Weirder bases, and what should you call the decimal point in other bases?

Now let's look at a fractional base.  Let's try base one-tenth.  What is

`\qquad\qquad\qquad 12.345_{\frac{1}{10}}`?

The wiggle equivalent is

 `\qquad\qquad\qquad 12.345_{\frac{1}{10}}  =  1_{\frac{1}{10}}2;_{\frac{1}{10}}3_{\frac{1}{10}}4_{\frac{1}{10}}5.`

Telescoping from the left, we have

`\qquad\qquad\qquad 1_{\frac{1}{10}}2;_{\frac{1}{10}}3_{\frac{1}{10}}4_{\frac{1}{10}}5,`
`\qquad\qquad\qquad\qquad  =  ((1 xx \frac{1}{10}) + 2);_{\frac{1}{10}}3_{\frac{1}{10}}4_{\frac{1}{10}}5,`
`\qquad\qquad\qquad\qquad  =  (2.1);_{\frac{1}{10}}3_{\frac{1}{10}}4_{\frac{1}{10}}5.`

Telescoping now from the right, we have

`\qquad\qquad\qquad  =  (2.1);_{\frac{1}{10}}3_{\frac{1}{10}}(4 + \frac{5}{\frac{1}{10}}),`
`\qquad\qquad\qquad  =  (2.1);_{\frac{1}{10}}3_{\frac{1}{10}}(4 + 50),`
`\qquad\qquad\qquad\qquad =  (2.1);_{\frac{1}{10}}3_{\frac{1}{10}}54,`
`\qquad\qquad\qquad  =  (2.1);_{\frac{1}{10}}(3 + \frac{54}{\frac{1}{10}}),`
`\qquad\qquad\qquad\qquad   =  (2.1);_{\frac{1}{10}}(3 + 540),`
`\qquad\qquad\qquad\qquad   =  (2.1);_{\frac{1}{10}}(543),
`\qquad\qquad\qquad\qquad   =  (2.1); + (5430),`
`\qquad\qquad\qquad\qquad   =  5432.1.`

So

`\qquad\qquad\qquad 12.345_{\frac{1}{10}}  =  5432.1,`

and numbers in base one-tenth are just the reverse of numbers in base ten, with the same digit (here a two) marked by the point.  If it doesn't look to be exactly the reverse (check!), that is because you are thinking of the point as positioned between `2` and `3` or `1` and `2`, rather than written just to the right of the `2`.

This suggests an answer to the vexed question of what to call the decimal point when the base is not ten.  Some people call it the binary point in binary, the ternary point in base three, as so forth.  A few have tried for a more generic solution by calling it the basal point.  Still others call it the fraction marker, because, for whole number bases greater than one, the part before the point represents a whole number, but the part afterthe point represents a fraction less than one.

But what it really does is tells us which digit appears at face value.  It should probably be called the 'face value point' or 'unit point'.  Because what it does is mark which digit appears at face value.  In some respects, it would be better if it were a mark under the face value digit, rather than just to the right.  If, for instance, we used an underbar to mark the face value digit, than we would have


`\qquad\qquad\qquad 1\ul{2}345_{\frac{1}{10}}  =  543\ul{2}1_{10},`

and we would see even more clearly the relation between base ten and base a tenth.

The common usage of the decimal point is, nevertheless, here to stay.  For that reason, it is probably best to learn to use it, and just think of it as marking the digit to it's left as appearing at face value.  The use of the semicolon to mark the face value superdigit can be justified similarly.