Chapter 1: History of Mathematics
To think the thinkable—that is the mathematician’s aim. (C.J. Keyser)
Page 1: Origins & Primitive Counting
1. A Sense of Number
The term “Mathematics” comes from the Greek word mathemata, which originally meant “any subject of instruction or study.” Later, the Pythagoreans (600 B.C.) restricted it to mean Arithmetic and Geometry.
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.
Page 2: Sticks, Knots & The Fire
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).
Page 3: The Egyptian System
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:
| Value | Symbol Object |
|---|---|
| 1 | Vertical Stroke |
| 10 | Heel Bone (Arch) |
| 100 | Coiled Rope |
| 1,000 | Lotus Flower |
| 1,000,000 | God Heh (Hands raised) |
Example: To write 1,232, they would draw:
1 Lotus + 2 Coils + 3 Heel Bones + 2 Strokes.
Page 4: Greek Numerals & Gematria
4. The Alphabetic System
The Greeks used their 27-letter alphabet (24 standard + 3 obsolete: digamma, koppa, sampi) to represent numbers.
| Ones | Tens | Hundreds |
|---|---|---|
| α = 1 | ι = 10 | ρ = 100 |
| β = 2 | κ = 20 | σ = 200 |
| γ = 3 | λ = 30 | τ = 300 |
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.
- 666: The “Number of the Beast” is likely the sum of the Hebrew letters for “Nero Caesar”.
- Amen: The Greek letters for “Amen” (αμην) sum to 99 (α=1, μ=40, η=8, ν=50).
- Pompeii Graffiti: “I love her whose number is 545.”
Page 5: Indian Math & Ramanujan
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.
When the mathematician G.H. Hardy visited Ramanujan in the hospital, he remarked that his taxi number, 1729, was dull. Ramanujan immediately replied:
“No, it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways!”
1729 = 13 + 123 = 93 + 103
He left behind 3 notebooks filled with thousands of theorems on infinite series and number theory that mathematicians are still studying today.
Page 6: Hall of Fame (Summary Table)
6. Mathematical Giants
A quick summary of the prominent figures and inventions covered in this chapter.
| Person | Place | Invention / Contribution |
|---|---|---|
| Thales | Greece | First proofs in Geometry. |
| Pythagoras | Greece | Pythagorean Theorem (a2+b2=c2), Irrational Numbers. |
| Euclid | Alexandria | Wrote The Elements (Geometry Axioms). |
| Archimedes | Greece | Volume of Sphere, Value of π (Pi). |
| Aryabhata | India | Aryabhatiya, Value of π, Earth’s Rotation. |
| Brahmagupta | India | Invented 0. |
| Al-Khwarizmi | Persia | Father of Algebra (Al-Jabr). |
| Fibonacci | Italy | Fibonacci Sequence (1, 1, 2, 3…), Spread Hindu-Arabic numerals. |
| Ramanujan | India | Number Theory, Number 1729. |
Test Your Knowledge
Assess your understanding of Chapter 1: History of Mathematics.