Instigator / Pro
3
1536
rating
18
debates
58.33%
won
Topic

# The Binomial Theorem Is The Most Beautiful Theorem In Mathematics

Status
Finished

All stages have been completed. The voting points distribution and the result are presented below.

Arguments points
0
3
Sources points
2
2
Spelling and grammar points
1
1
Conduct points
0
1

With 1 vote and 4 points ahead, the winner is ...

User_2006
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Miscellaneous
Time for argument
Two days
Voting system
Open voting
Voting period
One month
Point system
Four points
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Characters per argument
10,000
Contender / Con
7
1435
rating
48
debates
37.5%
won
Description
~ 0 / 5,000

No information

Round 1
Pro
For reference: nCr=nC(n-r)
Therefore 4C0=4C4 and 4C1=4C3
A simple combinatorial identity derived from Pascal's Triangle

The Binomial Theorem is hard not to love.

It gives us a simple way to find the value of the expansion of algebraic binomials when raised to a certain power.

The Binomial Theorem tells us that:

(x+y)^n=nC0*x^n*y^0+nC1*x^(n-1)*y^1+nC2*x^(n-2)*y^2+...+nC(n-2)*x^2*y^(n-2)+nC(n-1)*x^1*y^(n-1)+nCn*x^0*y^n

Or written in summation form

(x+y)^n=sigma(r=0)(n) nCr*x^(n-r)*y^r

Where C represents the choose function in combinatorial mathematics.

The reason this works is simple if you are familiar with combinations.

Let's take the expression (x+y)^4. Which can also be written as:

(x+y)(x+y)(x+y)(x+y)

I am sure we are all familiar with FOIL. The Elementary method that teaches how to multiply two algebraic binomials together. Representing First, Outside, Inside, Last.

Each term in a series of algebraic binomials must be multiplied with each other term in that series except for the one it is in the parentheses with.

For example in this series:

(x+y)(x+y)(x+y)(x+y)

The first x must be multiplied by 3 other x's and 3 other y's in any combination but not the y that it is in the parentheses with. This applies to every x and y in this series.

Therefore, to find all the combinations of x's and y's we can use the choose function or combinations.

We know there must be a term which includes all 4 x's. Therefore, we must pick 4 x's from this series to form the term. This can be done in 4C0 ways, which is 1. This means that there is only one occasion of this term in the expansion which makes sense since there is only one distinct way to pick 4 x's.

Moving on to a term with 3 x's and 1 y. This can be done in 4C1 or 4 ways. We must only pick 3 sets of parentheses to take an x from and the unpicked one is where the y term comes from. This means that this term appears 4 times in the expansion so it's coefficient is 4.

And on and on.

I hope readers understand the beauty of this equation and how it combines algebra and combinatorics to create a beautiful equation.

A keen mathematician should have picked up on the fact that the coefficients of an expansion to power n correspond to row n of Pascal's Triangle. Yet this is obvious while doing the combinatorial work as the denominator of the combination increases by 1 in each expansion.
Con
PRO has not used any sources yet. I will.

I, as a 14-year-old, may have a shallower understanding of mathematics compared to PRO, Nmvarco, whom I perceive as at least a high school senior.

CON's contender is Euler's identity

I, as CON, find Fauxlaw's format of debating as pretty efficient.

I Argument: Sources

I.a Sources are listed below. In this section, sources are being presented in the format of [1], [2], etc, with the link itself embedded within the units. Experienced voters would be used to this format and can tell for themselves.

I.b I completely agree with PRO on the beauty of his theorem, and no rebuttals should be made just yet. However, PRO Carries the Burden of Proof of proving that the Binomial Theorem is the most beautiful theorem in mathematics, while mathematical theorems of any kind could be brought up by CON as long as it is a good enough counter against PRO's case.

I.c It became a no-brainer that one vote does not over-trump the world's general opinion unless the former is the person that has a much larger say in this topic. The topic is Mathematical theorems, so unless PRO has presented evidence that he has a much bigger say than most people(Including mathematicians!) in this field of knowledge, his argument counts as only one vote.

I.c.1 I have many counts of evidence that Euler's identity is voted the most beautiful theorem in mathematics. The plainest example, plain as day, is Wikipedia, whom conduct research in all fields of knowledge. Wikipedia states, "A poll of readers conducted by The Mathematical Intelligencer in 1990 named Euler's identity as the "most beautiful theorem in mathematics".[1] In another poll of readers that was conducted by Physics World in 2004, Euler's identity tied with Maxwell's equations (of electromagnetism) as the "greatest equation ever".[2]"[3]

I.c.2 How reliable are those two sources, The Mathematical Intelligencer and Physics World? Well, the former was said to conduct frontline research in mathematics[4], and the latter consists of scientific innovations and discoveries from all fields of scientific research[5]. It is even more bizarre that Euler Identity was said to be one of the greatest and most beautiful equation/theorem ever, in not just mathematics, but the WHOLE SCIENTIFIC FIELD.

I.c.3 Wikipedia also states that human brains are naturally attracted to mathematical theorems such as Euler's theorem. The original text is: "A study of the brains of sixteen mathematicians found that the "emotional brain" (specifically, the medial orbitofrontal cortex, which lights up for beautiful music, poetry, pictures, etc.) lit up more consistently for Euler's identity than for any other formula.[6]"[7] Biologically, CON's statement would stand better when coming from a biological viewpoint that evidently suggests how humans find things beautiful.

