Number spirals and numeral systems

Number spirals and numeral systems

Beginning: Number spirals introduction

Numbers on number spirals can be represented in different number systems. For example, consider the numbers on the main axis of some spirals in different number systems. Table 1 shows the numbers on the main axis of the 2-spiral in binary, decimal and hexadecimal numeral systems.

Main axis of 2-spiral

Table 2 presents the numbers on the main axis of the 10-spiral in the same numeral systems.

Main axis of 10-spiral

Similarly, Table 3 presents the numbers on the main axis of the 16-spiral.

Main axis of 16-spiral

As can be seen from the tables above, each a-spiral in the number system with the base a will consist of separate turns with numbers that have the same number of digits in the positional system. Each subsequent turn of the numerical spiral for the number a in the number system with the base a adds one digit in the positional system. Each n-turn consists of numbers written using n+1 number of digits.

If we introduce the rule that unit arcs on one turn must be of the same length, then the number spiral will turn into a set of concentric circles. Each circle will contain numbers with the same number of digits. The introduction of such a rule violates the visual continuity of natural numbers.

Thus, each a-spiral is a graphical representation of natural numbers in the numeral system with the base a, which is written in the numeral system we have chosen. By default we use the decimal numeral system.

Continued: Number spirals and prime numbers.