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.
Extras from this interview: • The Four 4s (extra foo...
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[@isaak.studio]

May the fours be with you.

[@anon8857]

Isaak haha ! genius

[@danjbundrick]

Isaak - heyoooo!

[@SHUBHAMGI]

hahahaha

[@brandonkaravite1754]

Isaak finally a not mathematical comment thank you

[@spootot]

Star Fours The Fours Awaken

[@pete275]

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

[@Tigershark1989]

The trick is that they think a square root is okay, but exponating with (1/2) actually is using extra numbers.

[@JNCressey]

Introduce the incremenet operator, ++, and it becomes super easy! 0=4-4+4-4. 1=++4-4+4-4. 2=++++4-4+4-4. 3=++++++4-4+4-4. 4=++++++++4-4+4-4. 5=++++++++++4-4+4-4. etc Since, hey, we're allowed as many operators as we want.

[@mmdrdd]

Would something like (((4!)!)!)! an infinite amount of time work too ?

[@xzilla5915]

No, you would have to use all four 4s

[@bluishsnowflake]

How about (((4!)!)!)!x(((4!)!)!)!x(((4!)!)!)!x(((4!)!)!)! ? Add as many ! as you want.

[@indianseibel123]

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

[@batuozer7179]

Then, unite for better: four 4s can get together to make any real number

[@MayorVideo]

Then introduce variables

[@MayorVideo]

Then exploding numbers and teleporting numbers

[@BorgOvermind]

That's similar suggestion to what Fakestein did in physics.

[@LudwigvanBeethoven2]

lol

[@kloguy9945]

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

[@basilkehayas9659]

Kloguy hhbbh

[@javierjimenopresas1039]

Omg didnt laugh so much at a comment in a while 😂😂😂

[@imbbgamer1012]

XD

[@canaDavid1]

I see, a fellow pannenkoek fan

[@alexmurphy4277]

yea i agree

[@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.

[@sergiogalanmedina8851]

Was looking for this comment (Y)

[@KUNDANNOVELIST]

square root(x) and x^(1/2) aren't the same thing.They are Different.

[@takatotakasui8307]

Fleak Exponents are operations. They're the "E" in PEMDAS.

[@Morgormir]

Fleak I disagree, consider (4)^(1/4) * (4)^(1/4). Properties of powers hold here, so this becomes (4)^(1/2) = sqrt 4

[@Tsskyx]

What are you saying? Of course it is an operation. It's just a shortcut for writing inverted exponents.

[@dalitas]

concatenation is a silly "operator"

[@ThunderChunky101]

Dalitas D It is indeed.

[@OsamaRana]

Think of it as no operator.

[@EANTYcrown]

it can be understood as times 10 +. so 4 concatenated with 4 would be a short way of saying. 4(10)+4, and add another 0 to the ten depending on the position in the concatenation that the number will have

[@nicholasosullivan831]

Dalitas D this sketch has become entirely too silly. Month python ftw

[@cryptexify]

Computer Scientists disagree.

[@cumcad]

Instead of writing [4/4]% he should have written [4/(4%)]. That way there's no confusion that it's genuinely 100. 4/(4%) = 4/0.04 = 100 Versus [4/4] as a percentage = 100 percent = evaluates to 1.

[@noobpro5674]

Exactly...

[@cumcad]

@@tifsa I think he makes a distinction between round parentheses and square brackets. By [4/4]% he meant "calculate the brackets, but evaluate it as a percentage."

[@neerajbhatt700]

Exactly , I thought the same thing

[@aadityasharma8013]

Yes I also thought that

[@pedrosso0]

@@cumcad %=0.01 so (4/4)%=1*0.01=0.01=1%

[@stechuskaktus8318]

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

[@RazvanMaioru]

That's just 44×4+4. Easier way of doing it, still with 4 4s

[@stechuskaktus8318]

@@RazvanMaioru Well sure, but it's not about being easy. Easy is boring. 44 base 44 is at least different and creative.

[@CYXXYC]

How about (4-4)/(4-4)?

[@t.487]

Creative & Random its 1.

[@xhonyalla6013]

Creative & Random its a secret bit it's 1

[@Reydriel]

TyloniumTV 0/0 is NOT = 1. It is indeterminate :P

[@vlad-ioantir9651]

no its not :P

[@MrTuas]

Creative & Random well thats easy, that equal to *undefined answer* pretty simple.

[@micheleferrari1803]

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.

[@Mystery_Biscuits]

sqrt(4)/4! = sqrt(4)/24 = 1/12. Be careful not to accidentally math when expressing mathematical opinions. XD (I'm using math a verb intentionally)

[@micheleferrari1803]

True! Didn't notice that. Let's use (4/4) / ((4/4)+(4/4)) then. :D

[@filipsperl]

Well technically if you follow this kind of reason, multiplication isn't a 'real' operation either. 4*4 is essentially 4+4+4+4, and therefor i've used my four fours just to do this simple multiplication.

