An Introduction to Mathematical Reasoning Numbers, Sets and Functions

Numbers originally arose for the purpose of counting. This makes use of the positive integers ℤ+ = {1,2,3,…}, often called the natural or counting numbers. In order to write down numbers we need some system of numerals, a system of symbols to represent numbers. At first the notation used was simply some sort of tally or repeated mark but in time more elaborate systems of numerals were devised leading eventually to our present Hindu–Arabic decimal system of numerals which makes use of ‘cypherization’ (i.e. different symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 for different numbers) and a ‘place value system’ (e.g. in 121 the first ‘1’ denotes 100, the ‘2’ denotes 2 × 10 = 20, and the second ‘1’ denotes 1 so that the number represented is the sum 100 + 20 + 1).

An effective place value system requires a symbol to denote the absence of a positive digit and it is for this purpose that a symbol like ‘0’ first appeared (so that, for example, 102 and 12 represent different numbers). In due course zero was recognized as a number in its own right and then much later negative integers were introduced giving the complete system of integers ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …} in which any two elements may be added or subtracted.

However, numbers are also used for purposes of measurement, originally of lengths but then of many other quantities.