While rounding may have been an ancient tool, it can help us to understand how calculation was carried out and the educations of scribes themselves. First, in the case of the grain storage bureau , which was composed of merchants and other professions, truncation of measured values to reduce measurement inconsistency helps us to realize that measurements were performed in and around the city of Larsa itself. This may seem obvious, but measurement is seldom stated in the texts so that even this simple act needs to be proven. This is clear from the discussion in Chap. 5, where it was shown that some apparent measurement values were probably the results of an estimation by the scribe and not observation of measurement or measurement instruments. Indeed, in Chap. 7 it was seen that some assessments of value were based on a sample measurement followed by a change rate calculation. Estimation presents a potential observation, not an actual observation.

In YBC 07473 , rounding was based on measurement values as they appeared on metrological tables and their SPVN transformations. The author of YBC 07473 , Itti-Sîn-milki , the merchant overseer of Zarbilum , probably rounded 4 one-12th še up to 5 še and not 4 1/2 še because to round one-12th še to 1/2 še would have added a value outside of the SPVN cycle in which the author was working. Thus, rounding up to 5 še made more sense. Moreover, the author of YBC 07473 probably rounded this value because one-12th še is not found on any extant metrological list or table. While a scribe could work with values outside of these lists , to do so was often difficult, so that rounding in either direction was preferred.

To the modern observer, rounding reveals practices associated with calculated approximation , and this can help us to understand how a scribe was educated. This is seen with YBC 07473 as well. The importance of metrological tables , which the scribe learned in his elementary education , becomes apparent with this rounding: Itti-Sîn-milki was familiar with numerical tables learned in the elementary stage of the scribal curriculum because he carried out a series of divisions by means of multiplications with reciprocals. Most of these reciprocals are found in the standard reciprocal table . However, 1:50, the SPVN transformation of 1 bariga 5 ban, is not. It is a non-regular number of which the factors are 11 and 10. He was probably familiar with non-regular reciprocals and, if he rounded up to reduce roundoff error , he probably trained with texts similar to M 10 . Because he was working with equivalencies, this merchant probably participated in an advanced, perhaps professional education of some form which involved equivalency calculations as described in Chap. 8. This latter is confirmed with AO 08464 , presented in Chap. 5, which was used to show conformity of equivalency rates .

The city of Larsa and its surroundings, including Zarbilum , where Itti-Sîn-milki was active, is well represented in the texts. The education of Itti-Sîn-milki , which is witnessed in the texts that bear his name, provides an image of education in just one of the towns and small cities that made up the hinterland of the city of Larsa . The texts produced by Itti-Sîn-milki , and the discrepancies associated with these texts, explain some aspects of a high-ranking merchant’s education in Larsa’s hinterland. A fairly complete picture of the education of a merchant at and around Larsa can be produced because many of the texts attributed to Larsa or the Larsa area were produced by merchants . Indeed, it is clear that Itti-Sîn-milki’s education, including the use of the reciprocals of non-regular numbers , must have been somewhat standard. The equivalency rate of 1 bariga 5 ban was stated as ‘kar’ or ‘fixed rate’. This kind of rate was probably fixed by the local merchant community so that within the city of Zarbilum , which was probably reflective of Larsa’s education system due to its proximity to Larsa , the ability to work with the reciprocal of a non-regular number would have been assumed by this community. That is to say, if prices were fixed by this community then they must have agreed upon the rate 1 bariga 5 ban so that members of this community would be familiar with non-regular numbers and their approximate reciprocals .

SPVN was probably used while calculating this equivalency with YBC 07473 , as stated above. Chapter 5 showed that the use of SPVN was probably fairly standard when computing with equivalency rates, as well as wage rates , throughout the kingdom of Larsa . This was corroborated by NBC 08014 , probably from Larsa as well, where a bureau official probably rounded an equivalency to simplify transformation between SPVN and measurement values. Thus, equivalencies were not limited to a merchant’s education.

Equivalencies, as a form of estimation , rely on multiplication to carry out a calculation. The numbers and values used to calculate probably came from two sources: other texts or prior operations, as well as direct observations of the tools or performance of a measurement. YBC 04224 speaks to the derivation of multiplied values from prior operations. This text is a merchant account potentially dated to the reign of Gungunum of Larsa and with a suggested provenance of around Larsa. Lines 15 through 25 state various sesame expenditures for which the total found on line 26 is rounded down. This total serves as the basis for an equivalency evaluation in silver. The stated equivalency was probably the result of an actual calculation by the scribe who produced this text because the rounded sesame value was used to produce the equivalency , not the expected sesame value. The scribe added measurement values together, rounded and then made the equivalency .

