2566: Decorative Constants - Revision history

155.246.151.38: /* Explanation */ removed a citation needed. there's no joke here; this isn't a comically obvious fact

2025-10-15T16:03:22Z

‎Explanation: removed a citation needed. there's no joke here; this isn't a comically obvious fact

141.101.99.4: /* Explanation */ I had wondered. But bold/strong is out of place, making it italics/em instead.

2025-04-08T19:09:58Z

‎Explanation: I had wondered. But bold/strong is out of place, making it italics/em instead.

172.70.58.50: /* Explanation */ oops, emphasised the wrong word

2025-04-08T19:00:10Z

‎Explanation: oops, emphasised the wrong word

162.158.216.203: /* Explanation */

2025-04-08T18:53:28Z

‎Explanation

Numbermaniac: /* Explanation */ Missing full stop

2025-04-08T06:38:27Z

‎Explanation: Missing full stop

FaviFake: sorry, autocorrect

2025-04-06T12:37:52Z

sorry, autocorrect

172.69.43.181: /* Explanation */ I'm with you on that, but this is the US spelling.

2025-04-06T12:31:13Z

‎Explanation: I'm with you on that, but this is the US spelling.

FaviFake: /* Explanation */

2025-04-06T09:58:44Z

‎Explanation

Spacegreg: ungooglified Fluid Mechanics link

2023-10-25T20:57:25Z

ungooglified Fluid Mechanics link

FaviFake at 09:20, 25 August 2023

2023-08-25T09:20:23Z

@@ Line 14: / Line 14: @@
 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.
+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, 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.

@@ Line 31: / Line 31: @@
 *''e<sup>πi</sup>+1'' = ''0'' ({{w|Euler's Identity}}).
-Of these, only Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations). Euler's identity is massaged to include both 0 and 1 (since including both the additive and multiplicative identities is "more profound") since the interpretation of ''e<sup>iπ</sup>'' = ''-1'' is otherwise not necessarily intuitive; using the ratio of a circle's circumference to its '''radius''' (𝜏) rather than its diameter results in ''e<sup>i𝜏</sup> = 1'' which can be interpreted as a statement that "one trip around the circle leaves you back where you started".
+Of these, only Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations). Euler's identity is massaged to include both 0 and 1 (since including both the additive and multiplicative identities is "more profound") since the interpretation of ''e<sup>iπ</sup>'' = ''-1'' is otherwise not necessarily intuitive; using the ratio of a circle's circumference to its ''radius'' (𝜏) rather than its diameter results in ''e<sup>i𝜏</sup> = 1'' which can be interpreted as a statement that "one trip around the circle leaves you back where you started".
 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.

@@ Line 31: / Line 31: @@
 *''e<sup>πi</sup>+1'' = ''0'' ({{w|Euler's Identity}}).
-Of these, only Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations). Euler's identity is massaged to include both 0 and 1 (since including both the additive and multiplicative identities is "more profound") since the interpretation of ''e<sup>iπ</sup>'' = ''-1'' is otherwise not necessarily intuitive; using the ratio of a circle's '''circumference''' to its radius (𝜏) rather than its diameter results in ''e<sup>i𝜏</sup> = 1'' which can be interpreted as a statement that "one trip around the circle leaves you back where you started".
+Of these, only Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations). Euler's identity is massaged to include both 0 and 1 (since including both the additive and multiplicative identities is "more profound") since the interpretation of ''e<sup>iπ</sup>'' = ''-1'' is otherwise not necessarily intuitive; using the ratio of a circle's circumference to its '''radius''' (𝜏) rather than its diameter results in ''e<sup>i𝜏</sup> = 1'' which can be interpreted as a statement that "one trip around the circle leaves you back where you started".
 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.

@@ Line 31: / Line 31: @@
 *''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).
+Of these, only Einstein field equations have been spiced up with decorative indices (which actually hide a system of ten nonlinear partial differential equations). Euler's identity is massaged to include both 0 and 1 (since including both the additive and multiplicative identities is "more profound") since the interpretation of ''e<sup>iπ</sup>'' = ''-1'' is otherwise not necessarily intuitive; using the ratio of a circle's '''circumference''' to its radius (𝜏) rather than its diameter results in ''e<sup>i𝜏</sup> = 1'' which can be interpreted as a statement that "one trip around the circle leaves you back where you started".
 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.

@@ Line 37: / Line 37: @@
 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]]"
+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. 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

@@ Line 8: / Line 8: @@
 ==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 labelled 4-15:
+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:

@@ Line 62: / Line 62: @@
 ==Trivia==
-*This was the first comic that came out after the [[Countdown in header text]] started.
+This was the first comic that came out after the [[Countdown in header text]] started.
 {{comic discussion}}