## 25 Nov 2008

### 2/7 is 0.285714..., and 1/(1 - a) as 1 + a + a2 + a3 etc. From Hegel Science of Logic § 556

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Below is summary. At the end I cite the text in full. My interpretations not informed by a complete read of the text.]

from Hegel, Science of Logic, § 556:

The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it.

2/7 can be expressed as 0.285714..., because:

1/(1 - a) can be expressed as 1 + a + a2 + a3 etc., because we can represent it as a series of sums, and first we take out 1, that is, we subtract 1 from it (expanded as 1/1).

To obtain the same denominator, we multiply 1/1 by (1-a)/(1-a):

We combine them to obtain:

We distribute the negative in the numerator to get:

We subtract the ones to obtain:

We then pull the a out of the numerator:

This of course is equal to the original equation from which it was derived:

We then move the -1 over to the other side (and rotate the sides) to get:

But we see something interesting here, because we have the opportunity for a re-iteration. The left side is equal to a formula which includes that very same left side. So, we can substitute the whole right side in for the 1/(1-a) nested within the right side, to get:

We then distribute the a to both terms in parenthesis to obtain:

We may of course reiterate this process once again:

Thus we can see how 1/(1 - a) can be expressed as 1 + a + a2 + a3 etc.

From the original text:

The fraction 2/7 can be expressed as 0.285714..., 1/(1 - a) as 1 + a + a2 + a3 etc. As so expressed it is an infinite series; the fraction itself is called the sum, or finite expression of it. A comparison of the two expressions shows that one of them, the infinite series, represents the fraction no longer as a ratio but from that side where it is a quantum as anaggregate of units added together, as an amount. That the magnitudes of which it is supposed to consist as amount are in turn decimal fractions and therefore are themselves ratios, is irrelevant here; for this circumstance concerns the particular kind of unit of these magnitudes, not the magnitudes as constituting an amount. just as a multi-figured integer in the decimal system is reckoned essentially as an amount, and the fact that it consists of products of a number and of the number ten and powers of ten is ignored. Similarly here, it is irrelevant that there are fractions other than the example taken of 2/7 which, when expressed as decimal fractions, do not give an infinite series; but they can all be so expressed in a numerical system based on another unit.

Second demonstration based on explanation in:
http://broodsphilosophy.wordpress.com/2006/06/28/hegel-and-infinite-series/

Hegel. Science of Logic. Transl. A.V. Miller. George Allen & Unwin, 1969.
Text available online at:
http://www.marxists.org/reference/archive/hegel/index.htm