Decimals

Here
is what Macdonald has written in his notebook about decimals:

Decimals are fractions having for their
denominators the powers of the number 10, that is, unity with cyphers annexed; thus , , are decimal fractions. In the notation of decimals
the denominators are not written; but they are indicated by prefixing a point (∙)
before the numerators and in any given decimal, the denominator, in general,
consists of unity with as many cyphers annexed to it as there are figures after
the point; the decimal fraction , , are written , , .

Decimals consisting of the same number of figures,
have therefore the same denominator and in decimals consisting of different
numbers of figures, those figures at equal distances from the point will have
the same denominators; have the same denominator, viz. (100) and in
the decimals & the in the one and the in the other have the same denominator (viz.
1000).

If the decimal consist of 1 figure, its denominator
is 10; if of 2, 100; if of 3, 1000 &c – Hence the first figure of a decimal
may be called tenths; the second hundredths; the third thousandths; &c;
thus, we read the decimal , , , , , or hundredths of thousandths.

The number Unity is greatest than the greatest
possible decimal number, greater, for instance, than – hence, in numbers consisting partly of
integers and partly of decimals, that which has the greatest integer, is the
greater; thus is greater than .

Note. As in notation of integers, the values of the
figures increase in a tenfold proportion from the right hand to the left, so,
in decimals, or their values, decrease in a tenfold proportion from the left
hand to the right. The operations, in the arithmetic of decimals, are all
performed in the same manner as the corresponding operations in the arithmetic
of integers, only care must be taken in distinguishing the decimal part of
every result.