Analysis of number spirals

Beginning: Number spirals introduction

Movement along the turns of any a-helix is an addition table with the sequential addition of one unit to the previous number, starting from one. Movement along the rays of any a-spiral is a multiplication table with sequential multiplication of the number located at the beginning of the ray by the number a. These two movements are perpendicular, indicating the fundamental difference between addition and multiplication.

If we draw the axis of symmetry through the main axis of any a-spiral, then the symmetrical numbers on the turns form decomposition sums. For any a.n-turn this decomposition sum is equal to an+an+1. As an example, the decomposition sums are shown in 4.1-turn. The number of such sums, taking into account the sum of the numbers an and an+1 on the main axis, is equal to mn/2.

 Decomposition sums

This property of number spirals allows you to calculate the sum of natural numbers located on one a.n-turn. To do this, you need to multiply the sum of the decomposition by the number of such sums using formula (1), add a number located on the axis of symmetry, which is equal to half the sum of the decomposition, and subtract the number an+1, which refers to the following turn:

 Sum of numbers on a turn

Formula (3) allows us to derive the long-known formula for determining sums of all natural numbers from one to any number a. To do this, you need to determine the sum of the numbers on the zero turn of the a-spiral and add the number itself a:

 Sum of natural numbers

If we construct a number spiral for an infinitely large number (∞-spiral), the beginning of this spiral will differ from the number ray only in the absence of zero. The zero turn will consist of an infinitely large number of unit arcs, each unit arc will be limited by rays with an infinitely small angle value. The curvature of the coil will begin to appear in the region of very large numbers from the beginning of the spiral.

Continued: Number spirals and numeral systems.