Egyptian Mathematics Study Guide

Origins of Mathematics and Primitive Counting

• Primitive Man and Early Systems:

  • Primitive humans, functioning as hunter-gatherers, developed basic counting methods.

  • Evidence of counting includes notches made on wolf bones.

  • Groups of pebbles and stones were used as physical counters.

  • These methods led to the development of simple grouping systems.

• Transition to Early Civilizations:

  • The discovery of agriculture necessitates advanced societal tools.

  • A need for calendars was established to manage farming cycles.

  • Economic activities such as trading, bartering of services and goods, and production required formal accounting.

  • Observing the universe became a necessity, all of which mandated the use of mathematics (Lewinter and Widulski).

Egyptian Civilization and Society

• Historical Context and the Nile:

  • Egyptian civilization reached a high point very early, specifically by $3000\,B.C.$

  • Agriculture was developed by leveraging the wet and dry periods of the year.

  • The Nile flooded during the rainy season; predicting this flooding was vital, leading to the development of astronomy to provide calendar information.

• Societal Needs for Mathematics:

  • Administration, tax systems, and the maintenance of armies required mathematical support.

  • Increasing complexity led to the requirement for written records and computations for bartering.

  • The evolution of counting created a demand for formal writing and numerals to record transactions.

• Cultural and Technical Achievement:

  • Hieroglyphics: A writing system established for both words and numerals.

  • Papyrus: A medium for keeping written records. Key documents include:

    • The Rhind/Ahmes Papyrus.

    • The Moscow Papyrus.

  • Engineering: Complex structures such as the Pyramids and the Sphinx were built. The Great Pyramid at Giza, built around $2650\,BC$, is cited as an extraordinary feat of engineering requiring significant mathematical knowledge.

The Egyptian Numeral System

• Simple Grouping System:

  • The Egyptians used a hieroglyphic grouping system where specific symbols represented powers of ten:

    • Stick: $1$

    • Heel bone: $10$

    • Scroll: $100$

    • Lotus flower: $1,000$

    • Bent finger or snake: $10,000$

    • Burbot fish or tadpole: $100,000$

    • Astonished man: $1,000,000$

• Writing Numbers:

  • Large numbers were formed by grouping these symbols appropriately.

  • Example: The number $243,526$ would be represented by combining the corresponding number of symbols for hundred-thousands, ten-thousands, thousands, hundreds, tens, and units.

• Arithmetic Operations:

  • Addition: Ten of any symbol are replaced by one of the next higher symbol (carrying).

  • Subtraction: If borrowing is needed, one of the next higher symbols is replaced by ten of the symbols for the current place value.

Egyptian Multiplication and Division

• The Didactic (Doubling) Method for Multiplication:

  • Egyptian multiplication is based on repeated addition through doubling.

  • Successive powers of two are generated: $1, 2, 4, 8, 16, 32, 64, 128, \dots$ ($2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6, 2^7, \dots$).

  • Fundamental Principle: Any integer can be written as a sum of unique powers of two.

    • $11 = 8 + 2 + 1$

    • $23 = 16 + 4 + 2 + 1$

    • $44 = 32 + 8 + 4$

    • $158 = 128 + 16 + 8 + 4 + 2$

• Multiplication Process (Example: $12 \times 17$):

  1. Start with two columns, the first starting at $1$ and the second with the multiplicand ($17$).

  2. Double both numbers until the left column reaches the largest power of two less than or equal to the multiplier ($12$).

     • $1 \quad 17$

     • $2 \quad 34$

     • $*4 \quad 68$

     • $*8 \quad 136$

  3. Identify the powers of two that sum to the multiplier ($12 = 8 + 4$). Star these rows.

  4. Add the corresponding numbers in the right column ($136 + 68 = 204$).

  5. Therefore, $12 \times 17 = 204$.

  • Theoretical Basis: This works via the distributive law: $12 \times 17 = (8 + 4) \times 17 = (8 \times 17) + (4 \times 17)$.

• Division Process (Example: $25 \div 4$):

  1. Start with columns headed by $1$ and the divisor ($4$).

  2. Double both numbers until the right column gets as close as possible to the dividend ($25$) without exceeding it.

     • $1 \quad 4$

     • $*2 \quad 8$

     • $*4 \quad 16$

  3. Subtract the right-side numbers from the dividend starting from the largest ($25 - 16 = 9$; $9 - 8 = 1$).

  4. The remaining number ($1$) is the remainder.

  5. The quotient is the sum of the left-side numbers corresponding to the subtracted right-side values ($4 + 2 = 6$).

  6. Result: $25 \div 4 = 6\,R\,1$.

Egyptian Fractions

• Concept of Unit Fractions:

  • The Egyptians primarily used unit fractions (numerator is $1$), such as $\frac{1}{3}$ or $\frac{1}{8}$.

