Interesting mathematical statements (1 Viewer)

leehuan · Dec 26, 2015

A must know for the Ext 2 student:

Fundamental Theorem of Algebra
$\\ $Every non-constant single-variable polynomial always has at least one root$.$

leehuan · Dec 26, 2015

This has appeared elsewhere on this forum before:

Expressions for the Golden Ratio
$\\ \varphi =1+\frac { 1 }{ 1+\frac { 1 }{ 1+\frac { 1 }{ 1+\ddots } } }$

$\\ \varphi =\sqrt { 1+\sqrt { 1+\sqrt { 1+\sqrt { 1+\dots } } } }$

leehuan · Dec 26, 2015

If you square the first 9 numbers with only 1's in them:

$\\ { 1 }^{ 2 }=1\\ { 11 }^{ 2 }=121\\ { 111 }^{ 2 }=12321\\ { 1111 }^{ 2 }=1234321\\ { 11111 }^{ 2 }=123454321\\ { 111111 }^{ 2 }=12345654321\\ { 1111111 }^{ 2 }=1234567654321\\ { 11111111 }^{ 2 }=123456787654321\\ { 111111111 }^{ 2 }=12345678987654321$

Paradoxica · Dec 26, 2015

$$Ramanujan's continued fractions for algebraic numbers in terms of $e.$

$\sqrt[4]{5}\sqrt{\frac{1+\sqrt{5}}{2}}- \sqrt{\frac{1+\sqrt{5}}{2}} = \frac{e^{-\frac{2\pi}{5}}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-4\pi}}{1+\frac{e^{-6\pi}}{1+\ddots}}}}$

$\frac{\sqrt{6\sqrt{3}}-(1+\sqrt{3})}{4} = \frac{e^{-\frac{2\pi}{3}}}{1+\frac{e^{-2\pi}+e^{-4\pi}}{1+\frac{e^{-4\pi}+e^{-8\pi}}{1+\frac{e^{-6\pi}+e^{-12\pi}}{1+\ddots}}}}$

$\sqrt{\sqrt{2} -1} = \frac{\sqrt{2}e^{-\frac{\pi}{4}}}{1+\frac{e^{-2\pi}}{1+e^{-2\pi}+\frac{e^{-4\pi}}{1+e^{-4\pi}+\frac{e^{-6\pi}}{1+e^{-6\pi}\ddots}}}}$

leehuan · Dec 26, 2015

Very famous result on what e actually is:

$e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }$

Still famous but not as famous result on what e is, esp amongst HSC students:

$e=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } }$

Paradoxica · Dec 26, 2015

leehuan said:
Very famous result on what e actually is:

$e=\lim _{ n\rightarrow \infty }{ { \left( 1+\frac { 1 }{ n } \right) }^{ n } }$

Still famous but not as famous result on what e is, esp amongst HSC students:

$e=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n! } }$

$e^x \equiv \sum_{r=0}^\infty \frac{x^r}{r!}$

$\lim_{n \rightarrow \infty} \left (1+\frac{x}{n} \right )^n \equiv e^x$

$$\noindent By expanding the first statement and expanding the second statement using the binomial theorem for $n = \infty$, one can prove the two statements above are identical.$

leehuan · Dec 26, 2015

Tbh isn't the first statement really just a Taylor series?

InteGrand · Dec 26, 2015

leehuan said:
Tbh isn't the first statement really just a Taylor series?

Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.

Paradoxica · Dec 26, 2015

InteGrand said:
Yeah, but sometimes it's actually more convenient to define functions by their power series representation and then prove other things about them.

$The only other things we have to go off of are:$

$e^x $ is the fundamental solution to the differential equation $y' = y$

$e^x $ is it's own derivative$$

$e^x$ is the solution for the equation $x = \int_1^{e^x} \frac{\mathrm{d}t}{t}$

The first and second statements are nearly equivalent down to the family of solutions for any differential equation.

