1. A Sense of Number

Anthropologists tell us that early cultures often counted only to two. Any number larger than two was simply called “Many.” Some Bushmen in South Africa counted to ten using sums: 10 = 2 + 2 + 2 + 2 + 2.

To trade, they used Tallying (One-to-One Correspondence). The word “Tally” comes from the French tailler (“to cut”).

The Oldest Artifacts

• The Wolf Bone (30,000 B.C.): Found in Czechoslovakia. A 7-inch bone with 55 notches grouped in fives. It suggests early counting systems were based on the 5 fingers of a hand.
• The Ishango Bone (17,500 B.C.): Found in Africa. It features notches that include Prime Numbers (11, 13, 17, 19). It may have been a lunar calendar.

2. Wood & String Records

Before paper, people used wood and string to record debts and data.

The British Tally Stick

For centuries, the British Exchequer recorded loans on notched hazelwood sticks. A notch for £100 was as wide as a thumb; for £1000 it was as wide as a hand.

• The Foil: Kept by the Bank.
• The Stock: Kept by the lender (hence the term “Stockholder”).

Quipus & Mayans

• Inca Quipus: Knotted strings used in Peru. They used a Decimal System (10⁰, 10¹, 10²). The knot position indicated value. An empty space meant Zero.
• Mayans: Used a Vigesimal (Base-20) system. They used a Shell symbol for Zero and dots/bars for numbers (e.g., dot = 1, bar = 5).

3. Hieroglyphs & Papyrus

Egypt is “the gift of the Nile.” To measure land and tax the people, they developed a Decimal (Base-10) system around 3500 B.C.

The Symbols

It was an Additive System (non-positional). They repeated symbols to create numbers:

Example: To write 1,232, they would draw:
1 Lotus + 2 Coils + 3 Heel Bones + 2 Strokes.

4. The Alphabetic System

The Greeks used their 27-letter alphabet (24 standard + 3 obsolete: digamma, koppa, sampi) to represent numbers.

For large numbers (10,000+), they used the letter M (Myriad). e.g., βM = 20,000.

Gematria (Number Mysticism)

Because letters were numbers, every word had a numerical value. This led to a mystical practice called Gematria.

5. The Cradle of Zero

While Greece focused on Geometry, Ancient India revolutionized Arithmetic. The world owes the “Decimal Place Value System” and the invention of Zero (Shunya) to India.

Key Figures

• Aryabhata (476 A.D.): Wrote the Aryabhatiya. He calculated π ≈ 3.1416 and stated that the Earth rotates on its axis.
• Brahmagupta (628 A.D.): Invented ‘0’ and defined the rules for computing with Zero (e.g., A number minus itself is Zero).

India’s Prodigy: Ramanujan

Srinivasa Ramanujan (1887-1920) was a self-taught genius who attributed his mathematical insights to the Goddess Namagiri.

He left behind 3 notebooks filled with thousands of theorems on infinite series and number theory that mathematicians are still studying today.

6. Mathematical Giants

A quick summary of the prominent figures and inventions covered in this chapter.