Babylonian mathematics

Summary: Ancient Babylon had an advanced utilitarian mathematics from which we inherited sexagesimal timekeeping.

Our knowledge of Babylonian mathematics (2100–200 BCE) is based on extensive mathematical calculations found on clay tablets in the area of Mesopotamia (now Iraq), surrounding the ancient city of Babylon between the Tigris and Euphrates rivers. Because only a fraction of the tablets have survived—and only a small fraction of those have been translated—our knowledge of the depth and breadth of Babylonian mathematics is limited. Mathematics historian Otto Neugebauer likens the situation to tearing a few random pages out of a few textbooks and then trying to reconstruct a representation of modern mathematics. Nonetheless, Babylonian mathematics did involve complicated mathematics, and was used primarily to solve practical problems. These mathematical problems ranged from arithmetic calculations, to algebraic rules, to geometrical formulas, to numerical ideas.

Babylonian Number System

The Babylonian number system was sexagesimal, using both a place value notation based on powers of 60 and a base-10 grouping system for numbers between 1 and 59 within each place value.

Traces of their sexagesimal notation remain today in the recording of time (hours, minutes, seconds) and the measurement of angles (degrees, minutes, seconds). Their numbers were written in cuneiform, or the use of a triangular stylus to make wedges on a clay tablet. A vertical line represented unity and a horizontal wedge mark represented a 10.

For example, within each place value, the number 57 would be represented by 5 horizontal wedges and 7 vertical lines. Expanding the example, a cuneiform number represented in modern form as “3, 4, 57” was equivalent to

The Babylonians had neither a symbol for zero as a placeholder nor a symbol to designate the “decimal” point in their sexagesimal fractions. Writing and reading numbers required the Babylonian mathematician to understand the problem’s context and the use of a space to represent either an “empty” place value or shift to fractional place values. Thus, the previous number, “3,4,57,” possibly was equivalent to:

To avoid ambiguity, modern translations of these numbers would be first “3,0,4,57” or “3,4,57,0,0” or “3,4;57” or “0;3,4,57” respectively, where the semicolon separates whole numbers from fractional numbers. Tablets from the Seleucid period (300 BCE) did include a special symbol that played the double role of a placeholder (zero) and the separator between two sentences.

Babylonian Arithmetic

Using the sexagesimal system, the Babylonians were able to add, subtract, multiply, and divide numbers. Their computations were complemented by the use of extensive tables. Their multiplication tables had products ranging from 1×1 through 59×59, and seeming somewhat unusual, they had access to multiplications tables for “1,20” (or 80), “1,30” (or 90), “1,40” (or 100), “3,20” (or 200), “3,45” (225), and even “44,26,40” (or 160,000). Some of this can be explained by looking at their tables of reciprocals for working with fractions. For example, one table includes the deceptive notation 1÷1,21=44,26,40, with the latter value actually being “0;0,44,26,40.”

The Babylonians produced extensive tables of squares and cubes, tables of square sides and cube sides (square and cube roots), and sums of squares and cubes. When a table side-value was not available, the Babylonians approximated roots using an interpolation process based on averaging and division; this process was quite fast, producing 26-decimal accuracy in five iterations.

Babylonian Algebra

Though without an algebraic notation, the Babylonians solved numerous types of algebraic equations. Each solution involved the replication of a formulaic prescription represented by a step-by-step list of rules. In effect, their prescriptions invoked algorithms, which were usually specific to a stated problem and not generalized to a class of problems.

For example, consider this Babylonian problem: the area and two-thirds of the side of my square have I added and it is 0;35. In modern notation, their step-by-step solution was: 1, the unit, you take; two-thirds of 1, the unit, is 0;40: Its half is 0;20 and 0;20 you multiply 0;6,40, you add 0;35 to it and 0;41,40 has 0;50 for its square root. 0;20 that you multiplied with itself, from 0;50 you subtract and 0;30 is the side of the square.

In modern mathematics, this same problem would involve solving the quadratic equation:

The steps in this problem also can be interpreted using geometrical algebra, where the square is “completed” in a manner similar to the derivation of our general quadratic formula.