The scribe rounded down in YBC 04224 to simplify calculation. This is also one possible reason for the subtotal and possibly also the total in NBC 06763 from Larsa as well. The subtotal and total did not reflect the volumes stated in the texts, which are based on observations of man-days. Instead, they probably reflect prior calculations of volume from length by width by depth. If true, then lines 12 through 13 round down to produce 4 iku 26 sar 10 gin volume by means of truncation of added values during addition. This would mean that the subtotal and total allow comparison of a projected cost with an actual cost in man-days of labor, and thus reflect an additional calculation. The same can be said about the text’s total in line 15, where the rounded value in lines 12 through 13 is carried over to produce 4 iku 41 1/3 sar volume rather than 4 iku 41 sar 11 2/3 gin as expected.

By limiting the final numerical value associated with the lowest measurement unit to 1/3, the author simplifies multiplication: only a table for 20, the SPVN transformation of 1/3 sar, is needed to multiply the final digits in SPVN rather than the slightly more complex need to multiply using a table of 10 and table of 1:40 that was required prior to rounding (because 11:40 is the SPVN transformation of 11 2/3 gin). Conformity in YBC 04224 and NBC 06763 leads to simplification of calculation. This value would have been used to calculate labor and then wages following discussion in Chap. 8, especially problem 1 of YBC 04663 .

Volume measurement values like those used to produce the subtotal and total on NBC 06763 would have been used to evaluate labor and then wages as is shown explicitly with YBC 12273 , and as is suggested for NBC 11509 and its associated wage rate texts. Indeed, YBC 12273 shows that volume, and then probably labor, was calculated using SPVN just as is proposed for NBC 06763 when rounding may have simplified calculation in SPVN. Thus, the subtotal and then total reflect a further computation, labor and wages, so that the added value could well have been rounded to facilitate this, as well as to simplify calculation with a counting device and reduce perceived measurement inconsistency .

Some observation must have been the basis for all calculations. An observation, whether this was of a full measurement of a quantity or a sample measurement , would have been the input in any initial calculation, whether an addition, subtraction or multiplication. The measurement values of in-kind items which are evaluated in grain or silver must have been measured. Evidence that measurements were actually carried out and observed by scribes in and around Larsa is provided by the grain storage bureau . As stated in Chap. 4, this bureau was probably staffed by merchants , among other professionals. Discrepancies between measurement values taken before shipment and after delivery show that the scribes were aware of measurement inconsistency in so far as capacity was concerned. The degrees of granularity in each text show that they marginalized measurement inconsistency by truncating measured values. This was hypothesized as having been taught in the early elementary phase of scribal education when tabular lists were learned in school. At that phase, the education of merchants and other professions probably involved the memorization of metrological lists , which would have been a perfect time to explore measurement practice and inconsistency. Thus, the economic documents show that metrological lists and tables, as well as numerical tables , were probably incorporated into the education of scribes around Larsa . A similar phenomenon probably occurred in the territory of Isin after Rīm-Sîn’s conquest, based on the appearance of a value rounded up to decrease granularity in the difference of a balanced account .

One grain shipment text also shows that sample measurement to convert value between standards was used in this bureau. In YBC 07194 line 8, the phrase ‘9(aš) 2(bariga) gur ru-ub-bu-u2 ša gišba-ri2-ga’, ‘9 gur 2 bariga the increase of the bariga-standard vessel’ appears. It was proposed that this text referred to the sample measurement of grain and then multiplication by a change rate to produce the difference between the old standard and the new standard. Sample measurement underlines the importance of metrological lists and tables, while the change rate calculation suggests the importance of numerical tables in assessing approximate value out of measured value.

In the grain shipment texts , values were added and subtracted as well. Rounding can offer some insight into this process. Ashm 1923-340 , a tabular account from the grain production archive possibly from Larsa , even though it reports activity around Ur , presents addition of both grain and area carried out by bureau officials during the reign of Hammu-rābi . This text, discussed in Chap. 6, shows partitioning of measurement values into upper and lower levels. This is suggested by their appearance over two lines. The partitioning of numbers into upper and lower levels is probably due to an instrument such as an abacus being used to facilitate calculation. The scribe in question added each level up separately and then appended lower level calculation to upper level calculations. Thus, by rounding 5 ban up to 1 bariga, the author simplified the statement in the lower level, which in turn made it easier to append lower and upper levels together. The author of this text still divided lower and upper values between lines, even if all values would fit on one line, which shows that he did understand these as two separate numbers appended together.

