Editing 2566: Decorative Constants

{{comic
| number    = 2566
| date      = January 10, 2022
| title     = Decorative Constants
| image     = decorative_constants.png
| titletext = Arguably, the '1/2' in the drag equation is purely decorative, since drag coefficients are already unitless and could just as easily be half as big. Some derivations give more justification for the extra 1/2 than others, but one textbook just calls it 'a traditional tribute to Euler and Bernoulli.'
}}

==Explanation==
This is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a Math Tip.

Randall gives an example of a complex looking equation labeled 4-15:
:T = 𝔻m<sub>0</sub>(r<sub>out</sub> − r<sub>in</sub>)<sup>μ&#773;</sup>
But since 𝔻 and μ&#773; are "decorative", the equation can be reduced to 
:T = m<sub>0</sub>(r<sub>out</sub> − r<sub>in</sub>)
Here T is the net rate, m<sub>0</sub> the unit mass and (r<sub>out</sub> − r<sub>in</sub>) the flow balance.
	 
The decorative symbols can be interpreted as constants 𝔻 = μ&#773; = 1, in which case the implied operations of multiplication and exponentiation make sense. The 𝔻 is double-struck ("blackboard bold", thus in the comic only the vertical line is double). Mathematicians, who are always searching for more symbols{{citation needed}}, have taken to distinguishing things represented by the same letter by using different fonts, such as 𝑑, 𝐝, 𝒅, 𝐷, 𝐃, 𝑫, 𝒹, 𝒟, 𝖉, 𝕯, ∂, 𝕕, and 𝔻. The double-struck font is easier to write on a blackboard than a proper bold letter and often represents a set, such as ℝ for the set of real numbers or ℂ for the set of complex numbers. 𝔻 can represent the unit disk in the complex plane, the set of decimal fractions, or the set of split-complex numbers. 

μ is the Greek lowercase mu and has many uses in mathematics and science. Here it has a bar, μ&#773;, which could indicate a number of things, including the complex conjugate. Intriguingly, μ is the symbol in statistics for the population mean, and the overbar represents the sample mean, so this could represent a random variable which is the average of a sample of means μ<sub>i</sub> of different populations in some larger ensemble of populations.

Using a special version both of D and μ to even further spice up the formula all leads up to the math tip: 
:'''If one of your equations ever looks too simple, try adding some purely decorative constants.'''

Other examples of well known equations that are profound but look simple include
:''E'' = ''mc''<sup>2</sup> ({{w|Special Relativity}}),
:''PV'' = ''nRT'' (the {{w|Ideal Gas Law}}),
:''F'' = ''ma'' ({{w|Newton's Second Law}}),
:''V'' = ''IR'' ({{w|Ohm's Law}}), and
:''G<sub>μν</sub>'' + Λ ''g<sub>μν</sub>'' = ''κT<sub>μν</sub>'' ({{w|Einstein field equations}}), and
:''e<sup>πi</sup>+1'' = ''0'' ({{w|Euler's Identity}}).

Of these, only the Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations).

In the title text Randall mentions the {{w|Drag equation}}, which is attributed to {{w|Lord Rayleigh}}. In {{w|fluid dynamics}}, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is ''F''<sub>d</sub>&nbsp;=&nbsp;½''ρu''<sup>2</sup>''c''<sub>d</sub>''A''. Here ''F''<sub>d</sub> is the drag force, ρ the mass density of the fluid, u the relative flow velocity, ''c''<sub>d</sub> the {{w|drag coefficient}} and A is the area.

Randall jokes that the factor of ½ in the equation is meaningless and purely decorative, since the drag coefficients, ''c''<sub>d</sub>, are already unitless and could just as easily be half as big thus leaving out the ½ in front of the equation. The ½ is thus just an example of a "decorative constant." The usual reason for including the factor of ½ is that it is part of the formula for kinetic energy that appears in the derivation of the drag equation, i.e. ½''ρu''<sup>2</sup>. However, modern treatments are so condensed that this factor of ½ is often smuggled in with no explanation. 

Since we can choose the constants to be whatever we want, it could be possible to absorb the ½ into the drag coefficient ''c''<sub>d</sub>, but that does not mean it is unmotivated, since it comes from the kinetic energy. Still, Randall quotes Frank White's ''[https://www.amazon.co.uk/Fluid-Mechanics-Frank-White/dp/007119911X Fluid Mechanics''] textbook, [https://books.google.com/books?id=wGweAQAAIAAJ&q=traditional%20tribute&redir_esc=y which two times] calls it "a traditional tribute to Euler and Bernoulli." According to White, the factor of ½ rather comes from the calculation of the projected area of the object being dragged. Randall has brought up this point before, in his book, "[[How To]]"

The line from White probably refers to renowned mathematicians {{w|Leonhard Euler}} and {{w|Daniel Bernoulli}}. Euler who is held to be one of the greatest mathematicians in history worked directly with Daniel and was a friend of the {{w|Bernoulli family}}, that produced eight mathematically gifted academics.

Daniel Bernoulli is known for modifying the definition of ''vis viva'' (what we now call kinetic energy) from ''mv''<sup>2</sup> to ½''mv''<sup>2</sup>, as motivated by the derivation from the impulse equation. In 1741, he wrote
:[Define ''vis viva''] esse ½ ''mvv'' = ∫''pdx''.
That is, "define ''vis viva'' to be ½ ''mv''<sup>2</sup> = ∫''p''d''x''," where ''p'' is the force (from ''pressione'') and d''x'' is the differential of position (infinitesimal displacement). Today, this equation says that the kinetic energy imparted to an object at rest equals the work done on it.

In the drag equation ½ ρu<sup>2</sup> represents the dynamic pressure due to the kinetic energy of the fluid, and hence the 1/2 makes sense to keep in the equation, and could thus easily be argued not to represent a decorative constant.

The title text is pretty much word-for-word a repeat from Randall's book *How To*. In Chapter 11: *How to Play Football*, he misuses the drag equation, and mentions this fact in more depth, in a footnote.

==Transcript==
:[A small panel only with text. Written as an excerpt from a mathematical text book. Begins with a number for an equation, then follows the equation written in larger letters and symbols. And below are explanations of each term in the equation. The μ has a bar over the top and the D has a double vertical line.]
:Eq. 4-15
:<big>T = 𝔻m<sub>0</sub>(r<sub>out</sub> - r<sub>in</sub>)<sup>μ&#773;</sup></big>
:T: Net rate
:m<sub>0</sub>: Unit mass
:(r<sub>out</sub>-r<sub>in</sub>): Flow balance<br>
:𝔻, μ&#773;: Decorative

:[Caption below the panel:]
:Math tip: If one of your equations ever looks too simple, try adding some purely decorative constants.

==Trivia==
This was the first comic that came out after the [[Countdown in header text]] started.

{{comic discussion}}

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