[@micheleferrari1803]

Well, that's not my point. I know that 4*4 = 4+4+4+4, but the symbol * itself does not contain any "number information" while you write it. The symbol "√" does contain that kind of information: the “2” index of the root is simply omitted, but it’s clearly there. Something similar happens when you write “ln( )”: the base e of this logarithm is just omitted, but it would be wrong saying the symbol “ln( )” has nothing to do with e. I hope I made myself clear :)

[@mathyoooo2]

Michele Ferrari a shorter version of 1/2 = 4/(4+4)

[@The1wsx10]

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

[@Luka_D_Snots]

Your comment is now 4 years old. If my reply popped in your notification, congrats, you’re seeing your past comments in 2022

[@The1wsx10]

@@Luka_D_Snots oof

[@MarigoldIsMelancholy]

And your comment has 44 likes (as of this comment)

[@MarigoldIsMelancholy]

4 replies now ;)

[@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. (sqrt(sqrt(sqrt(sqrt(4^-4!))))+4)*4! = 99 In fact all numbers up to 100 and far beyond can be made with just +, -, *, /, (), a||b, a^b, !, sqrt.

[@nathsmath2871]

You are a genius! How did you come up with this?

[@TheZoneEater]

XDCrown "-4!"?

[@XDCrown]

Zone-E -(4!)

[@pedronunes3063]

Nath's Math I wondering as well, he really is a genius

[@tuckersmith7991]

The first part : (sqrt(sqrt(sqrt(sqrt(4^-4!))))+4) = 4.125 the second part : 4! = 24 the product of these two 4.125 * 24 = 99 I have no idea how he came up with it tho...

[@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.

[@BonzaiThePenguin]

Nick Clinite I mean, obviously the four base operators can't go to infinity on their own, you can only get to 4*4*4*4. Incidentally you can't get to infinity with any other operators either, it relies on the hidden (1/2) in the square root operator.

[@animowany111]

The other problem, is that sqrt is a unary operator, that simplifies to the binary exponentiation operator with a fixed argument. If I claim the successor function succ(x)=x+1 as an operator, that also defeats the puzzle.

[@JJJMMM1]

He never said that.

[@pausallent4597]

Obviously you can just get to 4^4 with the four base operators

[@Architector_4]

4/4/(4-4)=1/0=infinity. YESS IM GUNIUS

[@youtou252]

"let me introduce the operator ++" ++4 = 5 ++++4 = 6 ++++++4 = 7 ++++++++4 = 8 ... wow, I can write all numbers up to infinity just with the operator ++ !

[@nsapre]

And --, you can got to 3,2,1,0 ...

[@4345ghee]

for( c=0; c < infinity; c++) { int i=4; i++; } // 0wN3d

[@nsapre]

Several problems with code. 1. infinity - undeclared identifier, your program won't build. You can just leave it blank, which means there is no condition to exit the loop. Even then every integer (signed, unsigned, 32 bit, 64 bit) in C/C++ has a range, you can only go to the upper limit of c. Technically i will never go to infinity but repeat after the range is hit. 2. Say you can go up to infinity, your code is incorrect. int i = 0 should be outside the for loop or declared as static variable. In this case it will always be value of 5 for infinite number of times.

[@nedasbolevicius8676]

Nikhil Sapre infact, I'm pretty sure you can't use int twice on the same variable because it is already declared.

[@AlehenriqueGamer]

+Nikhil Sapre Pwned

[@benjaminvandenberghe9489]

Brilliant ! Although I find it annoying that we have to use 4s to build the base of log operator, but we are exempt of building the power of the squareroot, which is considered implicitly as 2.

[@user-fp7jz4ot6f]

its becaues the root is by default 2 and the base of the log is by default 10 or something (some pepole say that it have no default)

[@avadrumm]

@@user-fp7jz4ot6f the default logarithm is logₑ (at least according to most people and wolfram alpha)

[@missseaweed2462]

I'm lost in a forest of mathematics. I swear I've seen that log several times already.

[@GameVerseGR]

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

[@kickowegranie3200]

Basically you've had a FOURgasm

[@cellenttri2272]

Lol

[@Night_Hawk_475]

Here now because this video and your comment are 4 years old, and it's started showing up in my recommended randomly. I think the universe is telling me something.

[@brianmchaney7473]

4⬆️⬆️⬆️4⬆️⬆️⬆️4⬆️⬆️⬆️4

[@baptistebauer99]

Dammit that'd be a lot :o

[@devinpohl5532]

That's a... that's a big number.

[@enzy9864]

Brian McHaney is that the Ackerman function of 4?

[@MamboBean343]

Ilan Goldman the Ackermann-Péter function (the most common form) takes 2 arguments

[@UndefinedBehavior]

Ilan Goldman It's Knuth's up-arrow notation (en.wikipedia.org/wiki/Knuth's_up-arrow_notation)

[@terrowincheeseman5228]

Guido Mista: *TRIGGERED*

[@GMDThread8]

I knew there will be this comment XDDDDDD

[@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--)--.

[@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.

