Having the luxury to just wander about in mathematics, I asked if there is a number system with a base (radix ?) that is not an integer. A short search on the internet provides a yes answer. The one article I skimmed showed the golden mean can be used as a base. The golden mean is irrational, thus, a base can be irrational. My question then is are irrational numbers in base 10 irrational in other number systems with an irrational base?
I first thought of this question for more specific numbers such as Pi and e. So, the number represented by the symbols 10 is an integer in base e, for example. At least it looks like an integer. It also appears to be rational.
It may well be that many of the numbers, I will call them counting numbers to identify them, in a base e system are irrational. I realized that I think I can not write what is 1 in base 10 in base e. If I go outside and pick up a rock and put on the picnic table, then there is a symbol that represents the number of rocks on the table. If I put a second rock on the table, I can still see a symbol in my mind that represents those rocks. It is when I pick up another rock and put it on the table that things get complicated. What is that number? Is it rational?
So, in general, are numbers that are irrational in base 10 numbering, irrational in other number systems that have an irrational base?
Best Answer
Being irrational (or not) is an intrinsic property of a real number, which is to say a property that the number has completely independently of what symbols you choose to use to describe it. Either the number is a ratio of integers, or it isn't, and using a different number base (even an irrational one) isn't going to change that.