This post describes the duality between the usual (series) addition and the dual parallel addition. This duality is normally considered in electrical circuit theory and combinatorics, but it has a much wider applications. In Part I of this post, the focus is on developing series-parallel dual formulas—in contrast to the usual focus on formulas using only the series sum.

From the viewpoint of pure mathematics, the parallel sum is “just as good” as the series sum. It is only for empirical and perhaps even some accidental reasons that so much mathematics is developed using the series sum instead of the equally good parallel sum. There is a whole “parallel mathematics” which can be developed with the parallel sum replacing the series sum. Since the parallel sum can be defined in terms of the series sum (or vice-versa), “parallel mathematics” is essentially a new way of looking at certain known parts of mathematics.

Exclusive promotion of the series sum and prejudice against the parallel sum is series chauvinism. Before venturing further into the parallel universe, we might suggest some exercises to help the politically incorrect reader to combat the heritage of series chauvinism. Anytime the series sum seems to occur naturally in mathematics with the parallel sum being seemingly invisible, it is an illusion due to series chauvinism. The parallel sum has a “parallel” role that has been unfairly neglected.

In Part II of the post, series-parallel duality is applied to financial arithmetic and is shown to underlie a certain duality of formulas that has long been noted in the literature on valuation and appraisal. Hence this instance of intellectual trespassing applies concepts best known from electrical circuit theory to the operations of financial arithmetic. Moreover in economic theory, the much-used duality of convex functions and their conjugates is the “integral” of series-parallel (SP) duality, or, to put it the other way around, SP duality is the “derivative” of convex duality.

When resistors with resistances a and b are placed in series, their compound resistance is the usual sum (hereafter the series sum) of the resistances a+b. If the resistances are placed in parallel, their compound resistance is the parallel sum of the resistances, which is denoted by the full colon:

$a:b=(a^{-1}+b^{-1})^{-1}=\frac{1}{\frac{1}{a}+\frac{1}{b}}$

The parallel sum is associative x:(y:z)) = (x:y):z, commutative x:y = y:x, and distributive x(y:z) = xy:xz. On the positive reals, there is no identity element for either sum but the “closed circuit” 0 and the “open circuit” $\infty$ can be added to form the extended positive reals. Those elements are the identity elements for the two sums, $x+0 = x = x:\infty$. That is, adding a short circuit in series to a resistor does not change the resistance, and adding an open circuit in parallel to a resistor does not change the resistance.

As an example of series chauvinism in elementary math, the series sum of fractions is expressed by the annoyingly asymmetrical rule: “Find the common denominator and then add the numerators.” The parallel sum of fractions restores symmetry since it is defined in the dual fashion: “Find the common numerator and then (series) add the denominators.”

$\frac{a}{b}:\frac{a}{d}=\frac{a}{b+d}$

The usual series sum of fractions can also be obtained by finding the common numerator and then taking the parallel sum of the denominators.

$\frac{a}{b}+\frac{a}{d}=\frac{a}{b:d}$

The parallel sum of fractions can also be obtained by finding the common denominator and taking the parallel sum of numerators.

$\frac{a}{b}:\frac{c}{b}=\frac{a:c}{b}$

The rules for series and parallel sums of fractions can be summarized in the following four equations which restore full symmetry.

$\frac{a}{1}+\frac{b}{1}=\frac{a+b}{1}$ $\frac{a}{1}:\frac{b}{1}=\frac{a:b}{1}$

$\frac{1}{a}:\frac{1}{b}=\frac{1}{a+b}$ $\frac{1}{a}+\frac{1}{b}=\frac{1}{a:b}$

For another example, a series chauvinist might point out that the series sum appears naturally in the rule for working with exponents xaxb = xa+b while the parallel sum does not. But this is only an illusion due to our mathematically arbitrary symmetry-breaking choice to take exponents to represent powers rather than roots. Let a pre-superscript stand for a root (just as a post-superscript stands for a power) so 2x would be the square root of x. Then the rule for working with these exponents is axbx = a:bx so the parallel sum does have a role symmetrical to the series sum in the rules for working with exponents.

### Series-Parallel Duality: The Reciprocity Map

The duality between the series and parallel additions on the positive reals R+ can be studied by considering the (bijective) reciprocity map

$\rho :R^{+}\rightarrow R^{+}$ given by $\rho(x)=1/x$.

The reciprocity map preserves the unit $\rho(1)=1$, preserves multiplication $\rho(xy)=\rho(x)\rho(y)$, and interchanges the two additions:

$\rho(x+y)=\rho(x):\rho(y)$ and $\rho(x:y)=\rho(x)+\rho(y)$.

The reciprocity map captures series-parallel duality on the positive reals.

Percy MacMahon called a series connection a “chain” and a parallel connection a “yoke” (as in ox yoke). A series-parallel network is constructed solely from chains and yokes (series and parallel connections). By interchanging the series and parallel connections, each series-parallel network yields a dual or conjugate series-parallel network. To obtain the dual of an expression such as a+b, apply the reciprocity map to obtain (1/a) : (1/b) but then, for the atomic variables, replace 1/a by a and so forth in the final expression. Hence the MacMahon dual to a+b would be a:b, and the dual expression to a+ ((b+c) : d) would be a : ((b : c) + d) (see below).