[@numberphile]

+Pseudorandomly that wasn't the lesson I took from the interview. I thought it was a nice look at a Victorian puzzle with a colourful past and a light-hearted insight into the way different people play with numbers - including Paul Dirac who some argue is the greatest mind of them all. ;)

[@pseudorandomly]

+Numberphile You'll get no argument from me about Paul Dirac -- and it _was_ an entertaining video, as are all of the Numberphile series. But it is rather like answering the question "how high can I throw this rock?" by saying "well, if I reshaped the rock to be aerodynamic and then attached a rocket to it with a really big fuel tank and a guidance system, I could hit the Moon". :-)

[@TEnduril]

Given that the final solution at least contained only operators and not actual other numbers, I'm inclined to give them this one. But I was worried partway through.

[@pseudorandomly]

Thraviol Enduril Well, I'd argue that things such as square root and percentage and factorial are functions, not operators. But I recognize that others can and will have differing opinions.

[@therabbits69]

the original games rules as stated is to NOT use any other numbers besides the four 4s you are given. That's it. You can use any amount of functions/operators as you'd like.

[@nathsmath2871]

I can make every integer by using only one 1. log(1)=0 -log(sqrt(1%))=1 -log(sqrt((1%)%))=2 -log(sqrt(((1%)%)%))=3 ...

[@dxjxc91]

And log base 10. And I didn't like the use of % because it really just means divide by 100.

[@nathsmath2871]

If the base has to be specified, then I have to use one more 1. log_(.1) (1)=0 log_(.1) (sqrt(1%))=1 log_(.1) (sqrt((1%)%))=2 ... This is not spoiling the puzzle because decimal, square root and percentage are really basic, and log is a pretty simple thing.

[@rohinijoshi-katekar7998]

LoSir MATH Wow!

[@karolakkolo123]

Nath's Math actually it's supposed to be log_10(...) but I know what you're saying

[@Hairregrowthjourney]

Amazing

[@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.

[@gamingguitarist6927]

what why not % is just divided by 100.

[@SubbuFortyFive]

a better way would be 4/(4%) - 4/4 imo

[@RedRad1990]

​@@gamingguitarist6927 exactly, "a percentage of something" is just what we usually use in our everyday life IMO 1% and 0.01 are the exact same thing

[@hamster8706]

A percentage has a value,100% then its equal to 1 techically. So you can subtract from a percentage

[@horaceaugustine9698]

Hello, i have a math problem and wonder if you can help?

[@tbabubba32682]

I can't believe Paul Dirac won the Nobel prize for solving this puzzle.

[@freshrockpapa-e7799]

Are you silly? It wasn't for solving this thing, it was for something unrelated

[@derstreber2]

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

[@Uncle_Yam]

You can't make every number with ∞

[@tidorith]

@xnotx123 Sure you can. ∞ - ∞ = 17, for instance.

[@jihoonkim9766]

Infinity is not a number, so you can't do any operations on it.

[@tidorith]

That's a matter of definition, and while there is a strong consensus on what typically constitutes a number, it's not universal. There are number systems that *do* allow the treatment of infinite quantities of numbers. Conversely, you can have a number system in which negative quantities cannot be represented as numbers. It's all a matter of what you want to be able to do with your numbers (and what you want to *not* be able to do with them).

[@jihoonkim9766]

Richard Nesbit Yeah that's true, but I was talking about the standard number system.

[@stellarfirefly]

Some comments keep insisting that square root isn't "really" an operation because it's just a shortcut to an inverted exponent of 1/2. I'd say that reasoning is silly, because the same reasoning may be used to insist that multiplication is just a shortcut to repeated additions, and thus 4*4 shouldn't be allowed because it is really 4+4+4+4. That's not even getting into unary ! being 4*3*2*1.

[@numberphile]

+stellarfirefly ssssh, that's far too sensible. Next you'll be suggesting this video is not about the internationally established rules of four fours, and rather a whimsical look at a puzzle dating back to Victorian times and some related numberplay by one of the greatest scientific minds of all time.

[@bidaubadeadieu]

I disagree. To me, the nature of the puzzle is that it only uses the number four, like that's what makes it so uniquely interesting. So multiplication is totally allowed, even though it's a shortcut, because ultimately it still only uses fours! And, personally, I think that factorials shouldn't count, because just as you said, they employ non-four numbers. I think if you allow factorials and you allow square roots and you allow operations defined by any number other than four then you might as well say "Hey, did you know that you can use some of *any* number to create any other number?" With your logic, what's to stop me form defining an operation like a @ b = a * 5b, and then I could suppose that 4@4 = 80. To me that seems like obvious cheating, you could just define an infinite number of operations such that you can create any number trivially. It's no fun that way.

[@epigeios]

sqrt() is definitely cheating. Because if sqrt() is allowed, we might as well count ++ as +1, and trivialize the whole problem to nothing.

[@Pangui008]

I'm confused. I always tought of square root as the 2nd root, so it means that it's an operator AND the number two... the same way that squaring is the same as raising to the power 2. So, according to the rules, you should be able use roots, but not the second root ("square root"), just the 4th root. What am I wrong?

