The Saga of Mathematics: Babylonian Mathematics and Cuneiform Systems

Historical Foundations of Babylonian Civilization

• Geographic Context: The Babylonians inhabited Mesopotamia, a fertile plain situated between the Tigris and Euphrates rivers.

• Civilizational Succession: Babylonian society eventually replaced the Sumerian and Akkadian civilizations.

  • The Sumerians: This group was responsible for building cities and developing comprehensive systems for law, administration, post, and irrigation.

  • The Akkadians: Invaded the region around $2300\,BC$ and integrated with the Sumerian population. They are credited with inventing the abacus and developing foundational methods for addition, subtraction, multiplication, and division.

• Power Shifts: The Sumerians revolted against Akkadian rule, regaining control by $2100\,BC$. By $1800\,BC$, Hammurabi, King of Babylon, established power over the entire empire of Sumer and Akkad, founding the first Babylonian dynasty.

• Chronological Definition: While the empire was not always the primary cultural center, the term "Babylonian" is used to describe the Mesopotamian region from $2000\,BC$ to $600\,BC$.

The Cuneiform Writing System

• Etymology and Style: The term "cuneiform" originates from the Latin word cuneus, meaning "wedge." The writing consists of wedge-shaped symbols made by pressing a stylus into wet clay tablets.

• Medium Constraints: Because curved lines were difficult to draw on clay, the use of a stylus led to the characteristic linear/wedge-shaped symbols.

• Survival and Decipherment: Thousands of baked clay tablets have survived.

  • Georg Friedrich Grotefend ($1775-1853$): A German schoolteacher who originally deciphered the script as a result of a drunken wager with friends.

  • H.C. Rawlinson ($1810-1895$): Formally re-deciphered the script in $1847$.

• Mathematical Record: Over $300$ tablets specifically containing mathematical content have been discovered.

The Babylonian Sexagesimal Number System

• Base System: The Babylonians utilized a positional system with a base of $60$, known as the sexagesimal system.

• Positional Concept: In a positional system, the value of a symbol is determined by the position it occupies within the numerical representation.

• Internal Grouping (1 to 59):

  • For numbers within the base group, they used a simple grouping system.

  • A single vertical wedge represents $1$.

  • A wide horizontal wedge (chevron) represents $10$.

  • Example: Fifty-nine ($59$) is represented by five chevrons followed by nine vertical wedges.

• Large Numbers: For values exceeding $59$, a "digit" is shifted to the left, with each position increasing by a factor of $60$. Therefore, the symbol for $1$ could also represent $60$ depending on its placement.

• Notation and Conversion:

  • Standard notation for transcription: $(d_n, ..., d_1, d_0; d_{-1}, ...)_{60}$.

  • Example: $(3, 25, 4)_{60}$ is calculated as $3 \times 60^{2} + 25 \times 60^{1} + 4 \times 60^{0} = 10800 + 1500 + 4 = 12304$.

System Limitations and Modern Representation

• Drawbacks:

  • Lack of a sexagesimal point (equivalent to a decimal point), creating ambiguity in scale.

  • No symbol for zero until approximately $300\,BC$, when a placeholder symbol was introduced.

  • Context was essentially required to determine the specific value of many numbers.

• Modern Transcription Standards:

  • Commas ($,$) separate "digits."

  • Semicolons ($;$) indicate the fractional part (the sexagesimal point).

  • Example 1: $(25, 0; 3, 30)_{60}$ calculated as $25 \times 60 + 0 + \frac{3}{60} + \frac{30}{60^{2}} = 1500.05833...$.

  • Example 2: $(10, 20; 30, 45)_{60}$ calculated as $10 \times 60 + 20 + \frac{30}{60} + \frac{45}{60^{2}} = 620 + 0.5 + 0.0125 = 620.5125$.

  • Example 3: $(5; 5, 50, 45)_{60}$.

  • Note: Neither commas nor semicolons existed in the original cuneiform script.

