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HSC Stuff => HSC Maths Stuff => HSC Subjects + Help => HSC Mathematics Extension 2 => Topic started by: jamonwindeyer on April 16, 2016, 02:53:04 pm

Title: Infinity = Infinity Plus One
Post by: jamonwindeyer on April 16, 2016, 02:53:04 pm: Hey all! Extension 2 can seem like some of the craziest math out there, trust me, it isn't ;)

I thought I'd start a little collection of super nerdy, super irrelevant to HSC math content, that if nothing else, is just cool and fun to think about. Anyone is welcome to add to the list and we can glory in the math (I'm hoping I'm not the only person who finds these sorts of things interesting) ;D

Here is a very interesting video on different types of infinite numbers.

https://youtu.be/SrU9YDoXE88
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 16, 2016, 05:34:53 pm: Fermat's last theorem

$\text{Consider the equation}\\{a}^{n}+{b}^{n}={c}^{n}\\\text{where }a,b,c,n \in \mathbb Z$
$\text{If we restrict }|n|>2\text{ there exists no solution. Otherwise there's an infinite number of solutions.}$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 16, 2016, 05:36:00 pm: $\text{Everyone knows the famous quadratic equation. There also exists a cubic equation, and a dreadfully long quartic equation.}$

$\text{However it has been mathematically proven that there cannot be a quintic equation.}\\\text{i.e. There is no algebraic way to determine the zeroes of a general polynomial of at least degree 5.}$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 21, 2016, 09:26:42 pm: $\text{Green-Tao theorem: The sequence of prime numbers contains arbitrarily long arithmetic progressions.}$

$\text{Examples:}\\ \begin{align*}\text{3 in a row}&\rightarrow 3,5,7\\\text{4 in a row}&\rightarrow 251,257,263,269\\\text{5 in a row}&\rightarrow 9843019, 9843049, 9843079, 9843109, 9843139\\ &\vdots\\\text{26 in a row}&\rightarrow 468395662504823 + 205619\cdot 223092870\cdot n, \, n\in \mathbb N, n \le 25 \end{align*}\\ \text{and so on (to be discovered)}$
Title: Re: Infinity = Infinity Plus One
Post by: jamonwindeyer on June 07, 2016, 11:34:41 pm: This video is a really cool TED talk linking to the notion of repetition. Music is very mathematical, so I suppose we can consider this piece perfectly unmathematical ;)
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on June 21, 2016, 12:28:48 pm: Banach-Tarski Paradox

$\text{Is it really possible to make two spheres out of one?}$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on July 14, 2016, 04:33:13 pm: Collatz Conjecture
$\text{Define a sequence in which the }n\text{th term }T_n\text{ satisfies:}\\ T_{n+1}=\begin{cases}3T_n+1 &\text{ if }T_n\text{ is odd}\\ \frac{T_n}{2}&\text{ if }T_n\text{ is even}\end{cases}\\ \text{and }T_1\text{ can be any natural number.}$
$\text{The sequence is conjectured to always converge into repeating three final terms }4, 2, 1$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on November 25, 2016, 06:13:10 pm: $i^i\text{ is a real number.}$
Title: Re: Infinity = Infinity Plus One
Post by: brenden on November 25, 2016, 06:24:47 pm: Wouldn't this contradict the law of identity?
Title: Re: Infinity = Infinity Plus One
Post by: jakesilove on November 26, 2016, 11:41:11 am: Quote from: brenden on November 25, 2016, 06:24:47 pm
Wouldn't this contradict the law of identity?

A is A, and $i^i$ IS a real number, written differently xo
Title: Re: Infinity = Infinity Plus One
Post by: brenden on December 02, 2016, 08:50:27 am: Quote from: jakesilove on November 26, 2016, 11:41:11 am
A is A, and $i^i$ IS a real number, written differently xo
Nah, I mean -- the infinity thing?
Title: Re: Infinity = Infinity Plus One
Post by: frog1944 on April 04, 2017, 06:11:03 am: $\infty ! = \sqrt{2\pi}$
Title: Re: Infinity = Infinity Plus One
Post by: jamonwindeyer on April 04, 2017, 09:05:27 am: $1+2+3+4+5+...=\frac{-1}{12}$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 04, 2017, 09:38:27 am: Quote from: frog1944 on April 04, 2017, 06:11:03 am
$\infty ! = \sqrt{2\pi}$
Quote from: jamonwindeyer on April 04, 2017, 09:05:27 am
$1+2+3+4+5+...=\frac{-1}{12}$
(https://i.warosu.org/data/sci/img/0085/83/1483719942334.jpg)
Title: Re: Infinity = Infinity Plus One
Post by: loxodromic on April 04, 2017, 12:19:14 pm: Quote from: jamonwindeyer on April 04, 2017, 09:05:27 am
$1+2+3+4+5+...=\frac{-1}{12}$

This one, I'm afraid, is just plain wrong.