[@Necronpharia]

Root is an operator, sure. But with the square root comes a 2 and we're just skimping it out of convenience. So the problem isn't really solved in my opinion.

[@ThatWarioGiant]

I hate concatenation because it is base-based

[@chaiguy1337]

Not only base based but position based. Its meaning changes based on what its input is.

[@angelmendez-rivera351]

Logan Murray That is not a problem, though. Most functions work like that. In many number systems, multiplication isn't even commutative. So there is nothing wrong concatenation being position based and base based. Also, y'all don't have problems with logarithms even though it by definition is based.

[@alax155]

@שחר א. Wait... what? xD

[@arnouth5260]

Logan Murray that’s true for most things, 5-4 is not the same 4-5. 5/4 is not the same as 4/5 5^4 is not the same as 4^5 The fifth root of 4 is not the same as the fourth root of 5. See what I mean?

[@LordQueezle]

To be honest, at the beginning I was skeptical that there was a "special" formula that would solve this... I stand corrected and am thoroughly amazed! Thank you Numberphile!

[@fejfo6559]

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

[@PM-vs3rh]

fejfo's games 1=100%. 0.01=1%.

[@fejfo6559]

yes I followed those rules the video didn't

[@quarkyquasar893]

% as a operator is used for percentage. How much percent 4 is of 4. While what you seems to think about is the percentage symbol.

[@franzschubert4480]

fejfo's games I agree with you. But % is pretty much the same thing as /100 so I don't consider it a valid operator for this problem in the first place.

[@PM-vs3rh]

fejfo's games (4/4)=100/100=1

[@CTJ2619]

using an unlimited number of square roots is a bit of a cheeky cheat

[@kolrabi]

I agree, especially when considering that there is a hidden "2" associated with each radix.

[@cxpKSip]

Do realize that a pair of square roots is the same as the fourth root, whose exponents can be written with 3 4's: 4/(4*4)=4/16=1/4.

[@CTJ2619]

Garrison Pendergrass yes i did know this.

[@van_Streek]

May the fours be with you

[@alexmurphy4277]

yea, i agree

[@sk8rdman]

Okay, I have to voice my objection to the use of the use of square root. Square root is just root two, and you're not allowed to use twos. The fact that we don't typically write the two is just common notation, but the two is always implied as a value that defines the operator's function. I will concede to the use of root4, but doing so should use a 4. Maybe it can still be done somehow, I don't know. Here you're not using just four fours and any number of operators. You're also using any number of twos, which is cheating.

[@hsheheishje9649]

actually, he isn't, he is stacking a stackable operator to get any number. that's like saying for example, the problem, 4x4+4x4. when you multiply four with four you are now dealing with sixteen and thus you are cheating as you aren't using four anymore.

[@daminkon246]

but... hes using one of the fours when he uses a square root? idk what you're talking about

[@sk8rdman]

@@daminkon246 My argument is that he's also using a 2, because saying the square root of 4 is the same as saying root 2 of 4. They're the same operation. Would he allow other radicals like root 3? I think not, and therefor root 2 should not be allowed either.

[@kentmartin9289]

@@sk8rdman you could take it a step further then and say using multiplication is cheating, because 4*4 is actually just notation for saying 4 + 4 + 4 + 4. Same with factorial, it's a notation for saying 4 * 3 * 2 * 1. I understand your point, but I think it's going a bit far. If it was ambiguous without the 2, then I would agree, but the symbol used is not ambiguous, despite the lack of putting the 2 in there.

[@slobodyanukma]

If solution could use operations which closure on some number except 4 then straight way Is define infinity set of constant functions numN(4, 4, 4, 4) which return any N. On other hand we can't avoid closure on 0 and 1 somewhere because they are basic constants for construct any number.

[@borix2600]

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

[@thomassynths]

And hidden 1's

[@Arkayjiya]

square root is the function that transform a number "x" into a number "y" such as "y*y = x" with x and y belonging to |R+. So you don't need digits to define square root. If you consider you need digit for square, how about needing digit for defining multiplication? Or simply needing to use "successors" (but if you start using successors, you can simply use an unlimited number of them to obtain every number after 4).

[@borix2600]

You do need digits to define multiplication, at least two of them.

[@jscarborough9928]

Aesahethr you do need a digit to define the square root. We're so used to not thinking about it, but the square root has an index of 2. It's the same as log in that regard. Most people think log is intrinsically root 10 because that's how the calculators work. If you needed to use 4's to define which log you want to use, then you need to use 4's to define which radical you want to use. Another way to think of it is that the square root is not its own operator. It is another way of writing the exponent "1/2"

[@Arkayjiya]

"Another way to think of it is that the square root is not its own operator. It is another way of writing the exponent "1/2"" Yes, thank you, everyone knows that, that's besides the point I was making. But regardless, I just gave you a definition of the square root without using the number "2" with only unknown, multiplication and equations so it CAN be defined without the digit. Using the digit is just an alternative way to define it. edit: Why did my answer appeared twice??? I only typed it in answer to one of your messages. KZbin is seriously bugging.