  • The only non-unit fraction used regularly was $\frac{2}{3}$.

  • Unit fractions were denoted by placing an "eye" symbol over the number.

  • Special symbols existed for $\frac{1}{2}$ and $\frac{2}{3}$.

• Representation of Other Fractions:

  • Standard fractions were written as the sum of progressively smaller unique unit fractions.

  • Example: $\frac{3}{4} = \frac{1}{2} + \frac{1}{4}$.

  • Example: $\frac{7}{8} = \frac{4}{8} + \frac{2}{8} + \frac{1}{8} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8}$.

  • This practice continued into the Middle Ages in Europe.

• Methods for Writing Unit Fraction Sums:

  1. The Egyptian Method: Multiply the denominator by standard unit fractions ($\frac{1}{2}, \frac{1}{3}, \dots$) to find values summing to the numerator.

     • Example: $\frac{5}{6} \rightarrow \frac{1}{2} \times 6 = 3$ and $\frac{1}{3} \times 6 = 2$. Since $3 + 2 = 5$, $\frac{5}{6} = \frac{1}{2} + \frac{1}{3}$.

     • Example for $\frac{7}{18}$: Denominator is $18$. $\frac{1}{3} \times 18 = 6$. Need $1$ more. Use the Unit Fraction Rule: If you need $\frac{1}{n}$, use $\frac{1}{n \times \text{denominator}}$. So for $1$, use $\frac{1}{18}$. Result: $\frac{7}{18} = \frac{1}{3} + \frac{1}{18}$.

  2. Using Proper Divisors: Examine the divisors of the denominator for numbers that sum to the numerator.

     • For $\frac{11}{18}$, divisors of $18$ are $1, 2, 3, 6, 9, 18$. Since $11 = 9 + 2$, $\frac{11}{18} = \frac{9}{18} + \frac{2}{18} = \frac{1}{2} + \frac{1}{9}$.

     • Renaming Trick: If sum is impossible, rename the fraction. For $\frac{11}{15}$, rename as $\frac{22}{30}$. Divisors of $30$ are $1, 2, 3, 5, 6, 10, 15, 30$. Since $22 = 15 + 5 + 2$, $\frac{11}{15} = \frac{15}{30} + \frac{5}{30} + \frac{2}{30} = \frac{1}{2} + \frac{1}{6} + \frac{1}{15}$.

  3. Sylvester’s Method (Fibonacci's Method): Subtract the largest possible unit fraction from the given fraction and repeat with the remainder.

     • Originally developed by Fibonacci ($1175$-$1250$) and rediscovered by J.J. Sylvester ($1814$-$1897$) in $1880$.

     • Multiplier rule: If a fraction is $\frac{b}{a}$, choose multiplier $c$ such that ca > b.

  4. The Modern Method: Similar to Sylvester's. Set up the equation: $(M)(N) = D + C$, where $N$ is numerator, $D$ is denominator, $M$ is a chosen multiplier, and $C$ is a constant. Then divide by $(M)(D)$.

  5. The Splitting Method: Write the fraction as a sum of unit fractions with repetitions, then apply the formula: $\frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)}$.

Practical Applications of Fractions

• Comparison of Fractions: Egyptian fractions simplify comparisons. To compare $\frac{4}{5}$ and $\frac{7}{10}$, write them as sums:

  • $\frac{4}{5} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10}$

  • $\frac{7}{10} = \frac{1}{2} + \frac{1}{5}$

  • This shows $\frac{4}{5}$ is larger by exactly $\frac{1}{10}$.

• Fair Distribution: Using unit fractions results in more equitable division. Distributing $5$ loaves among $6$ friends ($\frac{5}{6}$ each):

  • Modern cut: Each gets $1$ piece of size $\frac{5}{6}$ (one person gets several small scraps, causing arguments).

  • Egyptian cut ($\frac{1}{2} + \frac{1}{3}$): Each person receives one $\frac{1}{2}$ loaf piece and one $\frac{1}{3}$ loaf piece, which is visibly identical and fair.

Egyptian Geometry and Measurement

• Development of Area Units: The Egyptians invented a square unit of area derived from a unit of length.

  • A unit square tile (e.g., $1\,foot$ by $1\,foot$) has an area of one square foot.