The third statement is somewhat convoluted.

dan964 · Dec 26, 2015

987654321 is divisible by 9
987654312 is divisible by 8 (this particular one gives 123456789 btw)
987654213 is divisible by 7
987653214 is divisible by 6
987643215 is divisible by 5
987543216 is divisible by 4
986543217 is divisible by 3
976543218 is divisible by 2
876543219 is divisible by 1

0123456789 * 2 = 246913578 (a permutation of 0123456789)
And likewise up to 100; excluding multiples of 3

Soulful · Dec 26, 2015

There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people

Paradoxica · Dec 27, 2015

Soulful said:
There is approximately a 50% chance of two people sharing the same birthday in a room of 23 people

This is, of course, assuming that every birthday date is identically probable, and that leap years do not exist, and that birthdays are treated as truly random. Under these conditions, a group of 70 people have a 99.9% chance of two people sharing a birthday.

AAEldar · Dec 29, 2015

Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

Then I moved to stats.

Paradoxica · Dec 29, 2015

AAEldar said:
Certainly not a pure mathematician, but Cauchy's Integral Formula stands out for me.

Then I moved to stats.

That's not a bad thing, there are unsolved problems in stats applicable to the real world such as p-values, error values, correlation vs. causation, interpolation of data, etc.
Most of these are relevant to the sciences, as statistical methods are required for collecting any information. Even the medical fields and the humanities require it.

But I digress.

$$\noindent The curious improper integral $\int_{-\infty}^{\infty} \frac{1+x}{1+x^2} \mathrm{d}x$ is divergent, yet if you evaluate it normally as a limit towards infinity, it converges to exactly $\pi .$

$$Conclusion: $\pi = \infty$

braintic · Dec 29, 2015

There is a lot of highly theoretical mathematics here.
Does anyone have more real-world mathematical statements?
(The birthday problem was a good one)

Paradoxica · Dec 29, 2015

braintic said:
There is a lot of highly theoretical mathematics here.
Does anyone have more real-world mathematical statements?
(The birthday problem was a good one)

The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.

InteGrand · Dec 29, 2015

Paradoxica said:
The Kakeya Needle Problem: What is the smallest area of a parking lot in which you can have a needle of length 1 turn around 180 degrees and return to its starting position, pointing in the other direction?

Answer: The area can be made arbitrarily small through a series of divisions and transformations of the shape required for the needle to turn around. Hence, no smallest area exists.

This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ

Paradoxica · Dec 29, 2015

InteGrand said:
This isn't very "practical" though, since the car / needle needs to be made arbitrarily thin if you want the area to be arbitrarily small. For anyone interested, there is a Numberphile video on the Kakeya Needle Problem: www.youtube.com/watch?v=j-dce6QmVAQ

Well the problem never stated any width. It is a mathematical solution, after all.

$\noindent Simpson's Paradox: When two sets of data are analysed separately, there is a common trend between both sets of data. But when the data is combined together and analysed as a whole, the trend reverses.$

$\noindent One of the famous real life examples was the Berkeley Gender Bias case. The entire university's admissions were significantly biased towards men. However, when the data from the individual faculties were looked at separately, each one had a small, but statistically significant bias towards women.$

$Not sure if your mind has exploded yet, but I'll go get the sponge.$

InteGrand · Dec 29, 2015

Another good one is that there are always two opposite points on Earth with exactly the same temperature. This is proved quickly and accessibly in this video by Dr James Grime:

https://youtube.com/watch?v=5Px6fajpSio

InteGrand · Dec 29, 2015

Paradoxica said:
Well the problem never stated any width. It is a mathematical solution, after all.

$\noindent Simpson's Paradox: When two sets of data are analysed separately, there is a common trend between both sets of data. But when the data is combined together and analysed as a whole, the trend reverses.$

$\noindent One of the famous real life examples was the Berkeley Gender Bias case. The entire university's admissions were significantly biased towards men. However, when the data from the individual faculties were looked at separately, each one had a small, but statistically significant bias towards women.$

$Not sure if your mind has exploded yet, but I'll go get the sponge.$

Haha yeah, it is an interesting result, I just meant that it's not really real-world as braintic wanted.

Interesting mathematical statements (1 Viewer)

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