[@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.

[@Valvex_]

Yeah, since it's basically adding the power of 1/2, it's almost like adding a number.

[@MasterBecause]

It's just a symbol representing an operator, as long as you explicitly state its meaning in the given context, you can define it to mean anything.

[@fablungo]

Mr. B. but defining it to mean "times 100" then just feels like adding numbers.

[@fablungo]

Halil Aydin Yeah, exactly. I would feel more comfortable adding an operator like ^ to allow exponent using the four 4s but adding that restriction would add the need to find new solutions, different from those in the video. That proof shown here for being able to get up to infinity requires the square root operator. The problem doesn't have any hard rules as such and so you can just invent your own symbols to perform arbitrary functions and its going to be down to personal preference whether you think those operators are acceptable.

[@sevret313]

4/4 is the same as 100/100, I don't see your problem. n/n is 100%, for and non-zero value of n.

[@guytorie]

I have OCD and my main favorite number used to be four. I've always been in love with the idea of four fours in a row, but all this stuff is completely new information to me. This was one of the most satisfying videos I have ever seen, to the point where I was emotionally moved in a way math has never done to me before. I feel spiritually refreshed.

[@juanmartinmaffi3579]

Nice to watch this 4 years later, with Numberphile having 4M subscribers

[@RitzScythe]

2:04 the numbers look so happy.

[@trobin]

I got up to 60 back in 4th grade... that was fun

[@MVBit]

I could _count_ to 60 in freaking PRE-K.

[@justas423]

*Mista crying in the background*

[@dotaro3741]

Slimeustas does he need my _healing_

[@hachikiina]

shi

[@argonburnspink1441]

Bruno come pick me up I'm scared

[@everythingyoudoismuda92]

don't worry, i'll set it back to 0

[@randomperson1955]

@@everythingyoudoismuda92 not if the world over heaven has to say something

[@gogl0l386]

Hop out the four dour with a four four, it was one two three and four.

[@theleftuprightatsoldierfield]

GogL0L chillin’ in the corridor Your dad is forty four

[@targuscinco]

All you heard was papa don't hit me no more.

[@musictest9999]

i was impressed up until about 1:42

[@General12th]

What's wrong with concatenation?

[@TommiHimberg]

True, I found concatenation to be "cheat-y" as well, in this context. But it is an example of what kinds of extra rules you could use to make the four fours go further. And in the end, concatenation wasn't needed in the puzzle-busting solution, only log and a big box of square roots.

[@tennenrishin]

He didn't seem very happy himself at that point.

[@Ditocoaf]

Concatenation introduces a hidden "10" because of base 10. "x concatenate y" is basically defined to mean "x times 10 plus y", which means we're no longer just dealing with 4s. The decimal point has a hidden 10 as well. The square root also has a hidden 2 (or a hidden 1/2). Sure, allow roots, but the *fourth* root, using one of the 4s.

[@entropyzero5588]

+Ditocoaf Actually, "x concatenate y" is even more complicated than that, since you could also say "44 concatenate 44" this way. What you actually need to do is "x concatenate y" := x * 10^(⌈log_10(y)⌉) + y

[@bcn1gh7h4wk]

(4+4)/(4-4) = infinity. genius! :P

[@konstantin0sV]

lim [(x+x)/(x-x)] x->4 That make sense😉

[@stefanusefanruntunuwu8269]

8/0 is not equal to infinity, it's undefined

[@kayzero9689]

A better argument would be (4-4)/(4-4)=0/0....lets take 0/0=x notice any value if x solves this equation ...u got ur answer... Well not really.. But atleast now u understand the difference between infinity and not defined

[@wingsofmathematicsbytanush2507]

@Dark Pink Cow correct. It will be tends to or approaches to ∞.

[@newkid9807]

Nighthawk you’re scaring me...

[@matthewkoehler6079]

I loved that I discovered this 4 years later

[@otaku3OBSESSION]

We did an exercise in 5th or 6th grade using the four 4s. It was one of our most fun, and memorable worksheets.

[@connorcolestock4757]

This was great fun! My grandfather has a running joke of four "being the answer to anything," which I always took at nonsense out out context. I always reasoned it was likely possible with more than one operator or perhaps it was a note on relativity by which he meant you could factor four out of every number. maybe I'm just now getting in his level :) thanks Numberphile!!

[@cfgauss71]

We recreational mathematicians can thank Dirac for taking the joy out of this puzzle.

[@hoagie911]

You seem like the kind of guy who would try to convince me my laptop was out of date and I needed to buy a new one, and you would "generously" offer me a small sum in return for my current one.

[@SirHenryMaximo]

That's exactly how my teacher introduced The Four Fours to us. In fact, I learned the concept of factorial in the occasion. Edit: But later, the problem was presented to me with a rule prohibiting the use of log or root.