Rounding helps to illustrate that these partitions were based on natural splits within the capacity system in Ashm 1923-340 , and this was supported by a similar division in Ashm 1922-277 , as well as mistakes in carrying numbers between columns in YBC 04224 . Granularity was limited to sila measurement units in Ashm 1923-340 and sar measurement units in Ashm 1922-277 , while a split was shown at mana in YBC 04224 . The scribes who produced Ashm 1923-340 and Ashm 1922-277 preferred to work with values found on the capacity and area metrological lists , rather than with gin and še measurement values which were present on the metrological list of weight but used to subdivide capacity and area as well. The scribe who produced YBC 04224 exhibited different mistakes at and above fractions of mana and at and below gin, suggesting that gin measurement values and below were treated differently from mana measurement values. All this helps to confirm that, at Larsa , metrological lists and probably tables of capacity started at fractions of sila measurement units rather than gin, as at Nippur . Perhaps the lists of area at Larsa commenced at fractions of sar measurement units, as expected.

Rounding numbers shows practices limited to individual archives as well. NBC 05474 and NBC 06339 describe brick disbursements to various foremen as part of a canal construction project, while NBC 09050 describes the production of bricks during this canal’s construction. Because these texts are part of the Lu-igisa archive, which belongs to the bureau of irrigation and excavation , brick production, disbursement and then construction probably fell within the parameters of this professional setting. This suggests a professional environment which involved both brick deliveries and earth transport in the bureau of irrigation and excavation . Rounding in this archive suggests standard values were used. With NBC 05474 , 47 5/6 sar seemed to be rounded down to 47 2/3 sar. The author perhaps preferred to work in halves and thirds of sar rather than sixths of sar. Thus, rounding here was based on archival preference.

Rounding also helps to show the utility of coefficients. The use of coefficients in calculation suggests conformity as well. This was especially clear for the bureau of irrigation and excavation . There, standard labor rates exhibited in mathematical texts were exploited to assess labor in YBC 12273 , although it is equally clear that labor rates used in YBC 12273 differed slightly from the mathematical text in how they were defined. Standard wage rates appeared in Riftin 1937: no. 114 and 116 to help assess project costs. NBC 06763 and Ashm 1922-290 seemed to use standard rates in assessing volume from observed man-days. Both texts seemingly used rounding as a means to offset uncertainty associated with estimated volume measurement values, as witnessed in YBC 12273 .

Similarity was seen between interest revenue suggested in A.26371 , tax revenue in AO 08493 , change rate calculation in LB 1075 and conversion calculation in YBC 04265 . Interest was presented in mathematical texts like VAT 08521 . However, calculation with tax, change and conversion rates are not visible in any scribal curriculum . The same can be said of labor calculation and yield calculation—while labor is present in mathematical texts, it is never associated with yields in these texts. This suggests an economization of mathematical practice in which one algorithm that was associated with one situation was implemented in other, similar situations. While mathematical texts present, for instance, interest calculations , it can be hypothesized that a form of commentary accompanied the presentation of an algorithm , explaining the processes involved in carrying out interest calculations and then the applicability of this algorithm and its components as well as differences in setup and results. The table as a medium for practice assisted the teacher in expressing this economization of practice . Thus, a teacher presenting interest rates could follow this with a discussion of other value assessment methods like change rates , conversion rates and tax rates . This could easily account for much of the missing evidence of scribal education.

The algorithm presented with VAT 08521 is useful across milieu. A.26371 is attributed to Šēp-Sîn, the merchant overseer of Larsa , AO 08493 was probably produced by a conveyor working for Sîn-rāmā , LB 1075 is attributed to the archive of a notable named Sîn-iddinam , while YBC 04265 is attributed to Nabi-Šamaš B . Conformity is produced by the very algorithms presented to student scribes, while at the same time conformity is exhibited by the numbers and measurement values used in economic texts, betraying the importance of the various metrological and numerical lists and tables that were learned in the elementary scribal education. This suggests the universality of education in the kingdom of Larsa and begs the question as to how universal these educations were.

