At their elementary levels the two are mirror images of each other.
A number expressed as two squared can also be described as the area of a square with 2 as the length of each side.
In the earliest surviving traces of a counting system, numbers are built up with a repeated sign for each group of 10 followed by another repeated sign for 1.
Arithmetic cannot easily develop until an efficient numerical system is in place.
Many of the theorems derive from Euclid's predecessors (in particular Eudoxus), but Euclid presents them with a clarity which ensures the success of his work.
It becomes Europe's standard textbook in geometry, retaining that position until the 19th century.The most famous equation in classical mathematics is known still as the Pythagorean theorem: in any right-angle triangle the square of the longest side (the hypotenuse) is equal to the sum of the squares of the two other sides.It is unlikely that the proof of this goes back to Pythagoras himself.There he establishes a philosophical sect based on the belief that numbers are the underlying and unchangeable truth of the universe.He and his followers soon make precisely the sort of discoveries to reinforce this numerical faith.Of the two Babylon is far more advanced, with quite complex algebraic problems featuring on cuneiform tablets.A typical Babylonian maths question will be expressed in geometrical terms, but the nature of its solution is essentially algebraic (see a Babylonian maths question).Archimedes is a student at Alexandria, possibly within the lifetime of Euclid.He returns to his native Syracuse, in Sicily, where he far exceeds the teacher in the originality of his geometrical researches.Since the numerical system is unwieldy, with a base of 60, calculation depends largely on tables (sums already worked out, with the answer given for future use), and many such tables survive on the tablets.Egyptian mathematics is less sophisticated than that of Babylon; but an entire papyrus on the subject survives.