The parallel operator
\|
The parallel operator represents the reciprocal value of a sum of reciprocal values (sometimes also referred to as the "reciprocal formula" or "harmonic sum") and is defined by:
a\parallelbl{:=}
| 1 | |
| \dfrac{1 |
{a}+\dfrac{1}{b}}=
| ab | |
| a+b |
,
where,, and
a\parallelb
\overline{C
The operator gives half of the harmonic mean of two numbers a and b.
As a special case, for any number
a\in\overline{C
a\parallela=
| 1{2/a} | |
| = |
\tfrac12a.
Further, for all distinct numbers
|a\parallelb|>\tfrac12minl(|a|,|b|r),
with
|a\parallelb|
a\parallelb
min(x,y)
If
a
b
\tfrac12min(a,b)<|a\parallelb|<min(a,b).
The concept has been extended from a scalar operation to matrices and further generalized.
The operator was originally introduced as reduced sum by Sundaram Seshu in 1956, studied as operator ∗ by Kent E. Erickson in 1959, and popularized by Richard James Duffin and William Niles Anderson, Jr. as parallel addition or parallel sum operator : in mathematics and network theory since 1966. While some authors continue to use this symbol up to the present, for example, Sujit Kumar Mitra used ∙ as a symbol in 1970. In applied electronics, a ∥ sign became more common as the operator's symbol around 1974. This was often written as doubled vertical line available in most character sets (sometimes italicized as //), but now can be represented using Unicode character U+2225 ( ∥ ) for "parallel to". In LaTeX and related markup languages, the macros \| and \parallel are often used (and rarely \smallparallel is used) to denote the operator's symbol.
Let
\widetilde{\C}
\widetilde{\C}:=\C\cup\{infty\}\smallsetminus\{0\},
\varphi
\C
\widetilde{\C}
\varphi(z)=1/z.
\varphi(zt)=\varphi(z)\varphi(t),
\varphi(z+t)=\varphi(z)\parallel\varphi(t)
This implies immediately that
\widetilde{\C}
\C.
The following properties may be obtained by translating through
\varphi
As for any field,
(\widetilde{\C},\parallel, ⋅ )
It is commutative under parallel and multiplication:
\begin{align} a\parallelb&=b\parallela\\[3mu] ab&=ba \end{align}
It is associative under parallel and multiplication:
\begin{align} &(a\parallelb)\parallelc=a\parallel(b\parallelc)=a\parallelb\parallelc =
| 1 | |
| \dfrac{1 |
{a}+\dfrac{1}{b}+\dfrac{1}{c}}=
| abc | |
| ab+ac+bc |
,\\ &(ab)c=a(bc)=abc. \end{align}
Both operations have an identity element; for parallel the identity is
infty
\begin{align} &a\parallelinfty=infty\parallela=
| 1{\dfrac1a | |
| + |
0}=a,\\ &1 ⋅ a=a ⋅ 1=a. \end{align}
Every element
a
\widetilde{\C}
-a,
a\parallel(-a)=
| 1{\dfrac1a | |
| - |
\dfrac1a}=infty.
The identity element
infty
infty\parallelinfty=infty.
Every element
a ≠ infty
\widetilde{\C}
| a ⋅ | 1a |
| = |
1.
Multiplication is distributive over parallel:
k(a\parallelb)=
| k | |
| \dfrac1a+\dfrac1b |
=
| 1 | |
| \dfrac1{ka |
+\dfrac1{kb}}=ka\parallelkb.
Repeated parallel is equivalent to division,
\underbrace{a\parallela\parallel … \parallela}ntimes=
| 1{\underbrace{\dfrac1a | |
| + |
\dfrac1a+ … +\dfrac1a}ntimes
Or, multiplying both sides by,
n(\underbrace{a\parallela\parallel … \parallela}ntimes)=a.
Unlike for repeated addition, this does not commute:
a/b ≠ b/a.
Using the distributive property twice, the product of two parallel binomials can be expanded as
\begin{align} (a\parallelb)(c\paralleld) &=a(c\paralleld)\parallelb(c\paralleld)\\[3mu] &=ac\parallelad\parallelbc\parallelbd. \end{align}
The square of a binomial is
\begin{align} (a\parallelb)2 &=a2\parallelab\parallelba\parallelb2\\[3mu] &=a2\parallel\tfrac12ab\parallelb2. \end{align}
The cube of a binomial is
(a\parallelb)3=a3\parallel\tfrac13a2b\parallel\tfrac13ab2\parallelb3.
In general, the th power of a binomial can be expanded using binomial coefficients which are the reciprocal of those under addition, resulting in an analog of the binomial formula:
(a\parallelb)n=
| an | |
| \binomn0 |
\parallel
| an-1b | |
| \binomn1 |
\parallel … \parallel
| an-kbk | |
| \binomnk |
\parallel … \parallel
| bn | |
| \binomnn |
.
