Additions of rates

Abstract

It is easily shown that there are two kinds of addition for rates: arithmetic addition and harmonic addition. The kind of addition required depends on whether the variable in common has the same units as the denominator or numerator. This is shown and illustrated with rates of speed and velocity. Several examples are given including the speed of light, which is shown to be classically invariant for all inertial observers.

Introduction

A rate is any ratio between two related variables with different units[1]. For measurements, one variable is set to a value or series of values and then the other variable is measured relative to it. The variable whose value is held constant while measurement occurs is the independent variable, and a variable whose value is to be determined is the dependent variable. An independent variable is often the denominator of a rate but can be the numerator.

Rates are averaged with either the arithmetic or harmonic mean, both of which entail adding rates. Rates with the same units are added over a variable in common. Addition of multiple rates equals a rate of the total of the measured variables and the value of the variables in common.

The kind of addition required depends on whether the variable in common is the denominator or numerator of the rates. If addition is over a common denominator, then arithmetic addition is used, but if addition is over a common numerator, harmonic addition[2] is required. This is shown below and illustrated with speed and velocity.

Arithmetic addition

There are two ways in which rates with a common denominator are added arithmetically. One way is the combination w_c of rates with a common denominator. The combination of rates of speed, u and v, with numerators x₁ and x₂ respectively, with the denominator t in common, equals the total distance traversed, x₁ + x₂, divided by the time interval in common:

$w_{c}=\frac{x_{1}+x_{2}}{t}=\frac{x_{1}}{t}+\frac{x_{2}}{t}=u+v$

Because numerators are added, this could be thought of as numerator addition.

The other way rates are added arithmetically is with rates added over a variable in common that has the units of the denominators. The sum w_t of rates of speed, u and v, over a time interval t in common, equals the total distance traversed divided by the time interval in common:

$w_{t}=\frac{tu+tv}{t}=u+v$

If u and v are velocities, then vector addition applies.

The common denominator variable is often time but may be another variable such as distance. An example is the inverse speed, also called pace[3], which is the elapsed time divided by the distance traversed.

Rates with different values of a variable in common can be averaged with a weighted arithmetic mean, which equals the total distance traversed divided by the total elapsed time:

$\bar{w}_{t}=\frac{t_{1}u+t_{2}v}{t_{1}+t_{2}}= \left (\frac{t_{1}}{t_{1}+t_{2}} \right ) u + \left (\frac{t_{2}}{t_{1}+t_{2}} \right ) v$

A mean speed averaged over time is called a time mean speed[4].

Harmonic addition

There are two ways in which rates with a common numerator are added by harmonic addition. One way is the combination w_c of rates with a common numerator. The combination of rates of speed, u and v, with denominators t₁ and t₂ respectively, and common numerator x, equals the distance in common traversed x divided by the total elapsed time, t₁ + t₂:

$w_{c}=\frac{x}{t_{1}+t_{2}}=\frac{x}{t_{1}} \boxplus \frac{x}{t_{2}}= u \boxplus v$

with the operation of harmonic addition u ⊞ v defined as[5]:

$u \boxplus v = 1/((1/u)+(1/v))$

where

$1/a :=\left\{\begin{matrix} a^{-1} & \mathrm{if}\; a\neq 0\\ 0 & \mathrm{if}\; a=0 \end{matrix}\right.$

Harmonic addition inverts, adds, then inverts again. With a change of variables, it is equivalent to arithmetic addition. Because denominators are added, harmonic addition could be thought of as denominator addition.

The other way in which rates are added by harmonic addition is with rates added over a variable in common that has the units of the numerators. The sum of rates of speed, u and v, with a distance traversed x in common, equals the distance traversed divided by the total elapsed time:

$w_{x}=\frac{x}{xu^{-1}+xv^{-1}}=\frac{1}{u^{-1}+v^{-1}}=u \boxplus v$

If u and v are velocities, then vector harmonic addition applies[6].

Rates with different values of a variable in common can be averaged with a weighted harmonic mean, which equals the total distance traversed divided by the total elapsed time:

$\bar{w}_{t}=\frac{x_{1}+x_{2}}{x_{1}u^{-1}+x_{2}v^{-1}}= \left (\frac{x_{1}+x_{2}}{x_{1}} \right ) u \boxplus \left (\frac{x_{1}+x_{2}}{x_{2}} \right ) v$

A mean speed averaged over distance is called a space mean speed⁵.

Examples of adding rates are given below. Remember that all values must be either measured or calculated.

Electrical circuits

Two kinds of addition are well known in resistor-inductor-capacitor (RLC) networks, though they are stated as ad hoc rules rather than consequences of adding rates. Electrical resistance is a rate of voltage difference and current, which from Ohm’s law, the resistance R equals the voltage difference V divided by the current I.

