base seven

Exploration between

Base 7 and Base 10

In this exploration, we investigate the definition, reason, quality, and quantity within elementary arithmetic,

as well as how to easily characterize the substantial differences between

Base 10 and Base 7.

Base 10

Used since the time of the Sumerians, the Base 10 system emerged because humans found it practical to simplify calculations using their hands.

Although other numerical systems have been used throughout history, humans have largely insisted on using Base 10 because it aligns naturally with the ten fingers on our hands. Today, Base 10 is the universal standard, forming the foundation of the mathematical language that governs almost every action in our daily lives.

The numbers in Base 10 are

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

The most common operations

ADDITION

SUBTRACTION

MULTIPLICATION

DIVISION

It is worth noting that these basic operations—whether simple or complex—can be performed using any numerical base, not just Base 10.

For example, they can be applied in other base systems, such as binary, octal, or hexadecimal.

Here is the Multiplication Table

from 1 to 10 of BASE 10

Inside the Yellow Table we note that the numbers are present in different quantity

In order to understand better, we begin to analyze the position of the numbers in the yellow table

Now we explore the graphic position of numbers on the previous yellow table

2025 - Base 10 Diagonal position numbers.jpg

Now we can explore and compare

the position that each single number

occupies in the Yellow Times Table

2025 - Base 10 Grapich position numbers Migliore.jpg

Base 7

The Base 7 system is composed of the following numbers

0 , 1 , 2 , 3 , 4 , 5 , 6

This is the Table from 1 to 100 in Base 7

Since we first started counting, we have been inclined not to question the concept of mathematical perfection. To clarify, let’s consider an example: the sum of 6+2 cannot be anything other than 8, and any different result is instinctively rejected. However, we must take into account that 8 is the resulting sum when the calculation is performed in Base 10.

The first encounter with a Base 7 number system is initially marked by the fact that a simple sum like 6+2 equals 11 instead of 8. This may lead to initial difficulties in performing mental arithmetic, but these challenges will be easily overcome once one becomes familiar with the mathematical properties of the Base 7 system."

This is the Multiplication Table

for the numbers that form Base 7

In examining the Yellow Table, we notice that each of the six numbers that make up Base 7 appears exactly six times. Meanwhile, zero is not involved in this numerical grid with its distinctive characteristics. In order to understand better, we begin to analyze the position of the numbers in the yellow table.

2025 - Base 7 Grapich position numbers.jpg

In Base 7, the positions of the numbers

reveal a unique symmetry:

the positions of 1 are equals

but opposites to those of 6

the positions of 2 are equals

but opposites to those of 5

the positions of 3 are equals

but opposites to those of 4

The number 0 is not present

within the inner structure

but instead appears

only on the outer perimeter

These characteristics mirror some of the patterns observed in Base 10, but Base 7 offers a distinctive harmonic numerical equilibrium. Using a Base 7 numerical table, composed of the digits 0 to 6, we observe the following results: Six numbers appear, each repeated six times, while the 0 exists only along the outer border, creating a natural boundary line. This harmonic structure is unique to Base 7.