## I.4 and II.1: What Do You Find in a Mathematical Paper? and The Origins of Modern Mathematics, From Numbers to Number Systems, Numbers in Early Mathematics, Pages 73 – 77.

In a mathematical paper one is trying to establish mathematical statements.  The most important mathematical statements are called theorems, there are also propositions, lemmas and corollaries.  A theorem is a mathematical statement that is regarded as intrinsically interested.  A proposition is like a theorem but is not as interesting as a theorem.  Lemmas are statements that are subgoals intermediate in the argument of the paper.  A corollary of a mathematical statement is a statement that follows easily from the mathematical statement.

A proof is used to establish these statements.  A proof is a sequence of mathematical statements written in a formal language with the first few statements the initial assumptions or premises, each remaining statement follows from the earlier ones by means of simple and clear logical rules and the final statement in the sequence is the statement that is to be proved.

Definitions, problems and conjectures are also included in a mathematical paper.

Prime palindromes are used to show a problem that is rather artifical from a mathematical point of view.  It depends upon the base that is used to solve the problem and is not even well defined, as trailing zeroes become leading zeroes of the reversed palindrome.  Thus, due to trailing zeroes, the reverse of the reverse of a palidrome may me a different number than one started with.

The next 80 pages are concerned with the origins of modern mathematics.

Numbers are initially used as adjectives.  When the same number or adjective is used on multiple nouns then it eventually takes a life of its own.  In this way it becomes an entity in itself and finally as a member of a system of such numbers.

In early mathematics fractions were not trivial to deal with.  In Egypt fractions tended to be thought of as sums of recipricals of integers.  In Mesopotamia fractions were written in base 60.  This is called a sexagesimal place value system.  Both of these systems had their issues.  An exact value for 1/7 was not posssible in the sexagesimal system.  An egyption surviving papyri includes the number 14 and 1/4 and 1/56 and 1/97 and 1/194 and 1/388 and 1/679 and 1/776 which in modern notation is the number 14 and 28/97.  And 4,000 years later I confirmed this, in a few seconds, with the software Mathematica!

The fact that we still have 60 minutes in an hour and 60 seconds in a minute goes back to the Babylonian sexagesimal fractions of about 4,000 years ago.