Babylonian Arithmetic and Multiplication

• Arithmetic Tools: Tablets show evidence of highly developed math including squares (for $1$ to $59$), cubes (up to $32$), square roots, cube roots, sums of squares and cubes, and reciprocals.

• Addition and Subtraction: These operations functioned similarly to modern methods, but "carrying" and "borrowing" occurred in units of $60$ rather than $10$.

  • Example addition: $(10, 30; 50)_{60} + (30; 40, 25)_{60}$.

  • Since $40 + 50 = 90$, which is $(1, 30)_{60}$, you write $30$ and carry the $1$ to the next column.

• Multiplication Tables: Tables often listed multiples $1p, 2p, ..., 20p$, and then skipped to $30p, 40p, 50p$.

  • Distributive Law: To find a value like $47p$, they would sum the results for $40p$ and $7p$: $a \times (b + c) = a \times b + a \times c$.

• Multiplication via Squares: Products could be found using formulas:

  • $ab = \frac{(a+b)^{2} - a^{2} - b^{2}}{2}$

  • $ab = \frac{(a+b)^{2} - (a-b)^{2}}{4}$

  • Example: $11 \times 12$. Using the second formula: $\frac{(11+12)^{2} - (11-12)^{2}}{4} = \frac{23^{2} - 1^{2}}{4} = \frac{529 - 1}{4} = 132$. In sexagesimal: $(2, 12)_{60}$.

Babylonian Division and Reciprocals

• Method: Division was viewed as multiplication by the reciprocal of the divisor: $a \div b = a \times (1/b)$.

• Reciprocal Tables: Babylonians maintained tables of reciprocals for calculation.

  • $2 \div 3 = 2 \times (0; 20)_{60} = (0; 40)_{60}$.

  • $44 \div 12 = 44 \times (0; 5)_{60} = (3; 40)_{60}$ (Note: $5 \times 44 = 220$, which is $3 \times 60 + 40$).

  • $25 \div 9 = 25 \times (0; 6, 40)_{60} = (2; 46, 40)_{60}$.

• Repeating Sexagesimals: For divisors with no finite representation (e.g., $1/7$), they used approximations and noted them as such (e.g., "7 does not divide").

  • $1/7 \approx (0; 8, 34, 17, 8)_{60}$.

Babylonian Algebra: Linear and Simultaneous Equations

• Capability: They could solve linear equations, systems of equations, quadratic equations, and some cubic equations.

• Problem Examples:

  • Linear Stone Problem: "I found a stone but did not weigh it; after I added to it $1/7$ of its weight and then $1/11$ of this new weight, I weighed the total $1\,mina$. What was the original weight?"

  • Units: $1\,mina = 60\,sheqels$; $1\,sheqel = 180\,se$.

  • Equation: $(x + x/7) + \frac{1}{11}(x + x/7) = 60\,sheqels$.

  • Result: $48.125\,sheqels$ ($2/3\,mina, 8\,sheqels, 22.5\,se$).

  • Simultaneous Rings Problem: Involving two silver rings where fractional parts are broken off to equal $1\,sheqel$ and the remaining weights are equal: $\frac{1}{7}x + \frac{1}{11}y = 1$ and $x - \frac{1}{7}x = y - \frac{1}{11}y$.

  • Result: $x = 4.375\,sheqels$, $y = 4.125\,sheqels$.

Quadratic Equations and Completing the Square

• Methodology: Quadratic equations were solved using the method of "completing the square."

• Case Study: $x^{2} + \frac{2}{3}x = 0; 35$.

  • Step-by-Step Procedure:

    1. Take half the coefficient of $x$: $\frac{1}{2} \times \frac{2}{3} = \frac{1}{3} = (0; 20)_{60}$.

    2. Square it: $(0; 20)_{60}^{2} = (0; 6, 40)_{60}$.

    3. Add the right-hand side constant to the square: $(0; 6, 40)_{60} + (0; 35)_{60} = (0; 41, 40)_{60}$.

    4. Find the square root: $\sqrt{(0; 41, 40)_{60}} = (0; 50)_{60}$.

    5. Subtract half the coefficient of $x$: $(0; 50)_{60} - (0; 20)_{60} = (0; 30)_{60}$.

  • The side of the square is $(0; 30)_{60} = 0.5$.