Conjugate Series-Parallel Networks

If each variable a, b, … equals one, then the reciprocity map carries each expression for the compound resistance into the conjugate expression. Hence if all the “atomic” resistances are one ohm, a = b = c = d = 1, and the compound resistance of a series-parallel network is R, then the compound resistance of the conjugate network is 1/R [MacMahon 1881, 1892; reprinted in: 1978]. With any positive reals as resistances, MacMahon’s chain-yoke reciprocity theorem continues to hold if each atomic resistance is also inverted in the conjugate network (i.e., if we just apply the reciprocity map).

MacMahon Chain-Yoke Reciprocity Theorem

The theorem amounts to the observation that the reciprocity map interchanges the two sums while preserving multiplication and unity. The fundamental intuition is that the series-parallel dual gives reciprocals or multiplicative inverses.

### Dual Equations on the Positive Reals

Any equation on the positive reals concerning the two sums and multiplication can be dualized by applying the reciprocity map to obtain another equation. The series sum and parallel sum are interchanged. For example, the equation

$\frac{1}{3}(5+\frac{2}{5}+\frac{3}{5})=2$

dualizes to the equation

$3(\frac{1}{5}:\frac{5}{2}:\frac{5}{3})=\frac{1}{2}$

The following equation

$1=(1+x):(1+\frac{1}{x})$

holds for any positive real x. Add any x to one and add its reciprocal to one. The results are two numbers larger than one and their parallel sum is exactly one. Dualizing yields the equation

$1=(1:\frac{1}{x})+(1:x)$

for all positive reals x. Taking the parallel sum of any x and its reciprocal with one yields two numbers smaller than one which sum to one.

For any set of positive reals x1,…,xn, the parallel summation can be expressed using the capital P:

$P_{i=1}^{n}x_{i}=(\sum_{i=1}^{n}x_{i}^{-1})^{-1}$.

Parallel Summation

### Series and Parallel Geometric Series

The following formula (and its dual) for partial sums of geometric series (starting at i = 1) are useful in financial mathematics (where x is any positive real).

$\sum_{i=1}^{n}(1:x)^{i}=(1:x)\sum_{i=0}^{n-1}(1:x)^{i}$

$=(1:x)\frac{1-(1:x)^{n}}{1-(1:x)}=x(1-(1:x)^{n})$

Partial Sums of Geometric Series

Dualizing (and some algebra) yields a formula for partial sums of the parallel-sum geometric series. The dual of the series subtraction a – b where a > b is the parallel subtraction $x\ominus y=(\frac{1}{x}-\frac{1}{y})^{-1}$ where x < y.

$P_{i=1}^{n}(1+x)^{i}=(1+x)P_{i=0}^{n-1}(1+x)^{i}$

$=(1+x)\frac{1\ominus(1+x)^{n}}{1\ominus(1+x)}=\frac{x}{1-(1+x)^{-n}}$

Partial Sums of Dual Geometric Series

Dualization can also be applied to infinite series. Taking the limit as $n\rightarrow \infty$ in the above partial sum formulas yields for any positive reals x the dual summation formulas for series and parallel sum geometric series that begin at the index i = 1.

$\sum_{i=1}^{\infty}(1:x)^{i}=x=P_{i=1}^{\infty}(1+x)^{i}$

The parallel sum series in the above equation can be used to represent a repeating decimal as a fraction. An example will illustrate the procedure so let z = .367367367… where the “367″ repeats. Then since 1/a + 1/b = 1/(a:b), we have:

$z=.367367\ldots=\sum_{i=1}^{\infty}\frac{367}{(1000)^{i}} =\frac{367}{P_{i=1}^{\infty}(1000)^{i}}$.

Taking y = x+1 for x > 0 in the previous geometric series equation yields

$P_{i=1}^{\infty}y^{i}=y-1$

for y > 1 which is applied to yield

$z=.367367\ldots=\frac{367}{P_{i=1}^{\infty}(1000)^{i}}=\frac{367}{1000-1}=\frac{367}{999}$.

For any positive real x, the beautiful dual formulas for the geometric series with indices beginning at i = 0 can be obtained by serial or parallel adding $1= (1:x)^{0}= (1+x)^{0}$ to each side.

$\sum_{i=0}^{\infty}(1:x)^{i}=(1+x)$

Geometric Series for any Positive Real x

$P_{i=0}^{\infty}(1+x)^{i}=(1:x)$.

Dual Geometric Series for any Positive Real x

### References

MacMahon, Percy A. 1881. “Yoke-Chains and Multipartite Compositions in connexion with the Analytical Forms called ‘Trees’ .” Proc. London Math. Soc. 22: 330-46.

MacMahon, Percy A. 1892. “The Combinations of Resistances.” The Electrician 28, 601-2.

MacMahon, Percy A. 1978. Collected Papers: Volume I, Combinatorics. Edited by George E. Andrews. Cambridge, Mass.: MIT Press.