Even in base π, π is still considered an irrational number; using an irrational based doesn’t change the fundamental identity of whole numbers or irrational numbers, it just changes the way we write them.
Kinda. Technicaly no since an irrational number is a number that cannot be defined as a ratio of 2 existing rational numbers. Any number that can be represented in any rational base can by definition be represented as a ratio of somthing/base^n. This ignore the case of an irrational base but its practically useless cos any rational and most other irrational numbers will be irrational.
What u think ur trying to say is that some numbers cannot be represented in one base but can in another for example 1/3 can be represented as a decimal in base 3 but cannot jn base 10 ie u get 0.333(3 repeating forever).
Tieing back to floating point which uses base 2 u end up with simmillar issues with base10 base2 conversions hence most of the errors with floating point errors (yes at very large and very small numbers u lose accuracy but in practice most errors arise from base convention).
But floating-point notation also can’t precisely represent irrational numbers…
What superior method do you propose?
Following Pythagoreanism and believing irrational numbers to be blasphemous. They’re represented by being struck down by the gods.
Symbolical computation is cool
But some irrational numbers are only so in base 10
What? That’s not true at all…
Base π: π=1
1 is always 1. It’s 1 × b⁰ where b is the base. Anything raised to the zeroth power is 1.
10 is the base. 1 × b¹ + 0 × b⁰
Writing the same number a different way does not make it rational. There are no two natural numbers p and q so that p/q = 1 base pi.
That doesn’t make it rational but simply makes it writable in 2 digits(10)
Also you should have 3.1415… “number of characters” in that base… The base becoming irrational will make the number irrational
Even in base π, π is still considered an irrational number; using an irrational based doesn’t change the fundamental identity of whole numbers or irrational numbers, it just changes the way we write them.
π = 10
in base 10, 10 = 10.
Kinda. Technicaly no since an irrational number is a number that cannot be defined as a ratio of 2 existing rational numbers. Any number that can be represented in any rational base can by definition be represented as a ratio of somthing/base^n. This ignore the case of an irrational base but its practically useless cos any rational and most other irrational numbers will be irrational.
What u think ur trying to say is that some numbers cannot be represented in one base but can in another for example 1/3 can be represented as a decimal in base 3 but cannot jn base 10 ie u get 0.333(3 repeating forever).
Tieing back to floating point which uses base 2 u end up with simmillar issues with base10 base2 conversions hence most of the errors with floating point errors (yes at very large and very small numbers u lose accuracy but in practice most errors arise from base convention).