In their solution of special types of algebraic equations, the Babylonians made extensive use of their tables of the sums of squares and cubes, especially if the equation was of the third or fourth degree. Some of their solutions to algebraic problems were quite sophisticated. For example, one problem involved a system of equations of the form

Its solution using substitution would normally lead to a single-variable equation involving x⁶, but the Babylonians solved it by viewing it as a quadratic equation in x³.

Babylonian Geometry

Dominated by their work with algebraic ideas, the Babylonians’ geometry focused on practical measurements such as the calculation of lengths, areas, and volumes. Again, the Babylonians used prescriptive formulas. For example, to calculate a circle’s circumference, they multiplied the diameter by 3, implying their value of π was 3. For the circle’s area, they squared the circumference and divided by 12, which is equivalent to our modern formula A=πr² if the correct value of π had been used.

Mathematics historians credit the Babylonians with the division of a circle into 360 degrees. Neugebauer suggests it is related to their Babylonian mile, a measure of long distance equal to about 7 miles. This measure evolved into a time unit, being the time it took to travel this distance. After noting that 12 of these time units equaled a full day or one revolution of the sky, the Babylonians subdivided their mile into 30 equal parts for simplicity, leading to 12×30=360 units in a full circle.

The Babylonians computed areas of right triangles, isosceles triangles, and isosceles trapezoids, as well as the volumes of both rectangular parallelepipeds and some prisms. They had difficulties with certain three-dimensional shapes, being unable to compute correctly the length of the frustum of a pyramid (they claimed it was the product of the altitude by the average of the bases).

The Babylonians did know some general geometric relationships. For example, they knew that perpendiculars dropped from the vertex of an isosceles triangle bisected the base, that corresponding sides of similar triangles were proportional, and that angles inscribed in a semicircle are right angles. The Babylonians used this knowledge to solve difficult geometrical problems, such as their determination of the radius of a circle circumscribing an isosceles triangle.

Evidence suggests that they knew a precursor of the Pythagorean theorem. One cuneiform tablet, known as Plimpton 322 (c. 1700 BCE) after American collector George Arthur Plimpton, includes sexagesimal numbers written along a square’s side (30) and diagonal (“42,25,35” and “1; 24, 51, 10”).

The latter number is both the product of the other two numbers and a good approximation of the square root of 2 (1.414214). Also, in the Plimpton 322 collection, some of the tablets contain tables of Pythagorean triples (a²+b²=c²), arranged with increasing acute angle of the associated right triangle.

In 2017, researchers publishing in the journal Historica Mathematica suggested that the Plimpton 322 tablet was in fact a form of a trigonometric table. If true, the discovery would mean that the Babylonians developed trigonometry at least a thousand years earlier than the field otherwise began to emerge, with similar tables not developed for three thousand years. The researchers determined that rather than using and angle-based approach with sines and cosines, Babylonian trigonometry was based on exact ratios, as in the lengths of triangle sides, within their base-60 system. However, other experts cautioned that the research was essentially speculative, and that it cannot be definitively proven that Babylonians understood what is today known as trigonometry.

Signs of Advanced Mathematical Thinking

For the most part, Babylonian mathematics was utilitarian, being tied to solving practical problems. Nonetheless, interpretations of some of the tables on the clay tablets suggest that the Babylonians occasionally explored theoretical aspects of mathematics. Examples include their tables of Pythagorean triples and tables of exponential functions (which perhaps were used to compute compound interest in business transactions). Also, the Louvre tablet (300 BCE) includes two series problems

but historians do not suggest the Babylonians knew general series formulas such as

Specific to number theory, mathematics historians point to the cumbersome nature of the Babylonians’ sexagesimal system, making it difficult to explore ideas such as factors, powers, and reciprocals. Some suggest that this is symptomatic of the Babylonian’s reasonable choice of 3 for π, rather than the fraction

equal to the more complicated repeating expression “3; 8, 34, 17, 8, 34, 17,… .”

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