[@Luka_D_Snots]

Your comment is now 4 years old. If my reply popped in your notification, congrats, you’re seeing your past comments in 2022

[@MaicahRu]

I've come up with a much simpler solution, you start with 4+4-4+4 to get zero, then use the ++ operator to get to any number you want

[@roflwaffles1001]

This video was brought to you by Wendy's(TM) Try the 4 for 4 combo meal today

[@neilisbored2177]

On 4/4/2004.

[@debhulbert]

OMG YES XDXD

[@LENGTHEATER]

If you introduce making the fours into letters, one could also make many words.

[@giorgioa.tsoukalos3653]

Alien mode: 4 4 4 4 = e the answer is truly easy.

[@WasickiG]

No Theory of the Ancient Astronauts required for this one: 4√((√4/4)!^4)=π

[@ayeariola]

Genius mode: sqrt(.(5!))*(5/5+5/5)=pi

[@bulgaria9003]

4^4-4-4 =248

[@isaacmerida6691]

(4^sqrt(4))||(4/4)=161

[@doormango]

I can do it all the way to infinity with only *ONE* additional operator. I call it "increment".

[@Jakromha]

100% - 1 = 0 100% = 1

[@davecrupel2817]

Jakromha 4

[@lauragoncalvesfranco5299]

He meant 4/(4%) = 100

[@nguyen7574]

OMG 99 likes :)))))

[@Greg-ix4nu]

Laura Gonçalves Franco I get that, but why'd he put the percent outside of the brackets?

[@derekmartin5340]

No he meant 4/4 as a fraction is a ooh I know what u mean

[@jojojorisjhjosef]

I bought this mans book, great read.

[@monitoringevaluasi9707]

Me: *watch a video called "The Four 4s" Also me: Yeah, its a big brain time.

[@newkid9807]

MONITORING EVALUASI We all hate you!

[@martinshoosterman]

sqrt((4-4)-(4/4)) I win.

[@kuskus_th13]

martinshoosterman I get it

[@want-diversecontent3887]

martinshoosterman *insert that equation here* 4+4+(4-4) some 3.141592653589

[@AnonYmous-xs8nx]

martinshoosterman sqrt of - 1? Can't have sqrt of a negative...

[@MVBit]

Anon Ymous Yes you can. sqrt(-16) = 4 *i* sqrt(-2) = 1.41... *i*

[@AnonYmous-xs8nx]

Dante Thompson OK, can't have a sqrt of a negative that involves real solutions. In this context, I don't think imaginary numbers are relevant. I see your point, though. Well done.

[@TwentySeventhLetter]

I can always count on mathematicians to make tally marks _reeeeally_ convoluted

[@Luka_D_Snots]

Your comment is now 4 years old. If my reply popped in your notification, congrats, you’re seeing your past comments now in 2022, do you still remember them?

[@TwentySeventhLetter]

@@Luka_D_Snots Not even a little! But this was funny to see in my notifications!

[@ValexNihilist]

@@TwentySeventhLetter Now its been 5 years!

[@loganisanerd5566]

​@@ValexNihilistYou're never gonna guess what just happened again...

[@ValexNihilist]

@@loganisanerd5566 :O Epic

[@WombatSlug]

4 years later, this pops up again on my feed...

[@joshyoung1440]

This reminds me of the SMBC joke about a Fourier transform meaning you take some numbers and you derive all the fours from them and see which one is fourier and whichever number has more fours is the fouriest.

[@gamer_kid_naz4942]

Take a shot every time he says 4

[@_yellow]

*dies at **1:57*

[@want-diversecontent3887]

Gamer_Kid_Naz *puts on music to become invincible* Done and alive. Seriously. These comments are getting annoying.

[@aircraftcarrier6789]

Now make all real numbers with just one 7.

[@batubatucp]

Your Crush *introduces the infinite operation*

[@TutukaBk]

I can do it with just 2 operations, try harder

[@want-diversecontent3887]

Aircraft Carrier 7-+_*€|+]€|£|€ = every number

[@Mars8765]

Aircraft Carrier Here is what I can make.. (with concentation and other things... (7 77 777 7777 77777 etc.) (7 .7 .77 .7777777777 .77.............????????????

[@Peter_Schluss-Mit-Lustig]

@@Mars8765 i can do infinity with 2 e's log_ln(sqrt(e))(ln(sqrt(sqrt(...e))...) n sqrt's in the second term =n

[@Blurksl]

Needed this for class thanks👀

[@hmekhtar]

This really helped me with my math homework

[@oscarmorales7406]

Great! i'm going to be the soul of the parties with this

[@__w__o__w__]

I thoroughly enjoyed this video. One of my favourites in a long time.

[@LaVictoria6751]

Fascinating! I am reviewing my elementary math!

[@BlueEyesWhiteTeddy]

If you do it properly you can't use roots except the 4th roots. This is because we remove the 2 for convenience, the 2 is still there.