The following identities hold:
| 1 | |
| log(ab) |
=
| 1 | \parallel | |
| log(a) |
| 1 | |
| log(b) |
,
| \exp\left( | 1 |
| a\parallelb |
\right)=\exp\left(
| 1 | \right)\exp\left( | |
| a |
| 1 | |
| b |
\right)
A parallel function is one which commutes with the parallel operation:
f\left(a\parallelb\right)=f(a)\parallelf(b)
For example,
f(x)=cx
c(a\parallelb)=ca\parallelcb.
As with a polynomial under addition, a parallel polynomial with coefficients
ak
| n | |
| \begin{align} &a | |
| 0x |
\parallel
| n-1 | |
| a | |
| 1x |
\parallel … \parallelan=a0(x\parallel-r1)(x\parallel-r2) … (x\parallel-rn) \end{align}
for some roots
rk
Analogous to polynomials under addition, the polynomial equation
(x\parallel-r1)(x\parallel-r2) … (x\parallel-rn)=infty
implies that for some .
A linear equation can be easily solved via the parallel inverse:
\begin{align} ax\parallelb&=infty\\[3mu] \impliesx&=-
| ba. \end{align} | |
To solve a parallel quadratic equation, complete the square to obtain an analog of the quadratic formula
\begin{align} ax2\parallelbx\parallelc&=infty\\[5mu] x2\parallel
| b | |
| a |
x&=-
| c | |
| a |
\\[5mu] x2\parallel
| b | |
| a |
x\parallel
| 4b2 | |
| a2 |
&=\left(-
| c | |
| a |
\right)\parallel
| 4b2 | |
| a2 |
\\[5mu] \left(x\parallel
| 2b | |
| a |
\right)2&=
| b2\parallel-\tfrac14ac | |
| \tfrac14a2 |
\\[5mu] \impliesx&=
| (-b)\parallel\pm\sqrt{b2\parallel-\tfrac14ac | |
| }{\tfrac12a}. \end{align} |
The extended complex numbers including zero,
\overline{C
l(\overline{C
infty
For every non-zero,
a\parallel0=
| 1{\dfrac1a | |
| + |
\dfrac10}=0
The quantity
0\parallel(-0)=0\parallel0
In the absence of parentheses, the parallel operator is defined as taking precedence over addition or subtraction, similar to multiplication.
There are applications of the parallel operator in electronics, optics, and study of periodicity:
In electrical engineering, the parallel operator can be used to calculate the total impedance of various serial and parallel electrical circuits.There is a duality between the usual (series) sum and the parallel sum.
For instance, the total resistance of resistors connected in parallel is the reciprocal of the sum of the reciprocals of the individual resistors.
| \begin{align} | 1 |
| Req |
&=
| 1 | |
| R1 |
+
| 1 | |
| R2 |
+ … +
| 1 | |
| Rn |
\\[5mu] Req&=R1\parallelR2\parallel … \parallelRn. \end{align}
Likewise for the total capacitance of serial capacitors.
In geometric optics the thin lens approximation to the lens maker's equation.
f=\rhovirtual\parallel\rhoobject
The time between conjunctions of two orbiting bodies is called the synodic period. If the period of the slower body is T2, and the period of the faster is T1, then the synodic period is
Tsyn=T1\parallel(-T2).
Question:
Three resistors
R1=270k\Omega
R2=180k\Omega
R3=120k\Omega
Answer:
\begin{align} R1\parallelR2\parallelR3&=270k\Omega\parallel180k\Omega\parallel120k\Omega\\[5mu] &=
| 1 | |
| \dfrac{1 |
{270k\Omega
The effectively resulting resistance is ca. 57 kΩ.
Question:
A construction worker raises a wall in 5 hours. Another worker would need 7 hours for the same work. How long does it take to build the wall if both workers work in parallel?
Answer:
t1\parallelt2=5h\parallel7h=
| 1 | |
| \dfrac{1 |
{5h}+\dfrac{1}{7h}} ≈ 2.92h
They will finish in close to 3 hours.
Suggested already by Kent E. Erickson as a subroutine in digital computers in 1959, the parallel operator is implemented as a keyboard operator on the Reverse Polish Notation (RPN) scientific calculators WP 34S since 2008 as well as on the WP 34C and WP 43S since 2015, allowing to solve even cascaded problems with few keystrokes like .
Given a field F there are two embeddings of F into the projective line P(F): z → [''z'' : 1] and z → [1 : ''z'']. These embeddings overlap except for [0:1] and [1:0]. The parallel operator relates the addition operation between the embeddings. In fact, the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each embedding has its addition a + b represented by the following matrix multiplications in M(2,A):
\begin{align} \begin{pmatrix}1&0\ a&1\end{pmatrix}\begin{pmatrix}1&0\ b&1\end{pmatrix} &=\begin{pmatrix}1&0\ a+b&1\end{pmatrix}, \\[10mu] \begin{pmatrix}1&a\ 0&1\end{pmatrix}\begin{pmatrix}1&b\ 0&1\end{pmatrix} &=\begin{pmatrix}1&a+b\ 0&1\end{pmatrix}. \end{align}
The two matrix products show that there are two subgroups of M(2,F) isomorphic to (F,+), the additive group of F. Depending on which embedding is used, one operation is +, the other is
\parallel.