7.1 (Numbers)

• The idea that language is a type of code is easy to digest.
• The idea the numbers are seems harder to digest because mathematics is called the universal language.
• Numbers are the most abstract codes we deal with on a regular basis

The Number 10

-> There’s nothing special about the number 10. * It’s because people count using their fingers which usually come in 5’s or 10’s * Thus digit can refer to fingers or toes as well as numbers ** Five and fist have similar etymological roots, for example

Most historians think numbers were originally invented to count things like people, possessions, and commercial transactions.

Roman numerals are the only early number system to still be in common use * at first because they were easy to add and subtract (for European bookkeepers) * but multiplying and dividing Roman numerals is hard

Ancient Greeks were good with geometry but struggled with algebra because their inadequate early number systems prevented them from working with numbers in a sophisticated way.

Today’s Number System

-> Known as the Hindu-Arabic / Indo-Arabic system * It’s of Indian origin * Brought to Europe by Arab mathematicians

The Hindu-Arabic number system was different from previous systems in three ways: # It’s positional # A particular digit represents a different quantity depending on where it is found in the number # Where digits appear in a number is more significant than what the digits actually are # Both 100 and 1,000,000 have only a single 1 in them, but we know a million is larger than a hundred. # It lacks a special symbol for 10, unlike other early number systems # Other early number systems lack zero.

7.2 (Zero and Positioning)

-> one of the most important inventions in the history of numbers and mathematics

• Zero supports positional notation because it allows differentiation of 25 from 205 and 250.
• It eases mathematical operations that are awkward in non positional systems, like multiplication and division.

The Structure of Hindu-Arabic Numbers

-> revealed in the way we pronounce them:

• • Example:**

4,825 = Four thousands Eight hundreds Two tens and five . 4,825 =

Positioning

-> Each position in a multi-digit number has a particular meaning. So seven places would allow us to represent any number from 0 to 9,999,999. * each position corresponds to a power of 10 * fractional quantities to the right of the decimal point also follow this pattern, only with dividing by 10 instead of multiplying by it

The beauty of the Hindu-Arabic number system is that adding decimal numbers of any length involves a procedure that breaks down the problem into steps, each of which involves nothing more complicated than adding pairs of single-digit numbers.

Positional systems of notation work well for counting systems not based on ten, like base-eight systems.