The portrait of education provided by rounding numbers , when combined with other errors and mistakes , is one of a single, uniform metrological and numerical system manifested in numerous microcultures within the kingdom of Larsa . A somewhat uniform elementary mathematical education, whether classroom or professional, appears or is alluded to throughout the kingdom of Larsa . This education consisted of at least familiarity with metrological lists , but more likely incorporated the memorization of metrological lists and tables as well as numerical tables . Familiarity with metrological lists and tables is suggested by the awareness of measurement inconsistency seen in the grain storage bureau , as well as the appearance of truncated values along natural splits in measurement systems. Indeed, these truncated values, as well as other rounding based on natural splits, showed training with tools such as an abacus probably occurred during this relatively universal elementary stage when metrological lists and tables were memorized. An abacus would also speak for the use of partial-SPVN . Familiarity with metrological and numerical tables was suggested with YBC 07473 and NBC 08014 to produce equivalencies, and especially YBC 12273 to calculate volume . Some training with non-regular numbers late in the elementary education or in an early advanced education was suggested with YBC 07473 . This is the image of elementary education produced from Larsa and its hinterland, including the city of Zarbilum .

The elementary education is evident across different milieus, in household , bureau and merchant environments. Outside of Larsa , evidence is much sparser. At Isin , scribes were at least familiar with metrological lists for capacity and measurement inconsistency associated with measurement values presented on these lists . For Ur , it can be difficult to tell what education was common because much of Ur’s administration was conducted by merchants and bureaucrats from Larsa , such as the notable Gimillum , as described in Appendix 2.G. This points to the possibility that both Ur and Larsa may have had similar educational processes. This would be because much temple estate administration was eventually moved to Larsa , so that officials and merchants from Ur would have been active in Larsa even while officials and merchants of Larsa would have been involved in administering Ur .

However, as was seen in Chap. 2, while numerical and metrological systems may have been based on a uniform mathematical culture, both metrological and numerical lists and tables differed between, and even within centers. Occasionally glimpses are offered into the format of these lists , such as at Larsa where the standard metrological list for capacity could very well have commenced with sila measurement units, rather than gin as at Nippur . The potential significance of these microcultures is witnessed in Chap. 7 where the ability of both student and master to deviate from these lists was observed. Metrological lists and tables formed the building blocks of measurement, as shown by measurement inconsistency in the grain storage bureau . Both metrological lists and tables as well as numerical tables formed the building blocks of calculation as seen in Chap. 5, and with NBC 08014 and YBC 07473 especially in Chap. 8. In Chap. 7, differences in standards also suggested different cultures and nuanced microcultures as well as the ability to cross between these cultures, both in measuring standards and estimating standards. The different cultures and microcultures affected how scribes evaluated both measured and calculated data, indicating that cultural and microcultural differences must have had an effect on economic activity.

Advanced education was less universal. The texts presented here show two possible forms of advanced education : an education which made use of mathematical texts evidenced in a school setting, and then a professional education that built on an elementary and perhaps even early advanced education in a school setting. Mathematical coursework probably centered around different algorithms and components of these algorithms that could be applied in multiple settings. This was quite visible when examining value assessments as seen in and around Larsa . Interest rates are well attested in the texts, while tax, change and conversion rates are not attested at all. The same can be said of labor rates with excavations and harvest. We might even speak of classes of algorithms as understood by the ancient scribes; in the cases presented here these would be rate and labor class algorithms based on the tables described in Sect. 8.4.

However, as noted in Chap. 8, commentary did not necessarily have to take place within a school setting. A student and teacher needed only to be familiar with an algorithm in order to be receptive to commentary . It may be hypothesized, for instance, that algorithms were taught in a limited advanced education , presented as rate and labor problems in a school education, and then the applicability of each, and the components of each, were explored within professional environments. With labor, as seen in excavation texts like YBC 04663 , a student would have become familiar with the need for and production of project statements to produce initial inputs into labor calculations , the use of labor rates to evaluate labor out of these project statements , and then the use of wage rates to evaluate wages out of this labor. These components of labor calculations could well be applied, with commentary , to the harvest of grain. This is, of course, conjecture, because of the nature of the proposed oral commentary —it would not be visible as mathematical practice in the scribal tradition, only in the economic texts. Thus, a specialized professional education is suggested, reflecting further microcultural activity as well.

This specialization is certainly seen in the economic texts, as has been described. A more universal curriculum is witnessed with merchant activity, as suggested by both equivalency and interest calculation , both of which may have made use of rate tables. Even these calculations, however, show knowledge not entirely exhibited in the academic texts. Thus, even here, some learning was probably acquired in situ rather than entirely in the classroom.

