4 ms·
If you were to count in base π, would the diameter be a transcendental number?
by russellbeattie 4y ago
If you were to count in base π, would the diameter be a transcendental number?
- roywiggins 4y agoIf the circumference is rational (in "base pi"), then the diameter is also rational, I think. Suppose the circumference is π, or 10ₚ (base pi). Then the diameter is 1₁₀. What's that in base Pi? Well, how many Pis are in 1? 1/π, or π⁻¹. But in base Pi you write that 0.1ₚ. Ditto something like π²+π+1 (111ₚ). The diameter is π+1+π⁻¹, or 11.1ₚ.
- russellbeattie 4y agoI'm both impressed with your answer, as well as your use of subscript/superscript in plain text! Thanks!
- roywiggins 4y agoIt's wrong after all that, 1 (base 10) is 1 base pi, of course. But the argument still works!
- kaashif 4y agoTranscendental means a number that is not a root of any polynomial with rational coefficients. The base you count in doesn't affect this definition, those are just different ways of writing down numbers, the numbers themselves don't change. The rational numbers and integers are still the same whatever base you use. 1 is still 1 in any base, there's only one multiplicative identity. In base pi, 10 (pi) is transcendental. I do get what you're trying to ask, the other reply has a more useful comment.