The Babylonian Square Root Algorithm

• Naming: Also known as Heron's method (after the $1st$ Century AD Greek mathematician) or Newton's method.

• History: Credited to the Babylonians around $1900\,BC$. A similar method was used in India as early as $800\,BC$.

• Algorithm Steps:

  1. Make a guess ($g_1$).

  2. Divide the original number by the guess ($n/g_1$).

  3. Find the average of the guess and the result ($g_2 = \frac{g_1 + n/g_1}{2}$).

  4. Repeat the process using the average as the next guess.

• Example ($\sqrt{37}$):

  • Guess $6$.

  • $37/6 = 6.166...$

  • Average: $(6 + 6.166...)/2 = 6.0833...$

  • Repeating $3$ times yields accuracy to $15$ decimal places with a percentage error of approximately $9.691 \times 10^{-16}$.

Tablet YBC 7289 and the Square Root of 2

• Description: This tablet shows a square with side labeled $30$ (interpreted as $0; 30 = 1/2$).

• Diagonal Calculation:

  • The diagonal is labeled $(1; 24, 51, 10)_{60} = 1.4142129$.

  • The value below is $(0; 42, 25, 35)_{60} = 0.7070647$.

  • Comparing to $\sqrt{2} = 1.41421356...$ and $1/\sqrt{2} = 0.707106...$, the tablet shows sophisticated approximation.

Babylonian Geometry

• Geometric Concepts: Babylonians linked algebra and geometry, using terms like "length" and "area" but often mixing dimensions (summing lengths and areas together).

• Known Formulas:

  • Area of rectangles, right triangles, isosceles triangles, trapezoids, and parallelograms.

  • The Pythagorean Theorem.

  • Proportionality of sides in similar triangles.

  • Property of isosceles triangles: The line from the vertex to the base midpoint is perpendicular to the base.

• Circle Approximations:

  • Common value for $\pi = 3$.

  • Circumference estimated as $C = 3 \times d$.

  • Area estimated as $A = C^{2} / 12$.

  • A more sophisticated estimate of $\pi$ was $(3; 7, 30)_{60} = 3.125$.

Plimpton 322: The Pythagorean Tablet

• Catalog Details: Catalog #$322$ in the G.A. Plimpton collection, Columbia University ($1900-1600\,BC$).

• Significance: Demonstrates knowledge of the Pythagorean Theorem over a millennium before Pythagoras. It contains $15$ rows and $4$ columns.

• Column Structure:

  • Far Right: Line numbering.

  • Middle Columns: Hypotenuses and one leg of integral-sided right triangles.

  • Left Column: Values of $(c/a)^{2}$, representing the squares of the secant of angle $B$.

• Notable Exceptions/Errors in Tablet:

  • Line 2: Unexplained error.

  • Line 9: $(541)$ recorded where $(9, 1)_{60}$ and $(481)$ where $(8, 1)_{60}$.

  • Line 13: $25921$ recorded ($161^{2}$).

  • Line 15: Recording $53$, which is half the correct value of $106$.

• Trigonometry: The tablet is considered the oldest record of trigonometric functions, serving as a secant table for angles between $30^{\circ}$ and $45^{\circ}$.

Theoretical Origins of Base 60

• Proposed Theories:

  1. Divisibility: Theon of Alexandria suggested $60$ has many factors ($1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$), making fractional representation more convenient.

  2. Astronomy: Originating from a $360$-day Babylonian year; a higher base of $360$ was used and later lowered to $60$.

  3. Cultural Merger: A result of the merger between two distinct peoples, one using a decimal system (base $10$) and one using a base-$6$ system.