The Four 4s - Numberphile

It was a famous problem for many years - until a physics genius solved it all the way to infinity.
Featuring author Alex Bellos - more links below.

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Videos by Brady Haran

[pete]

so the trick is you can use only 4 numbers, but infinite operators

[indianseibel]

Next we introduce shape shifting, now a 4 can turning into a 3 or a 5 .

[kloguy]

Today i'm going to tell you how to get infinity with just four fours but first we need to talk about parallel universes

[dalitas]

concatenation is a silly "operator"

[OKRolling]

The problem is, that sqrt isn't really an operation. If you write it as x^(1/2) then you're introducing another (non-four) number, so the puzzle doesn't work.

[stechuskaktus]

I remember this game. My personal favorite was 180 = 44 base 44.

[micheleferrari]

I obviously think this is cool, but it looks like "cheating" to me: √a is actually a^(1/2), so this operation contains implicitly the number 1/2. This means: each time I add a √ to the expression, I am using two 4s more, because 1/2 = √4/4!
I wonder how far one can get only using "pure" operations like sum, multiplication, logarithm (without fixed base), and so on.

[GameVerseGR]

I am a Jhin main and i had to watch this cause reasons

[IslanKleinknecht]

@1:35, whoaaaa, hold on now, I thought we were going to infinity with just the four base operators. You can't just change the rules like that mid-proof.

[Thewsx]

i want to see this video sped up by 4% every time he says 4

[XDCrown]

@2:40 He says that the number 99 can’t be made with “all of these”, meaning the operations addition, subtraction, multiplication, division, parentheses, concatenation, exponent, factional, decimal point, and square root.
BUT he is wrong, it can.

= 99

In fact all numbers up to 100 and far beyond can be made with just +, -, *, /, (), a||b, a^b, !, sqrt.

[pseudorandomly]

So basically the lesson is to successively redefine the problem until it becomes a problem you can solve. I daresay you can probably solve any problem with that technique.

[youtou]

"let me introduce the operator ++"
++4 = 5
++++4 = 6
= 7
= 8
...

wow, I can write all numbers up to infinity just with the operator ++ !

[fablungo]

That's incorrect use of the percent symbol. [4/4]% is 0.01, not 100 and even so, 1 "as a percentage" is 100% (i.e. 100/100). You have basically just defined the % symbol to mean "times 100".

My science teachers at school always used to incorrectly throw percents in and out of equations and it drives me crazy.

I am also not a fan of introducing the square root operator.

[ThatWarioGiant]

I hate concatenation because it is base-based

[kurzackd]

I don't like the 99. You can't really subtract from a percentage. That's an undefined mathematical operation. You can have a percentage OF something, not a percentage MINUS something that isn't a percentage itself.

[borix]

This is outright cheating: square root is a case of root function. To define it, you MUST spend digits. With this method, you just spoil your 4444 with lots of hidden 2

[fejfo]

I would say (4/4)%-(4/4) = 1%-1 = 1% - 100% = -99%

[derstreber]

"next we're going to introduce the symbol: ∞" see, that was easy.

[Nulono]

If you're allowed to include operations like square root (which has an implicit 2) and percentage (which has an implicit 100), then you might as well just add "increment" and "decrement" operators. Now 5 is just 4++, and 2 is just (4--)--.