The four fours.

[Amram]

Simple little game with a deceptive challenge to it, and it'll get you thinking.

the objective is to count as high as possible within the rules, which are:

1. You have only four fours, no other numbers are allowed, so 4x3 is a no go, you cannot use 0, 1, 2, 3, 5, 6, 7, 8, or 9.
   
2. You must use all four fours. Make it is convoluted as you like, but four fours, no more, no less and no others as per rule one.
   
3. You are free to use ANY operation you like, so long as you need not write a value other than a four. it can be implied, but it may not be written. So Square root is allowed, as the 2 is not meant to be written, its just √. Cube roots are out, because you must write a 3. Fourth roots ARE permitted, because 4 is allowed.
   
4. Constants are out, no using Pi for example. its has an inherent value, which is not 4, so rule 1 and 3 are both violated. Its not a four and has a value of its own other than four. Factorials are permitted, they have no value if they aren't first given a number on which to operate. So ! is meaningless, but 4! has a value, so its permitted, as you only used a four.
   
5. While not constants, ln() and rounding operations such as floor and ceil, are considered to be outside the spirit of the challenge. Ln() permits you to have any integer value you like.
   Rounding just makes a square peg fit a round hole by cutting off the corners. Or a round peg fit a square hole by gluing those corners on.
   Im going to use This definition for why rounding is not permitted. It is simply a notation to indicate that you have replaced the true result, with one that is approximately equal but shorter representation. The key is approximately equal, approximately is not precisely, they aren't equal. So a rounded answer, is not the answer.
   
6. Your answer must be the next integer in the sequence. So if 37 was last, your task is to figure out how to reach 38.
   
7. 44 and 444 are valid uses, as is .4 given that the zero need not be written because its implied and writing the zero has no influence on the value itself.
   
8. Aside from my quick set of 0-4 as a demo of how to play along, only one number per post. Multiple versions for the same number are acceptable. The idea is to avoid one person posting numbers 12 through 18 for example.
   
9. In line with the spirit of the previous rule, don't reply to yourself with the next number either, as then someone could provide 6 numbers in sequence on their own, which the previous rule was intended to prevent(and in part failed)
   
   1. small adendum, if its been more than 24 hours and you want to stick the next number up, feel free
      
      

If your going to just use google and search for an answer, then don't bother. If you can't take some pride in the power of your mind, then don't demean yourself by cheating.
A quick reference listing for anyone who wants a bit better notation without resorting to nifty little image makers such as this Online LaTeX editor
Windows Alt Codes for various Math symbols.
As mentioned in the rules, im taking on the first 5 as an example.
0 = 4+4-4-4
1 = (4/4)!^(4/4) overcomplicated on purpose. Could also just do 4/4 * 4/4, or (4+4)/(4+4)
2 = 4/4+4/4
3 = √(4*4) - 4/4
4 = √(4*4) * 4/4
Be wary of the order of operations while your at it.
Whose up for 5?
How high can we get? Edited February 14, 2014 by Amram
a bit of rule fine tuning/clarification. Doesn't affect any curent posts.

[KvickFlygarn87]

5 = 4/4⁴+4

[Amram]

5 = (4/4)⁴+4

Fixed that for you. Parenthesis are your friend.

6 = 4/4 * 4 + √4

[vexx32]

7 = 4/4 + √4 + 4

[Starwhip]

8 = 4_/4 - 4 + 4

The _/ is supposed to be a radical. How do you guys do that?

[Marcuzzi]

9 = (4/4) + 4 + 4

you mean this √?

alt+251 = √

[Superfluous J]

10 = (4*4+4)/(√4)

[Amram]

11 = 4! / √4 - 4/4

and because its worth a bit of elaboration

44 / (√4 + √4)

44 is a valid use of 2 fours.

Should .4 be allowed? Programming never cares if you omit the zero, excel doesn't though it corrects it for you, calculators don't. Im inclined to think yes in that the zero is implied much as the two from √, in as much that omitting the value that precedes the decimal implies there is no value preceding the decimal. I suppose a solid definition would be whether or not entry into a calculator would require a number other than four?

4/.4 + 4/4

Edited December 18, 2013 by Amram

[DisarmingBaton5]

12 = 4*(√4)+(√4)+(√4)

[Starwhip]

13 = 44 / 4 + √4

[vexx32]

14 = 4 + 4 + 4 + √4

[Lord_Potato]

15 = 44/4 + 4

[Amram]

15 = 4*4 - 4/4

[Superfluous J]

EDIT: More obscure and less parenthesis:

16 = 4*4*4/4

WAS:

16 = 4*4/(4/4)

[vexx32]

17 = 4*4 + 4/4

[Chrisd857]

18 = (4! + √4) - (4 + 4)

Edit:

For kicks and giggles

18 = Γ(4) * (4 - 4/4)

Edited December 19, 2013 by Chrisd857

[Duxwing]

19 = 4! - 4 - 4/4

-Duxwing

Edited December 19, 2013 by Duxwing

[Chrisd857]

20 = 4! - 4 + (4 - 4)

[Duxwing]

21 = 4! - √4 - 4/4

22 = 4*4 + 4 + √4

-Duxwing

Edited December 19, 2013 by Duxwing

[Chrisd857]

Edit: For 21, did you mean 4! - √4 - 4/4?

23 = 4! - (4/4)^4

[Duxwing]

24 = 4*4 + 4+4

EDIT: Yes Chris, I did. Thanks!

[Chrisd857]

25 = 4! + (4/4)^4

[Duxwing]

26 = 4! + √4 + 4 - 4

[Chrisd857]

27 = 4! + 4 - (4/4)

[Pursuedtank]

28= 4! + 4 + 4 - 4