Converting the Old Babylonian Tablet ‘Plimpton 322’ into the Decimal System as a Classroom Exercise: Practical Remarks on the Classroom Activity

Prerequisites. Numeral bases different from 10, fractions (where the numerator and the denominator can be algebraic expressions), Pythagorean Theorem, basic algebra (for Exercises 8 and 9 only).

Timing. The students can work in groups so that they can share (and check) their computations. In this case the activity should not exceed 90 minutes. The choice of whether or how much use of a calculator should be permitted is left to the instructor.

An introduction. The teacher can present some short base 60 to base 10 conversion exercises; for example, (0.(15)) ₆₀ = 15/60 = 1/4 = 0.25 and (0.(20))₆₀ = 20/60 = 1/3 = 0.33333… The second example is a rational number with a terminating sexagesimal representation that is not a decimal fraction, and hence it has infinitely many digits in base 10. This will help students see why we wish to construct the decimal analogue of Plimpton 322 (rather than just converting the original numbers to base 10): we want the displayed numbers to be decimal fractions so that they are not approximations, but exact values.

Bonus Exploration Section. Some of the more theoretical aspects of the algorithm appear at the end of the activity. Since the main part of the activity does not rely on them, some or all of these explorations could be assigned as required or optional exercises for individual or small-group completion; the instructor could also discuss them with the whole class or omit them altogether.

Discussion. The instructor may ask some or all of the following questions to prompt deeper mathematical reflection. These can be discussed first in small groups before leading a whole-class discussion that brings up the ideas we note below.

• What are some characteristics of fractions (or rational numbers) that have a terminating sexagesimal representation? Rational numbers with a terminating sexagesimal representation are those for which the least denominator divides a power of 60. This means that the prime factors of the least denominator can only be 2, 3, and 5. In contrast, the prime factors of the least denominator of a rational number with a terminating decimal representation can only be 2 and 5. For this reason, rational numbers with a terminating sexagesimal representation appear more frequently than decimal fractions, because in the least denominator they can also have the prime factor 3.
• What could have been the purpose of Plimpton 322? Historians have several different ideas about this. One idea is that it was a teaching resource, for example a source of drill problems or the solution to a problem. Or it may have been a multiplication table. Or a list of measurements. Or a list of Pythagorean triples. It could also have been a sort of trigonometric table. In the actual Plimpton 322 tablet, we have 15 rows giving a list of quantities which may have been associated with 15 non-skinny right triangles whose long leg is 1. For any non-skinny right triangle we may rescale its sides so that the long leg is 1 and then approximate it with one on the tablet for computational purposes.
• In the activity worksheet, both fractional and decimal representations of numbers were used. What are the advantages and disadvantages of these two different representations? Students will have different answers to this question, depending in part on how they prefer to do computations themselves. The main point to raise is that different computations may be easier to carry out, or to carry out correctly, in one representation than in the other.
• What are some of the arithmetical properties of base 10 representations? Do any of these also hold when we write our numbers in base 60? Sample answers might include the fact that the last digit of a natural number written in base 10 is 0 if and only if the number is divisible by 10. Base 60 is similar since the last digit is 0 (or blank) if and only if the number is divisible by 60; when written in cuneiform, however, it can be difficult to tell whether the last digit is 0 or not. In base 10, the last digit of a natural number can also be used to tell if the number is even or odd: the last digit is even if and only if the number itself is even. In base 60, the symbols written in the last place can also be used to determine if the number is even or odd, but this is complicated when we write out the number in cuneiform since that system has only two different symbols (or wedge-shapes) representing ten and one respectively. In contrast, there are 10 different digit symbols in the Hindu-Arabic base 10 system that we use today, one for each value between 0 and 9. Extending this idea to base 60, we would need 60 different digit symbols, one for each value between 0 and 59.