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==Explanation==
 
==Explanation==
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{{incomplete|Created by a DECORATIVE BOT - What is the formula representing when removing the two decorative constants? - Do NOT delete this tag too soon.}}
 
This is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a Math Tip.
 
This is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a Math Tip.
  
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Randall gives an example of a complex looking equation labeled 4-15:
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He gives an example of a complex looking equation (4-15 in the book it is taken from):
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:T = 𝔻m<sub>0</sub>(r<sub>out</sub> − r<sub>in</sub>)<sup>μ&#773;</sup>
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:T = Dm<sub>0</sub>(r<sub>out</sub> - r<sub>in</sub>)<sup>μ</sup>
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But since 𝔻 and μ&#773; are "decorative", the equation can be reduced to
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:T: Net rate
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:T = m<sub>0</sub>(r<sub>out</sub> − r<sub>in</sub>)
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:m<sub>0</sub>: Unit mass
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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.
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:(r<sub>out</sub>-r<sub>in</sub>): Flow balance<br>
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:D, μ: Decorative
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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.
 
  
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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.
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But since D and μ are decorative (and μ=1 is the only situation where it does not change everything), then the equation can be reduced to:
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:T = m<sub>0</sub>(r<sub>out</sub> - r<sub>in</sub>)
  
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Using a special version both of D and μ to even further spice up the formula all leads up to the math tip:
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Also note that the D is written with two horizontal lines and μ with a bar over the top, to further spice up the formula.
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:'''If one of your equations ever looks too simple, try adding some purely decorative constants.'''
 
  
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Other examples of well known equations that are profound but look simple include
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This all leads up to the Math tip of the day. If one of your equations ever looks too simple, try adding some purely decorative constants.
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:''E'' = ''mc''<sup>2</sup> ({{w|Special Relativity}}),
 
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:''PV'' = ''nRT'' (the {{w|Ideal Gas Law}}),
 
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:''F'' = ''ma'' ({{w|Newton's Second Law}}),
 
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:''V'' = ''IR'' ({{w|Ohm's Law}}), and
 
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:''G<sub>μν</sub>'' + Λ ''g<sub>μν</sub>'' = ''κT<sub>μν</sub>'' ({{w|Einstein field equations}}), and
 
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:''e<sup>πi</sup>+1'' = ''0'' ({{w|Euler's Identity}}).
 
  
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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).
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Other examples of well known equations that are profound but look simple are:
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:E = Mc<sup>2</sup> (from {{w|Special Relativity}})
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:PV = NRT (the {{w|Ideal Gas Law}})
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:F = ma ({{w|Newton's Second Law}})
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:V = IR ({{w|Ohm's Law}})
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These could all use some spicing up with extra constants.
  
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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.
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Maybe this was what {{w|Albert Einstein}} was thinking when he included the {{w|Cosmological Constant}} in {{w|General Relativity}}, but it ended up being meaningful.
  
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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.  
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In the title text Randall mentions the {{w|Drag equation}}. In 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.
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:The equation is:
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F<sub>d</sub> = 1/2ρu<sup>2</sup>C<sub>d</sub>A.
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Randall jokes about the 1/2 in front is meaningless and thus 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 1/2 in front of the equation. The 1/2 is thus just an example of a Decorative Constant according to Randall.
  
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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]]"
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He continues to claim that in some text books the derivations of the formula give more justification for the extra 1/2 than others. But finishes with an example of a book that just calls it "a traditional tribute to Euler and Bernoulli".  
  
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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.
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This is in reference of {{w|Leonhard Euler}} and the {{w|Bernoulli family}}, possibly mainly {{w|Daniel Bernoulli}} who Euler worked together with to formulate their {{w|Euler–Bernoulli beam theory}}. Euler was a friend of the entire family. Euler is held to be one of the greatest mathematicians in history. The  Bernoulli family was a patrician family, notable for having produced eight mathematically gifted academics. Both Euler and Daniel also worked on fluids.
  
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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
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But! First of all the drag equation is attributed to {{w|Lord Rayleigh}}, so that in it self has nothing to do with the two names mentioned.  
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:[Define ''vis viva''] esse ½ ''mvv'' = ∫''pdx''.
 
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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.
 
  
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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.
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Second the 1/2ρu<sup>2</sup> also called P<sub>D</sub> is the dynamic pressure due to kinetic energy of fluid experiencing relative flow velocity u with ρ being the mass density of the fluid. Thus this is analogues to kinetic energy E = 1/2mv<sup>2</sup>, thus the 1/2 comes from this.
  
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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.
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Third the force F is proportional to P<sub>D</sub>A where A is the area over which the pressure P<sub>D</sub> is applied. The proportionality factor is thus the one called C<sub>d</sub>, the drag coefficient. And thus the 1/2 belongs to a part of the equation, that could be taken out to give a specific value for the dynamic pressure. And it is thus relevant to keep it apart from the  otherwise unitless drag coefficient.
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So it seems that this is just another joke by Randall. But possibly he did read the statement in a text book, but a citation for that is needed.
  
 
==Transcript==
 
==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.]
 
:[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
 
:Eq. 4-15
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:<big>T = 𝔻m<sub>0</sub>(r<sub>out</sub> - r<sub>in</sub>)<sup>μ&#773;</sup></big>
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:<big>T = Dm<sub>0</sub>(r<sub>out</sub> - r<sub>in</sub>)<sup>μ</sup></big>
 
:T: Net rate
 
:T: Net rate
 
:m<sub>0</sub>: Unit mass
 
:m<sub>0</sub>: Unit mass
 
:(r<sub>out</sub>-r<sub>in</sub>): Flow balance<br>
 
:(r<sub>out</sub>-r<sub>in</sub>): Flow balance<br>
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:𝔻, μ&#773;: Decorative
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:D, μ: Decorative
  
 
:[Caption below the panel:]
 
:[Caption below the panel:]
 
:Math tip: If one of your equations ever looks too simple, try adding some purely decorative constants.
 
:Math tip: If one of your equations ever looks too simple, try adding some purely decorative constants.
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==Trivia==
 
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This was the first comic that came out after the [[Countdown in header text]] started.
 
  
 
{{comic discussion}}
 
{{comic discussion}}

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