Chapter 3: Notations and Numbers: Where our number system come from
Number Systems
· Roman Numerals: letters
· Babylonian sexagesimal system: base-60
· Egyptian number symbols
· Mayan base-20 numbers
· Greek numerals: did not use positional notation. They used their letters.
· Indian number symbols
o “Hindu-Arabic” numerals, ten symbols currently used to denote decimal digits.
o The first traces of what eventually became the modern symbolic system appeared around 300 BC in the Brahmi numerals. By AD 100 there are records of the full Brahmi system. They developed into Gupta numerals (because of the Gupta Empire expansion) and then into Nagari numerals.
o The positional notation was in use in India from about 400 onwards.
o Key Indian mathematicians: Aryabhata, Brahmagupta, Mahavira, and Bhaskara.
· The Hindu System
o Al-Khwarizmi’s On Calculation with Hindu Numerals of 825 made the Hindu system widely known in the Arab world.
· The Dark Ages? Introduction to Europe
o Leonardo of Pisa, also known as Fibonacci, published Liber Abbaci in 1202. In this book, he introduced Hindu-Arabic number symbols to Europe. It also includes and promoted the horizontal bar in a fraction.
o Around 1585, Dutchman Simon Stevin looked to the Italian arithmeticians of the Renaissance period, and the Hindu-Arabic notation transmitted to Europe by Fibonacci. He then tried to find a system that combined the best of both, and invented a base-10 analogue of the Babylonian system: decimals.
· Negative Numbers
o Early in the first millennium, the Chinese employed a system of “counting rods” instead of the abacus. Red rods for terms added and black rods for terms subtracted.
o Hindu mathematicians found negative numbers useful to represent debts in financial calculations.
“Just as you won’t learn to walk by always using a crutch, you won’t learn to think sensibly about numbers by relying solely on a calculator.”
*Rule of Three à From the oldest surviving Chinese mathematics text, Chiu Chang
· Roman Numerals: letters
· Babylonian sexagesimal system: base-60
· Egyptian number symbols
· Mayan base-20 numbers
· Greek numerals: did not use positional notation. They used their letters.
· Indian number symbols
o “Hindu-Arabic” numerals, ten symbols currently used to denote decimal digits.
o The first traces of what eventually became the modern symbolic system appeared around 300 BC in the Brahmi numerals. By AD 100 there are records of the full Brahmi system. They developed into Gupta numerals (because of the Gupta Empire expansion) and then into Nagari numerals.
o The positional notation was in use in India from about 400 onwards.
o Key Indian mathematicians: Aryabhata, Brahmagupta, Mahavira, and Bhaskara.
· The Hindu System
o Al-Khwarizmi’s On Calculation with Hindu Numerals of 825 made the Hindu system widely known in the Arab world.
· The Dark Ages? Introduction to Europe
o Leonardo of Pisa, also known as Fibonacci, published Liber Abbaci in 1202. In this book, he introduced Hindu-Arabic number symbols to Europe. It also includes and promoted the horizontal bar in a fraction.
o Around 1585, Dutchman Simon Stevin looked to the Italian arithmeticians of the Renaissance period, and the Hindu-Arabic notation transmitted to Europe by Fibonacci. He then tried to find a system that combined the best of both, and invented a base-10 analogue of the Babylonian system: decimals.
· Negative Numbers
o Early in the first millennium, the Chinese employed a system of “counting rods” instead of the abacus. Red rods for terms added and black rods for terms subtracted.
o Hindu mathematicians found negative numbers useful to represent debts in financial calculations.
“Just as you won’t learn to walk by always using a crutch, you won’t learn to think sensibly about numbers by relying solely on a calculator.”
*Rule of Three à From the oldest surviving Chinese mathematics text, Chiu Chang
Chapter 4: Lure of the Unknown: X marks the spot
“I found a stone but did not weight it… when I had added a second stone of half the weight, the total weight was 15 gin.”
- Old Babylonian Tablet
So, this chapter is about the beginning of symbolic math, or as we know it; algebra.
“Algebra is about the properties of symbolic expressions in their own right; it is about structure and form, not just number.” (The initial “al”, Arabic for “the”, indicates its origin.)
Al-jabr: adding equal amounts to both sides of an equation.
It called my attention how there were public battles to solve the cubic equations in Renaissance Italy (1500’s). Important figures: Niccolo Fontana (Tartaglia), Girolamo Cardano, Lodovico Ferrari.
Diophantus of Alexandria, around 250, was one of the first ones to use symbols in place of unknown numbers. (Arithmetica)
Other important figures in symbolic notation: François Vieta (consonants), William Oughtred (x for multiplication), Robert Recorde (= for equality; two parallel lines), Thomas Harriot (><), Descartes (√), Nicolas Chuquet, Gauss (x2).
- Old Babylonian Tablet
So, this chapter is about the beginning of symbolic math, or as we know it; algebra.
“Algebra is about the properties of symbolic expressions in their own right; it is about structure and form, not just number.” (The initial “al”, Arabic for “the”, indicates its origin.)
Al-jabr: adding equal amounts to both sides of an equation.
It called my attention how there were public battles to solve the cubic equations in Renaissance Italy (1500’s). Important figures: Niccolo Fontana (Tartaglia), Girolamo Cardano, Lodovico Ferrari.
Diophantus of Alexandria, around 250, was one of the first ones to use symbols in place of unknown numbers. (Arithmetica)
Other important figures in symbolic notation: François Vieta (consonants), William Oughtred (x for multiplication), Robert Recorde (= for equality; two parallel lines), Thomas Harriot (><), Descartes (√), Nicolas Chuquet, Gauss (x2).