Generation of a number system

The method to change from other base to the base 10 is

1. Take i=1, sum = 0 and the rightmost digit as the first digit
2. Repeat each step from 3 to 5 for digits from right to left
3. Calculate (i^th digit)×(base)^i-1 and store in digitvalue.
4. Store sum=sum+digitvalue
5. i=i+1 if (i+1)th digit from right exist else go to step 6
6. The value of the number in base ten is sum.

Now the problem arises when the base ten has no zero symbol. then the above procedure can be written as

1. Take i=1, sum = nil and the rightmost digit as the first digit
2. Repeat each step from 3 to 5 for digits from right to left
3. Calculate (i^th digit)×(base)ⁱ and store in digitvalue.
4. Store sum=sum+digitvalue
5. i=i+1 if (i+1)th digit from right exist else go to step 6
6. The value of the number in base ten is sum/base.

Let us generate a number system which has no zero in it.
Let the base of the number system be three (3). The symbols used are α, β and γ. Where α < β < γ. β = α + α, γ = β + α. The number system looks like this

⇒
⇓	Level 1	α	β	γ
	Level 2	αα	αβ	αγ
		βα	ββ	βγ
		γα	γβ	γγ
	Level 3	ααα	ααβ	ααγ
		αβα	αββ	αβγ
		αγα	αγβ	αγγ
		βαα	βαβ	βαγ
		ββα	βββ	ββγ
		βγα	βγβ	βγγ
		γαα	γαβ	γαγ
		γβα	γββ	γβγ
		γγα	γγβ	γγγ
	Level 4	αααα	αααβ	αααγ
		so.on.

In the above table the numbers increase from left to right and from top to bottom. First the numbers increase from left to right then move to one row below. The numbers in each succeeding level is the result of placing one digit before each number of the preceding level. First α is placed then β is placed then γ is placed. First level consists of 3 numbers, second level consists of 9 numbers and n^th level consists of 3ⁿ numbers. The process to convert the number to base 10 is the same as described above.


You can generate your your number system by this method. The number system may or may not have a zero.

1. Choose the symbols.
2. Define the law of precedence.
3. Describe the step size or relation between symbols.
4. Generate the number system in levels.
5. If the number contains zero. The first series in each level is same as the level before. Remove it.

Let us see how the method works for the above number system.
Step 1: The symbols are α, β and γ.
Step 2: α < β < γ.
Step 3: β = α + α, γ = β + α.
Step 4: Generate as in table.
Step 5: 0 is not present.

Let us convert some numbers from base three to base 10 where α=1.

βα = (β×γ^β + α×γ^α)/γ
βα = (2×3² + 1×3¹)/3
βα = (2×9 + 1×3)/3
βα = 21/3 = 7

γβα = (γ×γ^γ + β×γ^β + α×γ^α)/γ
γβα = (3×3³ + 2×3² + 1×3¹)/3
γβα = (3×27 + 2×9 + 1×3)/3
γβα = 102/3 = 34