The notation on the left: 1+2+3+.....
denotes the limit of the sequence of partial sums. It diverges.

The "proof" that you might see on youtube and elsewhere is nonsense.
If you go through the "proof" carefully you will see that it is fallacious.
Title: Re: Infinity = Infinity Plus One
Post by: jamonwindeyer on April 04, 2017, 12:29:29 pm: Quote from: loxodromic on April 04, 2017, 12:19:14 pm
This one, I'm afraid, is just plain wrong.

The notation on the left: 1+2+3+.....
denotes the limit of the sequence of partial sums. It diverges.

The "proof" that you might see on youtube and elsewhere is nonsense.
If you go through the "proof" carefully you will see that it is fallacious.

If you interpret it in the normal sense yes, but the methods behind that formula are not completely invalid. Not nonsense - It's just different :)

Not that I'm claiming to have a significant understanding of that mathematics mind you. Nor do I really mind if you think it is nonsense. Aha :P
Title: Re: Infinity = Infinity Plus One
Post by: loxodromic on April 04, 2017, 12:37:02 pm: Quote from: jamonwindeyer on April 04, 2017, 12:29:29 pm
If you interpret it in the normal sense yes, but the methods behind that formula are not completely invalid. Not nonsense - It's just different :)

The notation on the left is completely standard.
To say that we are using it to mean something else is disingenous and misleading.
The whole point about the (incorrect) equation is that it looks odd precisely because we interpret the left as an infinite sum.

If we say, okay, I'm using different notation, then great. But then the statment is essentially vacuous.
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 04, 2017, 12:39:00 pm: Quote from: loxodromic on April 04, 2017, 12:19:14 pm
This one, I'm afraid, is just plain wrong.

The notation on the left: 1+2+3+.....
denotes the limit of the sequence of partial sums. It diverges.

The "proof" that you might see on youtube and elsewhere is nonsense.
If you go through the "proof" carefully you will see that it is fallacious.
Essentially what Jamon said.

No matter who you tell, in the normal sense it's blatantly wrong, because it's obviously a (unboundedly) divergent series. However this is not the purpose of the Riemann-Zeta function. Methods of analysis used in Riemann-Zeta are used to analyse the specifc case of the Dirichlet function
$\sum_{n=1}^\infty \frac{1}{n^s}$

Well call it an analytic continuation of the function, specifically taken over a complex function with real part strictly greater than 1.

I am only a second year student so I do not claim mastery over all of these concepts. However, the methods of analysis used in this branch of mathematics is not intended to follow the conventional means of maths in society. Analytic continuation exists for the sake of assigning values to expressions that would otherwise be obviously divergent.

If you want to claim that 1+2+... = -1/12 is false, then in a similar way ∞! = √(2π) is wrong as well. I am unsure of why you targeted simply one expression over the other.

Also, on a personal note, I feel that if anything these comments have been used to provide humour to mathematicians for years. Just saying that it's wrong is ruining all of the fun for a lot of people.

Just because the expression isn't actually written in terms of the Riemann-Zeta function, does not mean we can have a bit of fun by writing it without it.
Title: Re: Infinity = Infinity Plus One
Post by: jamonwindeyer on April 04, 2017, 12:44:17 pm: Quote from: RuiAce on April 04, 2017, 12:39:00 pm
Also, on a personal note, I feel that if anything these comments have been used to provide humour to mathematicians for years. Just saying that it's wrong is ruining all of the fun for a lot of people.

This - We aren't submitting a paper for publishing here my friend - No one is misleading anyone, we're just having fun :)

For anyone who doesn't mind weird math, have a Google! It's fun ;D
Title: Re: Infinity = Infinity Plus One
Post by: loxodromic on April 04, 2017, 12:50:51 pm: Quote from: RuiAce on April 04, 2017, 12:39:00 pm
Essentially what Jamon said.

