The History of Numbers

~35,000 BCE: Tally Marks

The Lebombo bone from South Africa (dated ~35,000 BCE) contains 29 notches, possibly tracking lunar cycles. The Ishango bone from Congo (~20,000 BCE) has notches arranged in mathematical patterns, suggesting basic arithmetic understanding.

These bones are among humanity's earliest mathematical artifacts.

Early Counting: One-to-One Correspondence

Before abstract numbers, humans used one-to-one correspondence: one stone for each sheep, one notch for each day. Shepherds could tell if sheep were missing without knowing exact quantities—they'd match sheep to stones and notice leftovers.

~3400 BCE: Sumerian Numbers

The Sumerians in Mesopotamia developed one of the first written number systems. They used wedge-shaped marks (cuneiform) pressed into clay tablets. Their system was sexagesimal (base-60), which survives today in our 60-minute hours and 360-degree circles.

~3000 BCE: Egyptian Numbers

Ancient Egypt used hieroglyphic numbers with distinct symbols for 1, 10, 100, 1,000, etc. Their system was decimal (base-10) but not positional—the symbol for "10" meant ten regardless of position.

Example: 243 would be written as II ∩∩∩∩ ||| (2 hundreds + 4 tens + 3 ones) with symbols in any order.

~500 BCE: Greek Numbers

Ancient Greeks used two systems: acrophonic (similar to Roman numerals) and alphabetic (letters as numbers). α=1, β=2, γ=3, etc. This made Greek mathematics somewhat cumbersome—imagine doing multiplication with letters!

~500 BCE: Roman Numerals

The famous system using I, V, X, L, C, D, M. Elegant but impractical for calculation. Try multiplying XLVIII by XXIII mentally! Romans used abacus for actual math and numerals mainly for recording.

Limitation: No zero, no place value system, difficult arithmetic operations.

~500 CE: Indian Numerals & Zero

Indian mathematicians made revolutionary contributions:

• Place Value System: The position of a digit determines its value (ones, tens, hundreds, etc.)
• Zero as a Number: Not just absence, but a distinct mathematical entity. The Sanskrit word "shunya" (void) evolved into "sifr" in Arabic, then "zero" in English.
• Decimal System: Base-10 with only 10 symbols (0-9)

Mathematician Brahmagupta (628 CE) defined rules for arithmetic with zero and negative numbers.

~825 CE: Arabic Adoption

Persian mathematician Al-Khwarizmi wrote a treatise explaining the Indian decimal system, which Arab scholars adopted and refined. His name gave us the word "algorithm."

These became known as "Arabic numerals" in Europe, though they originated in India.

1202: Fibonacci's Liber Abaci

Italian mathematician Leonardo Fibonacci (Leonardo of Pisa) introduced Indian-Arabic numerals to Europe through his book Liber Abaci (Book of Calculation). He demonstrated how much easier arithmetic was with these numerals compared to Roman numerals.

Resistance was fierce! European merchants and scholars initially distrusted the new system. Italy even banned Arabic numerals for official documents for a time, fearing fraud.

1500s: Gradual Adoption

By the 1500s, Arabic numerals were winning out due to:

• The printing press making books more accessible
• Growing commerce requiring complex calculations
• Scientists needing efficient notation
• Obvious practical advantages

Negative Numbers (~600 CE onwards)

Indian and Chinese mathematicians first used negative numbers systematically, treating them as debts. European mathematicians were skeptical—how can you have "less than nothing"?

René Descartes (1637) called negative solutions "false roots." Full acceptance came only in the 1700s-1800s.

Fractions and Decimals

Fractions: Ancient Egyptians used unit fractions (1/n). Babylonians had sophisticated fractional notation.

Decimal Fractions: Chinese used decimal fractions by 400 CE. European adoption came much later, with Simon Stevin's De Thiende (1585) advocating decimal notation.

The decimal point itself wasn't standardized until the 1600s-1700s.

Irrational Numbers (~500 BCE onwards)

Greek mathematician Hippasus reportedly discovered that √2 cannot be expressed as a fraction—a shocking finding that allegedly got him thrown overboard by Pythagoreans who believed all numbers were rational!

Other famous irrationals: π ≈ 3.14159..., e ≈ 2.71828..., φ (golden ratio) ≈ 1.61803...

Imaginary Numbers (1500s-1600s)

What's the square root of -1? Italian mathematicians Cardano and Bombelli explored these "imaginary" numbers (called i). Combined with real numbers, they form complex numbers (a + bi).

Despite the name, imaginary numbers are essential for modern physics, engineering, and signal processing!

Mathematical Symbols Timeline

• + and − (1489): First appeared in German manuscript by Johannes Widmann
• × (1618): William Oughtred introduced multiplication symbol
• = (1557): Robert Recorde invented equals sign, using parallel lines "because noe 2 thynges can be moare equalle"
• ÷ (1659): Johann Rahn introduced division symbol (obelus)
• √ (1525): Christoff Rudolff first used radical sign
• ∞ (1655): John Wallis introduced infinity symbol
• π (1706): William Jones used π for pi; popularized by Euler
• e (1731): Euler introduced e for the natural logarithm base