I.d I have other sources that suggest my claim stands. They are listed below as [8], [9], and [10]. Self-explanatory. All of them list Euler's identity as the most beautiful theorem in mathematics.

I rest my case at least here.

[1]Wells, 1990

[2]Crease, 2004.

[6]Zeki et al., 2014.

Round 2
Pro
Forfeited
Con
PRO has failed to prove his case. I extend all arguments made last round.
Round 3
Pro
at least a high school senior
haha definitely not lmao
you're young enough for this too

PRO = nmvarco = The Binomial Theorem Is The Most Beautiful Theorem In Mathematics
CON = User_2006 = Euler's Identity Is The Most Beautiful Theorem In Mathematics

ARG 1.1: BASIS OF CON ARG
• CON's argument seems to be "x said y therefore y is correct."
• As opposed to providing statements as to why Euler's Identity is so beautiful, CON instead relies on second-hand sources to provide that information.
• (A real missed opportunity here -- Euler's Identity is formed from elements all over the field of mathematics)
ARG 1.2: BEAUTY OF BINOMIAL THEOREM REVISITED
• As Euler's Identity is formed from multiple different elements, the Binomial Theorem also combines algebra and combinatorics.
• These two fields barely ever have any interaction.
• Combinatorial identities are formed using committee-based proofs as opposed to algebraic.

Con
PRO actually wrote a response. Thank you.

haha definitely not lmao
you're young enough for this too
PRO's sources make sense but are otherwise irrelevant to the present debating topic.

PRO = nmvarco = The Binomial Theorem Is The Most Beautiful Theorem In Mathematics
CON = User_2006 = Euler's Identity Is The Most Beautiful Theorem In Mathematics
More accurately, CON=User_2006=The Binomial Theorem is not the most beautiful theorem in mathematics. BoP is on PRO.

As long as I disprove Nmvarco, I win. PRO never brought up a criterion that says CON MUST bring up a contesting theorem, and if I disprove his claim using any methods, I am the victor here.

ARG 1.1: BASIS OF CON ARG
• CON's argument seems to be "x said y therefore y is correct."
• As opposed to providing statements as to why Euler's Identity is so beautiful, CON instead relies on second-hand sources to provide that information.
• (A real missed opportunity here -- Euler's Identity is formed from elements all over the field of mathematics)
Bulletpoint 1 fails to disprove my argument. I have proved how reliable were the two sources of knowledge, and since there are, presumably, major math enthusiasts and even mathematicians participating in the polls, it would not make sense to declare another theorem as more beautiful, especially since PRO did not prove that he is a judgemental and vital mathematician within. PRO's opinion does not matter more than 2 groups of mathematicians and STEM enthusiasts.

Bulletpoint 2 fails to disprove my argument. I have proved that Euler's Identity is considered more beautiful because biologically, the brain reacts more consistently in a manner where people would observe when they see things that are beautiful. In fact, PRO did not even sufficiently prove that the Binomial Theorem is the most beautiful theorem, he merely asserts that his theorem is beautiful(Not the most beautiful!) without any backup evidence. I have used various studies to show that Euler's identity is not only beautiful, but it is also the MOST beautiful.

Bulletpoint 3 shows concession. Even PRO admits that Euler's identity is beautiful. Since he proved that the Binomial Theorem is beautiful(Not the MOST beautiful!), and he admits that Euler's identity, to some point, is beautiful.

Might as well throw the description of "Beauty" inside since PRO has failed to define it.
Beauty is the ascription of a property or characteristic to an animalideaobject, person or place that provides a perceptual experience of pleasure or satisfaction. Beauty is studied as part of aestheticsculturesocial psychology and sociology. An "ideal beauty" is an entity which is admired, or possesses features widely attributed to beauty in a particular culture, for perfection. Ugliness is the opposite of beauty.
The experience of "beauty" often involves an interpretation of some entity as being in balance and harmony with nature, which may lead to feelings of attraction and emotional well-being. Because this can be a subjective experience, it is often said that "beauty is in the eye of the beholder."[1] Often, given the observation that empirical observations of things that are considered beautiful often align among groups in consensus, beauty has been stated to have levels of objectivity and partial subjectivity which are not fully subjective in their aesthetic judgement.
PRO has failed to disprove the reason that humans think Euler's theorem is more beautiful, cranially.

ARG 1.2: BEAUTY OF BINOMIAL THEOREM REVISITED
• As Euler's Identity is formed from multiple different elements, the Binomial Theorem also combines algebra and combinatorics.
• These two fields barely ever have any interaction.
• Combinatorial identities are formed using committee-based proofs as opposed to algebraic.
These Bullet points do not make Euler's identity not as beautiful as the Binomial Theorem. PRO never tried to refute my reasons, such as scientific researches. Stating their fields does not make one more beautiful than the other, just like stating the background of the girls does not make one more attractive than the other.

Conclusions:
1. I have proved that Euler's identity is more beautiful through researches and PRO did not sufficiently refute them.
1. Reasons include:
1. One source credited Euler's identity as the most beautiful theorem in mathematics
2. Another source credited Euler's identity as one of the greatest theorems in the entire Scientific field
3. Euler's identity makes people's brains react more beautifully
2. PRO has not sufficiently proved that the Binomial theorem is the most beautiful theorem in the mathematical field.
I rest my case.

Sources