[@studybuddy7060]

exactly, square roots felt like cheating to me. It was basically a disguised exponent (which were allowed), so you were defining a symbol for a already defined operator but for a special case of n=½

[@pegy6384]

That was really cool, but I guess I misunderstood the problem when I heard about it the first time. I thought there was a constraint to only use one operator per number.

[@L1ft0ff]

Peg Y haha funny

[@matix676]

You can do whatever you want with those four 4's, as long as the only numbers you are using are those four 4's.

[@Cashman9111]

yeah, that would be way more challenging also, why is your comment 9 hours old if the video was uploeaded half hour ago ?

[@pegy6384]

Cashman9111 Patreon supporters often get sneak peeks at the videos before they officially go live.

[@lyrimetacurl0]

Graham's number: g(sqrt(4*4)*4*4) TREE(3): TREE((4+4+4)/4)

[@Vinny_3041]

Wut

[@hyrumtaylor9974]

Graham's number should be g(sqrt(4*4)*4*4)

[@lyrimetacurl0]

​@@hyrumtaylor9974 corrected

[@adum2ge4]

I have ocd. And I count in 4 4's with my procedures this video was a overload of goodness

[@151bar151]

this is absolutely brilliant, no irony

[@benjaminv3748]

just say infinity is an operating symbol already

[@mynewaccount2361]

But it isn't

[@rahulsbhatt]

Hey i know you from insta clash community . Sup

[@quin2910]

But you can't get any number with infinity

[@hypercube9531]

How to get infinity with 4 4s; (4/4)/(4-4)

[@khalil1626]

actually 0^0 isn't infinity... its more likely to be 1 but numberphile has a video on it -"the problem with zero" (or something like that).

[@hypercube9531]

Khalil okay, I fixed it. I messed up the equation when typing it. 🙂

[@user-rv9vk8by5i]

1/0 isn't infinity either, as infinity is not a number and cannot be used to answer an equation. Double check and fix again.

[@hypercube9531]

Can you try it? Because as I remember, any number divided by 0 is infinity because it keeps dividing without stopping. Anyway, cut me some slack, i'm a kid.

[@user-rv9vk8by5i]

It's okay, but what you remember is wrong. Or whoever told you that failed math. Here, lets use some basic algebra, where INF = infinity 1/0 = INF 2/0 = INF Therefore, 1=2 and 0 = 1*INF

[@isabellabornberg2153]

absolutely beautiful

[@AEIOUY234689]

I love the way he says "log" :-)

[@iankunesky6740]

All of the operations were fine except for square root and log. Log was technically fine too because you were expressing the base with 4, but a square root uses a base 2 not 4, so you would need to use root 4, or express root 2 in terms of 4, which would require more 4s

[@ludmilaharing2030]

I was thinking this the whole time too, because of this i think it is not a valid solution fro the problem.

[@kademate6888]

I appreciate the explanation of logs

[@vrusa9557]

4 4 4 4 4+4-4+4 4+0+4 404 ERROR: NUMBER NOT FOUND therefore, 8 does not exist (i haven't even watched the video, i just saw four fours and thought “oh, an alternating sum opportunity approaches”)

[@bunbunnbunnybun]

Discovering something that doesnt exist -> 8

[@danjbundrick]

"As a percentage" doesn't multiply a number by 100. 100% is one, not 100.

[@CraparellaSmorrebrod]

Or you could define the three operators "addition", "makes-it-zero", and "increments-by-one", and then just go to town with infinite applications of the latter. Boom, solved in 10 seconds :-) And I really believe that my solution has about the same level of lameness as the method illustrated in the video.

[@phyllisstewart5591]

You don't need logs. You DO need repeating decimal sign, which will multiply or divide by nine, just as .4 will mult and div by ten. Very useful.

[@BorgOvermind]

Shorter version, using Successor function: 1 = 44/44 2 = S(44/44) 3 = S( S(44/44) ) n+1 = NS (44/44)

[@hakachukai]

This concept is nonsense! You're not only using four 4's! When you use square root you're raising something to the 1/2 power, which means that you're no introducing digits other than 4 and also going outside of the original number of digits ( which is supposed to be 4 ). For the same reason, use of factorials should be ignored because it does the same thing. Basically all you're doing is disguising additional digits in the form of mathematical operators. If you take that apporach you can make four of ANYTHING equal anything you want... which means that it's not special in any way.

[@petrag0n]

The only nonsense here is the use of logs. The use of square roots is widely accepted by many websites out there. There are websites dedicated to this.

[@leonhardfrommhold8463]

hakachukai but with just the normal operators there is a limit to how far you can go because you can't use multiple ones in a row. It's impossible to reach numbers like 10^10^10 without multiples symbols in a row

[@quoteoftheday3847]

hakachukai ((4!)×(4!)×(4!)×(4!))!!!!!!........=infinity

[@quoteoftheday3847]

((4!)×(4!)×(4!)×(4!))!!!!!!........=infinity

[@techoo]

4*4*4*4=256

[@Mike-739]

Quick math

[@petergriffin8767]

Ok

[@plaguexdocta8638]

4^4 x 4^4

[@kornsuwin]

@@plaguexdocta8638 4^4^4^4

[@kornsuwin]

@1729 number blog P([4^4^4^4]!)

