Cosmopoeia
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Number theory

The tetractys and the Cartesian coordinate system

The four basic operations of arithmetic in the four quadrants of a coordinate system. To understand the primes, you have to switch back and forth between calculating and drawing.

A tetractys is also contained in the four basic operations of arithmetic. Draw a Cartesian coordinate system:

  • The two axes carry the natural numbers, in the full coordinate system the negative ones too. From the origin they correspond to addition and subtraction.
  • Each of the four quadrants is a multiplication table and a division table at once. Division and multiplication depend on each other.
  • The cells within the quadrants are coloured by the three kinds of divisibility, exactly as in the division table.
k = 1: the number itselfdivisiblecoprimecommon factor, not divisible

One quadrant: the division table, coloured by divisibility. In the full coordinate system this pattern repeats, mirrored, in all four quadrants.

The number line itself and the three kinds of divisibility together give the four basic figures from The key to the tetractys.

Painting: half-length portrait of a man with long dark hair, moustache and white collar
René Descartes, portrait after Frans Hals (Paris, Louvre). The Cartesian coordinate system is named after him. In his Géométrie (1637) he joined algebra and geometry by describing curves through equations.

Four quadrants, four signs

The coordinate system with four quadrants as we draw it today arose only after Descartes. He himself worked almost only with positive values, for in his time negative numbers were still regarded by many as “false” numbers. Only with them does the cross of the axes become complete, and the four quadrants acquire an order of their own, shown by the signs.

I+ · + = +II− · + = −III− · − = +IV+ · − = −xy

If one multiplies the two coordinates of a point, the quadrant decides the sign. The two quadrants with equal signs lie opposite each other and give plus, the two with unequal signs give minus. So the picture shows why minus times minus gives plus.

The axes divide the plane into four fields, and two of them belong together each time. In miniature this is again the scheme met often on this website: a centre, the origin, two opposites, plus and minus, and four fields in which they combine. A point in the third quadrant is the mirror image of a point in the first, reflected in the origin, and its product stays the same. What in calculation is a rule one learns by heart is in the drawing a symmetry one sees.

Points instead of digits

The message of this chapter is one of method. To really understand the primes, you have to switch in your mind from arithmetic to geometry and back. Numbers must be replaced by points or other figures that stand in a ratio to one another. Only in this way can you escape the illusions of a counting system. The Pythagoreans, who calculated with pebbles, already knew this. Pure calculation often obscures what a simple drawing reveals.

The coordinate system is reminiscent of Peter Plichta’s prime number cross and deals with the same subject, but is built quite differently.

Sources and further reading

Sources

  • René Descartes, La Géométrie, appendix to the Discours de la méthode, Leiden 1637

Further reading

  • Peter Plichta, Das Primzahlkreuz (The prime number cross), Düsseldorf 1991