Prehistoric Mathematics and Tools

Prehistoric Mathematics

• Early humans had a basic understanding of quantity, recognizing "one," "two," and "many."
• The concept of abstract symbols for numbers developed over time.
• Evidence of early mathematical thinking includes notched bones from Africa dated 20,000-25,000 years old.
• Mathematics existed before written records.

The Ishango Bone

• A bone artifact from Africa dated to 20,000 BC.
• Contains markings that may represent a calendar, a lunar phase counter, or a number system.

Paleolithic Period

• The longest part of prehistoric times.
• Humans lived in caves and used natural materials for shelter and clothing.
• Focus on survival through hunting and gathering.
• Development of tools like daggers, spears, and fire.
• Distinction between "one" and "two" emerged.

Neolithic Age

• Transition to food production (agriculture).
• Increased civilization with settlements and villages.
• Development of pottery, carpentry, weaving, and metalworking.
• Inventions like the wheel, wagon, hammer, tents, and huts.
• Concepts of trade, private property, and quantity emerged.

Egyptian Mathematics

• Used a system of hieroglyphics for numerals.
• Developed a base-10 system with symbols for 1, 10, 100, 1,000, and so on.
• Used unit fractions for division.
• The Rhind Papyrus is a key source for understanding Egyptian mathematics, written around 1650 BC.
• Demonstrates knowledge of arithmetic, geometry, and algebra.
• The Eye of Horus Papyrus shows their system of unit fractions, based on the Eye of Horus symbol.
• Egyptians approximated the area of a circle by multiplying the diameter by 8/9 and squaring the result.
• Constructed the pyramids, showcasing their understanding of geometry and volume calculation.

Egyptian Multiplication

• A method of multiplying numbers using doubling.
• Example: 12 x 15
  • Left side: 1, 2, 4, 8
  • Right side: 15, 30 , 60, 120
  • Select numbers on the left side that add up to the multiplicand (12): 8 + 4
  • Add the corresponding numbers on the right side: 60 + 120 = 180
  • Result: 12 x 15 = 180

Egyptian Division

• Fractional notation developed for market and trade needs.
• Eye of Horus symbol was used to represent fractions (half, quarter, eighth, sixteenth, thirty-second, sixty-fourth).
• Example: Divide 3 loaves between 5 people.
  • Each person receives 1/3 + 1/5 + 1/15 = 3/5 of a loaf.

Notable Egyptian Mathematicians

• Ahmes: Scribe who copied the Rhind Papyrus, providing valuable insights into Egyptian mathematics.
• Abu’l-Hasan Ibn Yunus: Prominent astronomer and mathematician, known for his meticulous calculations and the lunar crater named after him.

Babylonian Mathematics

• Mesopotamia, the "Cradle of Civilization."
• The Sumerians developed the oldest known writing system: cuneiform script.
• Early development of mathematics for land measurement, taxes, and calendrical calculations.
• Used a sexagesimal system (base 60) for their number system.
• Developed a place-value system for numbers, unlike other ancient civilizations.

Sumerian and Babylonian Number System:

• Used symbols for one, ten, and sixty for counting.
• Used cuneiform symbols for numerals, written with a stylus.
• Had a placeholder symbol functioning like zero.
• This system was advantageous for its divisibility by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 60.
• Continues to influence our current timekeeping and angle measurements.

Babylonian Clay Tablets

• Show evidence of their knowledge of geometry and calculation of areas, volumes, and quadratic equations.
• The Plimpton 322 tablet suggests possible knowledge of Pythagorean triples (a² + b² = c²), predating Pythagoras.

Babylonian Contributions

• Developed the place-value system, critical for modern mathematics.
• Invented the concept of zero as a placeholder.
• Made significant contributions to algebra, geometry, and astronomy.
• Laid the foundation for later mathematical advancements in various civilizations.