Mathematical Coincidences

In this video we take a look at some fun mathematical coincidences!
#math
#weird
#coincidence
#funfacts

[Kuvina]

CORRECTION: At 2:29, the identity should say 8pi*k^2, not 8pi^(k^2).

ANOTHER CORRECTION: At 1:36 I said the chance of getting 6 of any digit in a row within the first 768 digits is < 0.1%. However, I just ran a simulation (on 10 million 768 digit sequences), and I got 0.68446%. In my opinion a 0.69% chance is still notable enough to be in this video, but it's not quite as rare as I thought. The confusion comes from ambiguity in language. I thought the source meant "the chance that you get at least 1 of these patterns is <0.1%" but it actually meant "for any one of these patterns on its own, the chance you get it is <0.1%"

Also I got the notes wrong in the stack of 4ths. I did C F B E A when it should be C F B♭ E♭ A♭

Hi everyone, I hope you enjoy the video.

[FiniteSimpleFox]

In general the chance of a specific coincidence occurring is very low, however the chance of *some* coincidence occurring is very high

[notexactlysiev]

This is only half mathematical, but I like how the ratio of miles to kilometers (1.609344) is close to the golden ratio (1.61803...)
This means you can approximately convert those units using the Fibonacci sequence. 2 miles is about 3 km, 3 miles is about 5 km, 5 miles is about 8 km, etc.

[robo]

Here's another one. 82, 000 is in base 2, in base 3, 110001100 in base 4, and 10111000 in base 5. It is predicted to be the largest number to be represented using only 1s and 0s in all four bases and is thought to be a massive coincidence that a number so large even has that property to begin with.

[Karaokedad]

One that I was waiting to see if you mentioned:
10! seconds = exactly 6 weeks.

[robo]

My favourite mathematical coincidence is that if you look at space between e and pi on the number line and mark a point exactly two thirds across, that point is almost exactly the number 3 (3.000489)

[msman]

The first 40 seconds of the video is literally "How to memorize 15 digits of e"

[Yvelluap]

2:20 i envy your ability to convey this much raw happiness in a single drawing

[ryandupuis]

It should also be noted that these situations only occur in base 10, which is a human-based standard. Other bases may have coincidences like these, either more or fewer, though.

[Skybrg]

The fact that 2^31+1 is prime is one of the most useful coincidences in cryptography, since large primes are needed for the math aspect and modulo multiplication’s runtime is based on the number of 1s digits in binary which is useful for the calculation aspect.

[n]

My dude just causally explained the mathematical basis for music in the middle of this.

[johnbarnhill]

It’s worth mentioning that a lot of these coincidences arise because we use base 10. There might be other coincidences in other bases that we don’t know about

[parthhooda]

Math class initially: 6:40
me: *blinks for a nanosec*
7:40

[Relkond]

2 points about the 7th US president - he was elected in 1828, and served 2 terms.
If you draw a diagonal line across his square picture, you get a triangle with 3 angles: 45, 90, and 45

rewriting all that: 2. 7 1828 1828 45 90 45

[yellowmarkers]

One thing I like a bit more than the fact that π has a string of six 9s at digit 763 is the fact that 2π has a string of seven 9s at digit 837. It isn't the first instance of four characters in a row in 2π's decimal expansion (since there was a "1111" before it) but it's still the first instance of five, six, and seven characters in a row.

[kyay]

You note around 1:20 that the pie coincidence is a product of language, but it's important to also note that a lot of the coincidences in this video are a product of using a base 10 system, and that they thus are "arbitrary"

[themathhatter]

Other mathematical coincidences involving pi:
sqrt(2)+sqrt(3) ~~ pi
9/5+sqrt(9/5) ~~ pi
e^(pi*sqrt(163)) ~~ (640320)^3+744

Not involving exact mathematical numbers
(mile*Astronomical unit)/(inch*light year) ~~ 1

[Dissimulate]

I see average people being surprised by coincidences. I try to explain to them how with the number of things that they see, these coincidences are almost certain.

[pinkraven]

As an amateur musician it always fascinated me how actually lucky it is that 12 tone equal temperament (where each note is the previous one multiplied by 2^(1/12) can get you so close to the most important musical intervals such as 3/4 and 2/3. Sure, maybe that's not as surprising, because from all the possible reasonable divisions of an octave, like 13, 14, 15 notes, one of them should be good enough in approximating these crucial intervals, but, idk, it's very pleasing to me

[FZs]

My favorite coincidence is that the sines of the most commonly used angles (0°, 30°, 45°, 60°, 90°) follow a pattern:

sin(0°) = 0 = sqrt(0)/2
sin(30°) = 1/2 = sqrt(1)/2
sin(45°) = sqrt(2)/2
sin(60°) = sqrt(3)/2
sin(90°) = 1 = sqrt(4)/2

This doesn't work for any other value though.

Despite that, this is how I always memorized them in school (the cosines are the same but the other way around, because cos(x) = sin(90°-x), and tangents are just sin/cos).