  • For a room $30\,feet$ by $20\,feet$, area is computed as $30 \times 20 = 600\,\text{square feet}$.

• Geometric Formulas:

  • Area of Rectangle: $A = LW$

  • Area of Triangle: $A = \frac{1}{2} BH$

  • Volume of Rectangular Solid: $V = LWH$

  • Volume of Pyramid: $V = \frac{1}{3} HB^2$

  • Volume of Frustum (Truncated Pyramid): $V = \frac{1}{3} H(B_1^2 + B_1 B_2 + B_2^2)$

• Area of a Circle:

  • Rhind Papyrus Problem #50: For a field with diameter $9\,khet$.

  • Method: Subtract $\frac{1}{9}$ of the diameter ($9 - 1 = 8$). Square the result ($8^2 = 64$).

  • Egyptian Formula: $A = (\frac{8d}{9})^2 = \frac{64d^2}{81}$.

  • Implied value of $\pi$: $\pi = \frac{256}{81} = 3 + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} \approx 3.1605$.

  • Measurement vocabulary: $1\,khet = 100\,cubits$. $1\,meter \approx 2\,cubits$. A setat is a square khet.

The Moscow and Rhind Papyri

• The Moscow Papyrus ($\approx 1850\,B.C.$):

  • Unknown author; contains $25$ problems.

  • Problem 10: Features complex subtractions and multiplications ($\text{Line 11}: \text{Subtract } \frac{2}{3} + \frac{1}{6} + \frac{1}{18} \text{ to get } 7 + \frac{1}{9}; \text{Line 12}: \text{Multiply } 7 + \frac{1}{9} \text{ by } 4 + \frac{1}{2}$).

  • Problem 14: Volume of a frustum. Scribe directs squaring $2$ and $4$, adding the squares to the product of $2$ and $4$ ($4 + 16 + 8 = 28$), then multiplying by one third of the height ($6/3 = 2$). Result: $56$.

  • Problem 6: Rectangular enclosure with area $12\,setat$, width is $\frac{3}{4}$ of length.

  • Problem 7: Triangle height is $2.5$ times the base, area is $20$.

  • Problem 17: Triangle height is $\frac{2}{5}$ of the base, area is $20$.

• The Rhind/Ahmes Papyrus ($\approx 1650\,B.C.$):

  • Purchased by Scottish Egyptologist A. Henry Rhind in Luxor in $1858$.

  • Written by scribe Ahmes, copying a document $200$ years older.

  • Contains $85$ problems; claims to be a "thorough study of all things, insight into all that exists, and knowledge of all obscure secrets."

  • Problem 41: Volume of cylindrical granary (diameter $9$, height $10$).

  • Problem 43: Amount of grain in cylindrical granary (diameter $9$, height $6$).

  • Problem 48: Comparison of a circle of diameter $9$ and its circumscribing square.

  • Problem 51: Area of a triangle (side $10$, base $4$).

Egyptian Astronomy and Calendar

• Observations:

  • Noticed the periodic trajectories of heavenly bodies and the night/day progression.

  • New moons occurred at approximately $28$-day intervals, providing a basis for time measurement.

  • Observed sun rising in East and setting in West.

• Needs for a Calendar:

  • Planting/harvesting crops.

  • Predicting Nile flooding (tied to the helical rising of the star Sirius).

  • Recording events like the Pharaoh's birthday.

• Geographic and Geometric Models:

  • Imagined the sky as a huge hemisphere and Earth as a flat disc sharing a boundary (the horizon).

  • Tracking stars (like the North Star at $1:00\,am$) involved two angles: a point on the horizon and the "angle of elevation."

• Calendar System:

  • $360$-day calendar.

  • Three seasons tied to Nile cycles.

  • $4$ months per season, $30$ days per month.

  • Additional "yearly five days" added for feasting and celebration.

  • The "opening of the year" was marked by Sirius.

Eye of Horus and Final Topics

• Symbolism:

  • The Oudjat (Eye of Horus) was a talisman for wholeness, health, vision, and fertility.

  • The different sections of the eye represented specific fractions used for grain and liquid measurements.

  • Total sum of Oudjat parts: $\frac{63}{64}$.

  • The missing $\frac{1}{64}$ was believed to be supplied by Thot, the God of scribes.

• Additional Mathematical Concepts:

  • Ciphered System: Egyptian hieratic numerals serve as an example.

  • Method of False Position: Used for solving linear equations (Exercises $19$ and $20$).