The Four 4s (extra footage)

Featuring author Alex Bellos - more links below.

We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science.

NUMBERPHILE

Videos by Brady Haran

[jjtt]

Is this one video in a second channel that lasts three minutes, and is about fours?

[tempest_dawn]

Doing this in binary is a lot less fun. "One 1" and "Zero 0s" have very few ways they can be arranged.

[ddck]

It would be interesting to see if there's a similar formula for the n n's.

[Pooopers]

this guest was very fun, please do more with him :)

[TrimutiusToo]

it doesn't have to be 4... it works for any number except 1 and 0... because log base n of square root of n is always a half except for 0 and 1 where log base 0 and 1 just doesn't work even in complex plane... (though none of the points on unit circle of complex plane really work)

[Theraot]

Here is an idea: get any number using only four 4s, using any number of operators you want, but don't repeat operator.

Note: valid operators are any operator that has been ever used in a peer reviewed math publication prior to the current date.

[ethnofalcon]

I'm mad this wasn't four minutes

[illustriouschin]

If you thought that was amazing I'm going to blow your mind by just using a single 4 and a single operator which I invented called "₪" which turns the number 4 (and only the number 4) into infinity. 4₪ = ∞ .

[AlaskaSkidood]

I still want to know if this works for counting systems other than base 10.

[hosz]

Assume ArcSin() is a legitimate operator, then I claim the ultimate 1-line solution is this:
ArcSin(4-4)/ArcSin(4-4)
=(multiple Pi)/(n Pi)
= m/n
= any rational number.
Here I use only a standard multi-valued function, without borrowing, e.g. 2 from functions sqrt, or 1 from operators like C++. My solution applies to any 4 z's.
ArcSin(z-z)/ArcSin(z-z)
=any rational number

[alanhere]

X X's. With the first few binary hyper-operations and inverses, to yeald natural numbers, no big stacks of unity operations, no "concatenate" like operations:

No Nothings:

One One:

1


Two Twos:

2 - 2 = 0
2 / 2 = log(2, 2) = 1
2 + 2 = 2 * 2 = 2^2 = ... = 4


Three Threes:

3 + 3 - 3 = 0
(3 / 3) ^ 3 = 1
(3 + 3) / 3 = 2
(3 / 3) * 3 = 3
3 / 3 + 3 = 4
...
3 * 3 - 3 = 6
...
3 + 3 + 3 = 9
...
3 * 3 + 3 = 12
...
(3 + 3) * 3 = 18
...
3 * 3 * 3 = 27
...
(3^3)^3 = 19, 683
...
3^(3^3) = 7, 625, 597, 484, 987
...

[CircuzFunPants]

The cards seem to have a ridiculously small klickrate, only 50 views on this video but 4000 on the main one

[devinhead]

Hey can you check out this new game? Someone created a game out of conways game of life. It's called "The game of Life and Death". Really interesting game and I thought of you when I saw it.

[MathHacker]

at least he acknowledges that square roots are kinda cheating (I would say blatantly cheating)

[davidalearmonth]

It's completely cheating. Square root is Exponent of 1/2, so it should require a 1 and a 2. So yes, if you are allowed to use an infinite number of 1s and 2s, of course you can make Any Number.

It is an interesting solution, but completely unfair, in my opinion.