Han Erim October 7, 2015
PREFACE
Right Side number bases and numbers have been defined with this study and have won a basement.
When we make a definition as Right Side in mathematics, naturally at first it is needed to explain what Left Side means in mathematics. Left side represents normal order of number bases that we still use. The table below represents the order of numbers in left side. Normally we handle and use the number bases and numbers as we see in this classic table.
As you know, in our daily life we use the number base 10. Number base 10 consists of 10 digits (0,1,2,3,4,5,6,7,8,9). And also, Binary (Number
Base 2) and Hexadecimal (Number Base 16) are the ones that are commonly used in programming and mathematical calculations.
A numbers opening rule according to its base is as below:
NUMBER BASE 1
Even it is not used in Left Side, for Right Side number base 1 is very important. Number base 1 is composed of one element but a second helper number is needed to express it. For this, number “0” is used. The numbers in base 1 can be shown in two formats.
As seen on the table, in the first display form we write 1 back to back as the value of number. In the second display form we add 0 back to back as the value of number and we add 1 in front.
In this study, both of the display forms have been used. For instance, on the main table above, in respect of suitability to the table second display form has been preferred.
The numbers which are the elements of a number base always have smaller values than the value of base. For instance, Number Base 2 (0,1), Number Base 6 (0,1,2,3,4,5), Number Base 16 which is called Hexadecimal consist of numbers (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). But due to necessity, and as an exception, in number base 1, 1 is used both in number value and base value. So; in Base 1, representation is made in such way 1_{1 , }10_{1, }111_{1 }, etc.
it is unpractical for mathematical calculations. But although, we can make calculations using this number base. For example, the operation 3+2=5 is like below in Base 1:
According to 1st display format:
111_{1}+11_{1} = 11111_{1}
According to 2nd display format:
1000_{1} + 100_{1} = 100000_{1}
When base is shown in writing, it is not possible to fall into confusion about display format. 11111_{1}=100000_{1}
OBTAINING RIGHT SIDE NUMBER BASES
The comparative table below is showing the difference between Left Side number bases and Right Side number bases (The values on the table have been regulated according to the decimal system ignoring the number writing rule.). As can be seen the Right Side numbers forms with the rule:
1_{1} = 2 * 1_{2} = 3 * 1_{3} = 4 * 1_{4} = _{................} = (n1) * 1_{n1} = n * 1_{n} k
Below, we see the format prepared proper to number writing rule of the table above. Adding a geometric comment to the table, the numbers were placed vertical with respect to the values they carry.
All the numbers in the Right Side table stays in the length of number 1_{1} representing the base 1.
Right Side Numbers’ Location on the "Length 1" and Its Numerical Value:
A number element at right side locates in the ratio of the value it includes on a stable point on Length 1 and number value/base measure. For instance, number 2 belonging to Base 6, at the point 0,333... of Length 1, number 3 belonging to base 8 is at the point 0.375 of Length 1. Location point value is at the same time number’s real numeric value.
(Same number’s Left Side equivalent are in the format 2_{6 }= 2 and 3_{8 }= 3 )
On the table below we see the main characters of Left Side and Right Side numbers relatively.
Right Side Numbers’ Fractional Notation
At Right Side, fractional notation format is obtained by writing own base values to the number elements’ denominator values. On the table below, we see the 2 different notation types of Right Side Number Bases as natural and fractional. On the fractional notation, on the numerator and denominator Decimal number system is used for simplicity. But additionally, suitably to the number writing rule, numbers can be shown at Right Side, too.
Right Side numbers’ Natural Notation: 2_{6 }, 8_{13 }, 58_{66 }, ... Right Side numbers’ Fractional Notation: 2/6 , 8/13 , 58/66 , ...
THE CONNECTION BETWEEN LEFT SIDE AND RIGHT SIDE
With the aim of creating a bridge between both of the sides, the number 1_{1} belonging to the number base 1 at the Left Side and the number 1_{1} belonging to the number base 1 at the Right Side are assumed to be equal to each other. Consequently, "1 Length" of Right Side is equal to number 1 at Left Side.
A Number’s Left Side and Right Side Components
A number’s value that is equal to and bigger that 1 is Left Side component of the number and the values smaller than 1 is number’s Right Side components.
19_{10} = 19 (Left Side)
The part staying at right side of the comma and smaller than 1 is the value 0,375. Now let us show that the number 0,375 is a number belonging to Right Side number bases: 0,375 = 3/8
Because of that, in Right Side number bases, a number’s value occurs in the format number/base, the fraction 3/8, is equating to number 3 in Right Side Base 8.
So, according to Right Side: 3_{8}_{ }= 3/8 = 3/10_{8} =0,375.
19,375 = 19 + 3/8 = 19_{10} (Left Side)+ 3/10_{8}(Right Side)
19,375 = 19_{10} (Left Side) + 3_{8 }(Right Side)
Results:
A number’s value equal to and bigger than 1 is Left Side component, its value smaller than 1 is number’s Right Side component and the value of the number consists from the sums of these two components. Left Side and Right Side numbers all together include all Rational Numbers.
2,3333333..... = 2 + 0,333333.... = 2 + 1/3 =_{ } 2_{10 }(Left Side) + 1_{3 }(Right Side) 8,5 = 8 + 0,5 = 8 + 1/2 = 8_{10 }(Left Side) + 1_{2 }(Right Side)
By its definition, irrational numbers do not have a certain solution in Right Side. In irrational numbers, numerator and the denominator in the Right Side fraction take values extending to the infinity. Because of that, we cannot write the infinite numbers, we can express the irrational numbers in an approximate formation, ending the numerator and the denominator values at a point.
= 1,4142135623730......= 1 + 0,4142135623730...... ≈ 1_{ }+ 6625109/15994428 = 1_{10 }(Left Side) + 6625109_{15994428 }(Right Side)
PI = 3,1415926535....= 3 + 0,1415926535.... ≈ 3 + 29629644/209259755 PI=3_{10 }(Left Side) + 29629644_{209259755} _{ }(Right Side)
Some last sentences on Right Side
Number Bases and numbers
Han Erim


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