Are Not
If it possible to have bases that are not integers? For example, could we have a base b = 3/2?
Or a base b = 7/3?
(a) What would 110b represent in base b = 3/2?
(b) What would 1.11111111…b represent in base b = 3/2?
(c) How might one write the integer 2 in base b = 3/2?
Curiously, the answer is “yes” (though it is a very unpleasant system for representing numbers since it lacks the uniqueness properties that representations for integer bases have — see below — and arithmetic would be beastly to carry out): the digits for such a representation would range from 0 up to the greatest integer less than b.
For (a) 110_b represents 1*(3/2)^2 + 1*(3/2)^1 + 0 = 15/4
(b) 1.11111…. _3/2 represents 1 + (3/2)^-1 + (3/2)^-2 +. . . a geometric series with first term 1 and
common ratio 2/3, so 1/(1-2/3) = 3
for (c)
let x:= 2,
until x = 0 (which may not happen in finitely many steps)
find the largest power of 3/2 which is less than or equal to x, put a 1 in that column, and replace
x with x minus that power of 3/2.
So it starts with
10.01. . .
(when I left off x was 1/2 – 4/9)
But notice, if we used this method, we would get a very different representation for 3:
10.1000001. . .
And, if one is clever one might notice that 0.11111111. .. . (all 1′s after the 3/2-ary point) would also represent 2.
And consider adding 10 + 10 base 3/2, we know the answer should be a base 3/2 representation of the number 3, but how do you get either of the ones we just saw from that?
Cute idea, but not that useful.
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