *tefillin* squareness this accurately. Since *Chazal* have used the calculation of 1.4 or a ratio of 7:5, which we know is an estimate, to determine the correct diagonal of a square, there is no requirement to make one’s *tefillin* squarer than this, and it is perfectly fine simply to measure the length of each of the sides of one’s *tefillin* and its two diagonals to ascertain that the ratio between the diagonal and the side is 7:5.

In the above-cited responsum, Rav Moshe notes that he had heard that the Brisker Rav, Rav Yitzchak Ze’ev Soloveitchik, had ruled that it was preferable to check one’s *tefillin* in the most scientific method available. Rav Moshe writes that he finds this suggestion very strange and disputes its being *halachically* correct (*Shu”t Igros Moshe, Yoreh Deah *Volume3 #120:5).

Thus, according to these authorities, we have answered our previous question regarding the *halachic* significance of estimated values: Indeed, the purpose of *Chazal*‘s making these estimates was that observing *halachah* does not require that these calculations be mathematically precise, provided they meet the criteria that the *halachah* established.

**An Alternate Approach**

Although the majority of late authorities conclude that the calculations of *Chazal* are indeed part of the *halachos* of *shiurim*, this is not a universally held position. The *Tashbeitz*, a *rishon*, wrote a lengthy responsum on the topic, in which he presents two ways to explain why *Chazal* used estimates that are not precisely accurate. His first approach reaches the same conclusion as we have already found in the later *poskim*, that these measurements are included within the *halachos* of *shiurim* that are part of the *halachah leMoshe miSinai*.

The second approach of the *Tashbeitz*, however, differs with the above-mentioned *halachic* conclusion. In his second approach, he contends that all the above estimates were meant for pedagogic, but not *halachic* reasons. The rounding of pi to three and the diagonal of a square to 1.4 were provided to make the material easily comprehensible to all students, since every individual is required to know the entire Torah. Thus, when *Chazal* used these estimates in calculating the laws, their intent was to enable the average student to comprehend the *halachic* material, not to provide the most accurate interpretation. When an actual *halachic* calculation is made, it must be totally accurate, and any *halachic* authority involved would realize that he must use a highly accurate mathematical computation and then round either upward or downward as necessary for the specific application. (A similar position is held by *Chiddushim Uviurim*,* Ohalos *5:6.)

**Conclusion**

Certainly, the majority of late *halachic* opinions conclude that the estimates of *Chazal* are meant to be *halachically* definitive, and not simply pedagogic in nature. However, I leave it to the individual reader to ask his or her *posek* what to do when a practical question presents itself.