The economic texts can help to understand the specialized education of scribes active in the bureau of irrigation and excavation , just as seen above with merchants . Coefficients used to define man-days of labor and then pay to be expended in excavations must have been memorized as part of this education. The addition of a multiplication table for 2:13:20, the reciprocal of 27 found on MS 3974 , suggests that some additional numerical tables may have been memorized as well. It is difficult to say whether these were learned in a more universal elementary stage or as part of an advanced, professional education . Whatever the case, they were certainly incorporated into a professional environment. The coefficients used in YBC 12273 were similar to those found on the coefficient lists and problem texts, but depths associated with them differed, suggesting that part of a scribe’s training in this bureau was in how to apply these coefficients in the field. Finally, scribes memorized algorithms used to calculate the volume of an excavation, and then labor and pay associated with this volume, like the models present in YBC 04663 and YBC 07164 discussed in Chap. 8. Commentary associated with these texts could explain the apprenticeship texts BM 085211 and BM 085238 —they were produced in a professional environment, building on practices learned in an elementary and early advanced education . The numerous mistakes in transformation suggest the importance of this professional education , even with reference to the elementary education : it afforded the scribe time to hone skills learned in the classroom while introducing them to professional practice. In addition, the centesimal system was either learned with the professional education as well, or learned in the elementary scribal education and reinforced at this point. Education in the bureau of irrigation and excavation was probably a professional education that built on exercises common to the elementary and perhaps to some advanced educations but adapted through commentary for more specific purposes: irrigation and excavation works. This underlines the practicality of education as espoused by Michalowski (2012).

This phenomenon was not limited to the bureau of irrigation and excavation . A similar professional education existed in the grain storage bureau . Basic algorithms similar to interest rate problems were learned, which were then built on or adapted to calculations of sample measurement and change rate in an advanced, professional education . Text layouts were learned at this point. A similar structure could be applied to the scribes active in the grain production archive and grain harvest archive , where yield rates and then labor rates respectively must have been learned in an advanced, professional education , building on or adapting algorithms learned to compute labor, similar to those of the bureau of irrigation and excavation . Text layout was learned here as well. Indeed, each bureau or archive presented in Chap. 4 seems to have presented its own practical knowledge, whether this was in addition to a traditional advanced scribal education or in lieu of this advanced education . Bureaus and archives required adherence to local standards that each actor had to accept. Each bureau provides evidence of Michalowski’s hypothetical limited practical Old Babylonian scribal training , which produces the possibility of both a regional variety in education as well as variety based on profession and social status, and is largely borne out by the microcultural activity witnessed in this study.

How, then, was rounding numbers presented in the scribal curriculum ? How was it adapted for administrative purposes? On the one hand, rounding was a tool used by scribes to limit potential discrepancy in a text. The scribes themselves were aware of observational, conceptual and systematic errors in the forms of measurement inconsistency , estimation and the use of approximate reciprocals for non-regular numbers in the texts. Rounding helped to mitigate these errors. On the other hand, as YBC 04224 and NBC 06763 show, rounding could be used to simplify a current or future calculation. This is exhibited in YBC 04698 statement 14, as presented above, where 59:24:30, the SPVN equivalent to 5 ban 9 1/3 sila 4 1/2 gin, was probably rounded up to 1, corresponding to 1 bariga in order to simplify calculation. Rounding, as a means to estimate value, also assisted movement between the cultures and microcultures as they existed in the kingdom of Larsa . For instance, rounding by means of truncation would have limited measurement inconsistency associated with remeasurement of value by different standards. However, while attempting to mitigate discrepancy , these same rounded values showed the same uncertainties associated with error . Thus, rounding itself produced error . Errors and mistakes associated with approximation and rounding on YBC 07473 underline this.

The rather systematic use of rounding suggests that it was an important feature of record keeping, one with customs and practices associated with it, that existed to help navigate the uncertain realm of error , to simplify calculations, to mitigate possible discrepancies between expected and actual values and, perhaps, to cross between cultural and microcultural boundaries. Thus, rounding numbers was fostered in the education system by the nature of the system itself: rounding was a means to limit discrepancies associated with adapting the mathematical world of potential into the real world. Rounding numbers was a means to cope with the system of mathematics as it was taught in the various educational milieus, to offset errors of estimation and evaluation in this system in order to produce realistic and usable results. Rounding, which is associated with observational errors , conceptual errors and systematic errors , proves that some errors could be based on agreed customs and practices in order to maintain acceptable deviations from a truth.