Number Bases, Numbers

and

The Right Side of Mathematics

Han Erim

7 November 2015

FOREWORD

The Right Side in Mathematics is a different interpretation of number bases. In this interpretation, the number 1 in base 1 is the greatest number. The values of the number elements belonging to the bases are formed according to the rule number/base.

Right-side number bases and numbers are defined with this work and gain a foundation.

The Left Side of Mathematics

When we introduce a definition called the Right Side in Mathematics, it is naturally necessary first to explain what the Left Side in Mathematics means. The Left Side represents the normal ordering of number bases in the way we currently use them. The table below shows the ordering of numbers within the Left Side. Normally we handle and use number bases and numbers in the classical way that we see in this table.

As we know, in everyday life we use base 10. Base 10 consists of 10 elements (0,1,2,3,4,5,6,7,8,9). In addition, some bases such as Binary (base 2) and Hexadecimal (base 16) are number bases that are frequently used especially in programming and mathematical calculations.

Any number can be written in terms of any base. For example, the representations of the number "127" in some bases are as follows:

The expansion rule of a number with respect to its base is as follows:

In the examples, the expansions of the number 127 in bases 10 and 4 are shown.

BASE 1

Although Base 1 is never used on the Left Side, it is extremely important for the Right Side. Base 1 consists of a single element, but in order to express it we also need a second auxiliary number. For this purpose, the number 0 is used. Numbers in Base 1 can be represented in two ways.

As can be seen in the table, in the 1st representation we write as many 1’s in succession as the number value.

In the 2nd representation we append as many 0’s in succession as the number value and place a 1 at the front.

In this work, both representations are used. For example, in the main table above, the second representation is preferred for consistency with the table.

The digits that are elements of a given base always have values smaller than the base. For example, base 2 consists of (0,1), base 6 of (0,1,2,3,4,5), and base 16 called Hexadecimal of the elements (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F).

However, as a necessity and as an exception, in Base 1 the digit 1 is used both as the number value and as the base value. That is, in base 1, representations such as 1₁, 10₁, 111₁ are used.

Because it is impractical for mathematical calculations, base 1 is not used on the Left Side. However, it is still possible to perform calculations using this base.

As an example, the operation 3 + 2 = 5 in base 1 is as follows:

According to the 1st representation: 111₁ + 11₁ = 11111₁

According to the 2nd representation: 1000₁ + 100₁ = 100000₁

OBTAINING RIGHT-SIDE NUMBER BASES

The comparative table below shows the difference between Left-Side and Right-Side number bases.

(The values in the table are arranged according to the decimal system, ignoring the rules of number notation.)

As can be seen, Right-Side numbers are shaped by the rule 1₁ = 2×1₂ = 3×1₃ = 4×1₄ ... = (n-1)×1_n-1 = n×1_n.

Below is the same table prepared in accordance with the rules of number notation. By adding a geometric interpretation, the numbers are placed vertically in proportion to the values they carry.

Within the Right-Side table, all numbers remain within the length of the number 1₁.

The Position and Numerical Value of Right-Side Numbers on the Unit Length:

On the Right Side, a number is located at a fixed point on the Unit Length in proportion to its value and is positioned at numberValue/base.

For example, the number 2 belonging to base 6 is at the point 0.333... of the Unit Length, while the number 3 belonging to base 8 is at the point 0.375. The position point is at the same time the actual numerical value of the number.

(The counterparts of the same numbers on the Left Side are: 2₆ = 2 and 3₈ = 3.)

In the table below we can see the main characteristics of Left-Side and Right-Side numbers in comparison:

Fractional Representation of Right-Side Numbers

On the Right Side, the fractional representation is obtained by writing the base values of the digits as the denominator.

In the table below two different representations of Right-Side Number Bases can be seen: natural and fractional.

In the fractional representation, the numerator and denominator are written in the decimal system for convenience. However, they can also be written in accordance with number notation rules.

Natural Representation of Right-Side numbers: 2₆, 8₁₃, 58₆₆, ...

Fractional Representation of Right-Side numbers: 2/6, 8/13, 58/66, ...

The Connection Between the Left Side and the Right Side

In order to build a bridge between the two sides, it is accepted that the number 1₁ of base 1 on the Left Side is equal to the number 1₁ of base 1 on the Right Side.

Therefore, the Unit Length of the Right Side is equal to the number 1 on the Left Side.

The Left-Side and Right-Side Components of a Number

The value of a number that is equal to or greater than one is the Left-Side component of the number; the values less than one are the Right-Side component.

For example, in the number 19.375:

The integer 19 to the left of the decimal point is the Left-Side component: 19₁₀ (Left Side)

The value 0.375 to the right of the decimal point is the Right-Side component.

Since 0.375 = 3/8, it is expressed on the Right Side as 3₈.

Result:

19.375 = 19₁₀ (Left Side) + 3₈ (Right Side)

Other examples:

2.333... = 2 + 0.333... = 2 + 1/3 = 2₁₀ (Left Side) + 1₃ (Right Side)

8.5 = 8 + 0.5 = 8 + 1/2 = 8₁₀ (Left Side) + 1₂ (Right Side)

Right-Side Representation in Irrational Numbers

In irrational numbers, the numerator and denominator values in the Right-Side fraction extend to infinity.

Therefore, the fractions are expressed with approximate values.

≈ 1 + 0.4142 ≈ 1 + 6625109/15994428

That is:

≈ 1₁₀ (Left Side) + 6625109_15994428 (Right Side)

Similarly:

π ≈ 3 + 0.1415 ≈ 3 + 29629644/209259755

π ≈ 3₁₀ (Left Side) + 29629644_209259755 (Right Side)

A few final words on Right-Side number bases and numbers

The Right Side is a broad subject that needs to be investigated. Because of its interesting structure, I think it is possible to produce various algorithms and solutions using Right-Side number bases and numbers.

If mathematicians work on this subject, they will reach very interesting findings.

For me, the most beautiful result of Right-Side mathematics is that it led me to my work in physics that I named Alice Law.

Thank you for reading.

Han Erim