History of Numbers Flashcards
What are the key aspects of number systems?
- Symbolic structure
- Base of number system
- Place value or Symbol value
- Integers and Fractions
- Methods of calculation and arithmetic
- Influence of or on uses of mathematics
- Representation of zero negatives and irrationals
What is the earliest evidence of awareness of number?
Notches carved on (Lebombo) bone from 37000 years ago - which may record the phase of the moon
What was the earliest evidence of Egyptian use of numbers?
Tags showing quantities of commodities from around 3000 BCE
Why did writing develop in ancient Egypt?
Writing developed to record supplies when cities were built rather than to record spoken language
What are the two types of Egyptian writing?
- Hieroglyphics used for inscriptions on stone
- Hieratic script used on papyrus
What was the base of the Egyptian number system and did it use positional notation?
It was based on powers of 10 but did not use positional notation
How were numbers written in the Egyptian hieroglyphic system?
Each power of 10 had a different symbol and each symbol was repeated as often as necessary with no need for zero
How did the Egyptian hieratic script differ from hieroglyphic numbers? Why?
It used abbreviations instead of repeating many symbols. It was easier to create more complex symbols writing on papyrus than carving on stone.
How did the Egyptians represent fractions?
They used only unit fractions (1/2, 1/3, etc) except for two thirds (2/3) and represented reciprocals with an oval symbol (modern day representation is an overbar)
What were the two main surviving sources of Egyptian mathematics?
- Ahmes Rhind Papyrus from about 1640 BCE
- Moscow Papyrus from about 1890 BCE
What method did Egyptians use for multiplication and division?
They used repeated doubling or halving sometimes multiplying or dividing by 10
What was the Babylonian writing system and how was it recorded?
They used a cuneiform script written on soft clay tablets that were baked in the sun
What was the base of the Babylonian number system and how was it structured?
They used a sexagesimal base 60 place value system alternating base 10 and base 6
What was a major limitation of the Babylonian number system?
Before the fourth century BCE there was no placeholder for zero so numbers could be ambiguous
Why might the Babylonians have used base 60?
Possibly due to its easy factorisation or its use in astronomical records
What mathematical operations did the Babylonians use tables for?
Multiplication tables of reciprocals and square root approximations
Why was Greek mathematics mostly concerned with geometry?
The discovery of incommensurable lengths meant that numbers were inadequate for describing geometrical magnitudes
How were numbers represented in Classical Greek mathematics?
They used a ciphered system with successive letters of the alphabet representing numbers
What was the Chinese number system and what issue did it have?
It was a base 10 place value system but lacked a symbol for zero making it ambiguous
When and where did Hindu-Arabic numerals originate?
They developed in India around 600 BCE and were derived from symbols used in the third century BCE
Who introduced base 10 notation to Europe?
Leonardo of Pisa also called Fibonacci through his book Liber Abaci
How did the Renaissance contribute to the development of algebra?
Algebra was written in progressively more abbreviated forms leading to modern symbolic notation
Who found an algebraic solution to cubic equations?
Del Ferro in the early 16th century but he did not publish his results
How did Cardano contribute to solving cubic equations?
He published the general solution after learning of Del Ferro’s work despite promising not to publish Tartaglia’s results