No matter who you tell, in the normal sense it's blatantly wrong, because it's obviously a (unboundedly) divergent series. However this is not the purpose of the Riemann-Zeta function. Methods of analysis used in Riemann-Zeta are used to analyse the specifc case of the Dirichlet function
$\sum_{n=1}^\infty \frac{1}{n^s}$

Well call it an analytic continuation of the function, specifically taken over a complex function with real part strictly greater than 1.

I am only a second year student so I do not claim mastery over all of these concepts. However, the methods of analysis used in this branch of mathematics is not intended to follow the conventional means of maths in society. Analytic continuation exists for the sake of assigning values to expressions that would otherwise be obviously divergent.

If you want to claim that 1+2+... = -1/12 is false, then in a similar way ∞! = √(2π) is wrong as well. I am unsure of why you targeted simply one expression over the other.

Also, on a personal note, I feel that if anything these comments have been used to provide humour to mathematicians for years. Just saying that it's wrong is ruining all of the fun for a lot of people.

Just because the expression isn't actually written in terms of the Riemann-Zeta function, does not mean we can have a bit of fun by writing it without it.

Appealing to analytic continuation and invoking the Riemann zeta function misses the point.
It doesn't make the original expression any less wrong or misleading.

"maths in society" ?!
Maths is maths.

The truth of the expression doesn't depend on some consensus view.
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on April 04, 2017, 12:55:48 pm: Quote from: loxodromic on April 04, 2017, 12:50:51 pm

Appealing to analytic continuation and invoking the Riemann zeta function misses the point.
It doesn't make the original expression any less wrong or misleading.

"maths in society" ?!
Maths is maths.

The truth of the expression doesn't depend on some consensus view.
It does not matter. The original expression is only there for the sake of humour. If you cannot handle the fun that mathematicians try to invoke in their lives I recommend you ignore this thread. A straightforward reply to the expression would've been fine if all you seeked was to provide clarification, but no. You choose to use adjectives like 'plain' and 'disingenous' which appears much more like an insult, rather than a helpful message. Basically, killing Jamon's joke.

If anything, regardless of the truth of your statement the way you have produced it is highly offensive to other mathematicians like myself.

Exactly. Maths is maths. Nothing different to saying 'maths in society'. It's how we treat maths on an everyday basis. Nothing more.

Everyone would know that the expression is false. You are merely showing that you do not appreciate any sense of humour in maths. Once again, if you cannot handle an abuse of notation, please ignore this thread for your own benefit.
Title: Re: Infinity = Infinity Plus One
Post by: sudodds on April 04, 2017, 01:12:16 pm: oooo if you want to make the Riemann Zeta function really fun read the script for 'A Disappearing Number' by Theatre De Complicité :) Such a mind fck - based all around that equation.

(literally the only thing I could ever contribute to a maths thread - dw I'll leave now before I lower the collective maths IQ by 20 points haha)
Title: Re: Infinity = Infinity Plus One
Post by: Sine on April 04, 2017, 03:59:39 pm: $e^{ i\pi } + 1 = 0$
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on July 06, 2017, 10:33:52 am: The amount of integers in the world is equal to the amount of odd numbers in the world.
Quote from: Sine on April 04, 2017, 03:59:39 pm
$e^{ i\pi } + 1 = 0$
Cute.
$i^i=e^{\frac{-\pi}2}$
Title: Re: Infinity = Infinity Plus One
Post by: frog1944 on July 06, 2017, 12:17:05 pm: $\ln (-1) = i \pi$
Title: Re: Infinity = Infinity Plus One
Post by: frog1944 on July 06, 2017, 12:20:34 pm: Gabriel's Horn
A geometric figure which has an infinite surface area but finite volume.
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on July 06, 2017, 12:22:31 pm: Quote from: frog1944 on July 06, 2017, 12:20:34 pm
Gabriel's Horn
A geometric figure which has an infinite surface area but finite volume.
You do know it's actually possible right
Title: Re: Infinity = Infinity Plus One
Post by: frog1944 on July 06, 2017, 12:52:12 pm: In the "mathematical realm"? Yeah, I saw something similar where you find the area under the curve y=1/x (boundless), and then you find it's volume by rotating it about the x-axis (and it's finite).
It's pretty cool!
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on July 06, 2017, 12:57:33 pm: Yeah it's basically possible in an idealistic realm because we can actually treat everything on a continuous scale here. At the end of the day the physical world is still built of matter, which is discrete.
Title: Re: Infinity = Infinity Plus One
Post by: RuiAce on September 10, 2017, 02:42:58 pm: $1^3+2^3+3^3+\dots+n^3 = (1+2+3+\dots+n)^2$