[@vortexlegend101]

Love this!

[@bikramkoirala.]

Excellent.............................................

[@edoardograsso3391]

I think that the only problem here is not the infinite number of operators, because there are no restrictions about it. The problem is that a square root is a root with index 2, as you said not to be lecit doing 4^(1/2) for you're using an extra number, i think that to be completely lecit you should have used only roots with index 4 avoiding square roots too for their index is 2, which is an extra number. At least that's my opinion, great video anyway! Very intresting as always

[@Fircasice]

But that would still exceed the boundary on the amount of 4s there can be, which is 4.

[@zackfair54]

@@Fircasice, that's what he is saying.

[@TylerMatthewHarris]

4give me but I 4get what a 4's 4.

[@feynstein1004]

That was a bit 4ced.

[@TylerMatthewHarris]

Lol. I 4got all about this video. 4tunately I had push notifications turned on.

[@feynstein1004]

Tyler Matthew Harris Lol touche'.

[@cartervandenberg4771]

Watching this 4 years in the future

[@wallonice]

0:03 thanks for introducting the fours, as if no four understood it

[@alkenrinnstet]

Parentheses are not an operator. Mathematical operators have no intrinsic precedence. Parentheses are merely tools for explicitly notating precedence; to mistake them as comparable to mathematical operators betrays a fundamental lack of understanding, and serves only to mislead.

[@alkenrinnstet]

100% - 1 is not 99 100% = 1 100% - 1 = 0 What a joke.

[@BENBOBBY]

What do you mean mathematical operators have no intrinsic precedence? Never heard of BODMAS or PEMDAS?

[@AlexKnauth]

Alken: You're right. They're using the percent "operator" as a times 100 thing, when it really should be a divided by 100 thing. That's what it means: *per cent*, as in per = divided by, cent = hundred.

[@AlexKnauth]

BEN, Fester: Syntax is arbitrary, what really matters is the tree structure: add / \ 1 multiply / \ 2 3 It just so happens that parentheses are useful to encode some of this structure in one line of text

[@TosiakiS]

They have no intrinsic precedence for parentheses because "order of operations" is only an artifact of the way we write the operators. If we wrote multiplication as "×AB" and addition as "+AB" instead of "A×B" or "A+B" (such as +1 1 = 2 instead of 1+1=2), there would be no need for parenthesis. For example, in this way of writing it, you could write +×4 4×4 4=32 or ××4+4 4 4=128 and all this is clear without any parentheses whatsoever.

[@FunkyDocter]

i allmost managed to explain this function to my 7th grade student that i am doing extra curricular with. the video was a very nice way of summarising. Allso this game is awsome. How long time did you guys use to solve 31 (without this function) i used 8 ours. and only managed to get it right when i got drunk and couldnt get the girls, and just sat down thinking about 31 instead. and more than 1 way ? you know the tripple sqrt and the factorial

[@FunkyDocter]

you can kinda say my student logged in.

[@MrBob-bj6kk]

r/iamverysmart

[@cobaltbluesky2276]

I remember in my 7th grade math class my teacher had us write all the integers 0-20 with just 4 4’s and any operations we could think of (remember this was 7th grade). Our class was apparently the only one that, in the teacher’s 10 years of teaching, found every number. I was the one to find the last one, 13. To this day I’m still proud of that moment.

[@omatic_opulis9876]

(44÷4)+√4?

[@RSLT]

Amazing!

[@wfcyellow]

In primary school I only learnt to count to 3. So this is lost on me.

[@pronounjow]

Still better than Valve. LOLOLOL Ok bye

[@factsheet4930]

But what about real numbers, not just whole numbers...?

[@IIARROWS]

You can, with my WIW operator. The What I Want operator let me define any number. Do you want Pi? Well then, 4 WIW(Pi) = Pi. See? It's easy.

[@factsheet4930]

If you really want π, its easy, ((√4/4)!/(√4/4))^ notice the power sign alone would mean to the power of 2, (if it works for roots, it better for powers >w

[@TheOiseau]

You don't need to invent that rule with the implicit square. Just write π = [ (√4/4)! x √4 ] ^ √4.

[@factsheet4930]

I came up with it too quickly, but yeah you are right

[@freddiepage6162]

That's actually impossible - There are only a countably infinite number of ways to write expressions from a finite set of operations, yet uncountably infinite real numbers. This is related to the result (In fact I think it is the same) that almost all numbers are inexpressible.

[@Apex-tn5iv]

this help so much!

[@kylesmith7086]

Watching this on its fourth year on KZbin. Imagine the comments Infinity from now. 🤯

[@benwincelberg9684]

Great video! Seeing this kinda as how to get to any integer on my simple scientific calculator with 4 fours this is correct. I would just say the percentage example should be 4/(4%) instead of (4/4) “as a percentage” because that’s how the operator on my calculator works