If resistors are connected in series, the current is constant and the voltages add, so the equivalent resistance is their sum:

$R_{T}=\frac{V_{1}}{I}+\frac{V_{2}}{I}=R_{1}+R_{2}$

If resistors are connected in parallel, the currents add and the voltage is constant, so the equivalent resistance is their harmonic sum:

$R_{T}=\frac{V}{I_{1}}+\frac{V}{I_{2}}=R_{1} \boxplus R_{2}$

On the other hand, voltage appears in the denominator of the capacitance rate: capacitance is a rate of the charge stored to the voltage difference. If capacitors are connected in parallel, the voltage is constant and the charges add, so the equivalent capacitance is their arithmetic sum but if capacitors are connected in series, the charge is constant and the voltages add, so the equivalent capacitance is their harmonic sum[7].

Reduced mass

Mass is a rate of force and acceleration. In the classical two-body problem the forces are equal and opposite:

$m_{1}=\frac{\mathbf{F_{12}}}{\mathbf{a_{1}}}\; \mathrm{and} \; m_{2}=\frac{\mathbf{F_{21}}}{\mathbf{a_{2}}}= \frac{-\mathbf{F_{12}}}{-k\mathbf{a_{1}}}= \frac{ \mathbf{F_{12}}}{k\mathbf{a_{1}}}$

where m_i are the masses of the bodies, a_i are their accelerations, F_ij is the force between them (for i, j = 1, 2), and k = m₁/m₂. Because the numerators of the mass rates can be equalized, the combined mass is their harmonic sum:

$m_{c}=m_{1} \boxplus m_{2}$

which is called the reduced mass, which is shown to be a form of rate addition.

Relative velocity

With n runners over a racecourse of length d, the average speed of all the runners is a space mean speed that is the harmonic mean of their individual speeds. What is the speed of the runner with the best time, t₁, relative to the runner with the worst time, t_n? Since the runners traverse a distance in common, their relative speed is the harmonic difference between their individual speeds, v₁ and v_n:

$v_{1/n}=v_{n} \boxminus v_{1}$

Mean speed of light

Consider transmitting a ray of light along the x-axis a distance s and reflecting it back, for a total distance of 2s. With c as the mean speed of light, the total time is 2sc⁻¹.

For an inertial observer moving at velocity v relative to the laboratory, it is not the case in classical mechanics that the light ray travels at the velocity (c − v) one way and the velocity (c + v) the opposite way, because the light ray travels a given distance, not a given time interval[8]. To combine velocities over a distance in common requires harmonic addition. If the observer’s velocity is v, then in one direction the observed velocity u_a is

$u_{a}=c \boxplus \mathbf{v}$

and the velocity u_b in the opposite direction is

$u_{a}=c \boxminus \mathbf{v}$

with c ⊟ v = c ⊞ −v. The total time is

$\frac{s}{u_{a}}+\frac{s}{u_{b}}= \frac{s}{c \boxplus \mathbf{v}}+\frac{s}{c \boxminus \mathbf{v}}= s\left ( \frac{1}{c}+\frac{1}{\mathbf{v}} \right )+ s\left ( \frac{1}{c}-\frac{1}{\mathbf{v}} \right )=2sc^{-1}$

and the space mean speed of light equals c, which shows that the mean speed of light is invariant for any inertial observer.

Conclusion

As rates are averaged in two ways, with the arithmetic or harmonic mean, so also rates are added in two ways, with arithmetic or harmonic addition. Understanding this belongs with the elements of science, but the harmonic addition of rates has been underappreciated or overlooked for a long time.

^†) Electronic mail: rag2127@nyu.edu

[1] Webster’s New International Dictionary of the English Language, 2nd edition, Unabridged. Merriam Webster Co. (2016).

[2] Michael Grossman, Robert Katz. Non-Newtonian Calculus. Chapter 8: The Harmonic Family of Calculi. Lee Press, Pigeon Cove, MA (1972).

Harmonic addition is also known as reciprocal addition and parallel addition, e.g.:

James Williams. “A Multiplication-Free Characterization of Reciprocal Addition.” Pi Mu Epsilon Journal, 4 (4): 168-176 (Spring 1966).

Kent E. Erickson. “A New Operation for Analyzing Series-Parallel Networks,” IRE Transactions on Circuit Theory, Institute of Radio Engineers (IRE). CT-6 (1): 124–126 (March 1959).

N. Anderson, Jr., R. J. Duffin. “Series and Parallel Addition of Matrices,” Journal of Mathematical Analysis and Applications, 26 (3): 576-594 (June 1969).

[3] Victor L. Knoop, Introduction to Traffic Flow Theory: An introduction with Exercises, 2nd edition, Delft University of Technology, Delft, Netherlands (July 2018). p. 1.

[4] Reference 4, p. 5.

[5] Here the boxplus (⊞) symbol is used to shows harmonic addition as a form of addition; several other symbols have been used.

[6] W. N. Anderson, Jr, G. E. Trapp. “The Harmonic and Geometric Mean of Vectors,” Linear and Multilinear Algebra, 22: 199-210 (1987).

[7] Harmonic addition is here associated with connections in series, so it would be misleading to call it parallel addition.

[8] Contrary to many publications on the speed of light; notably, Albert A. Michelson, Edward W. Morley. “On the Relative Motion of the Earth and the Luminiferous Ether,” American Journal of Science 34 (203): 333–345 (1887).

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