How to count to 1

Math is funny sometimes, you can spend half a day working on something, and if you don't take the right steps you end up showing that 1=1. Most of the time it's not that bad but can still be a little disappointing.

Case in point, in an effort to find a more pleasing shape for the superellipse, i decided to use phi, also known as the golden ratio or golden section, and is approximately 1.618. When i used phi in the obvious way the ellipse was not very appealing - too diamond shaped. So the next idea was to take 1/phi + 1/phi of the remainder + 1/phi of that remainder... ad infinitum. Well, now i need to find out how to add all those addends up. I spent a couple of hours using what i remember of sums and series, but the problem didn't conform to the formulas i knew. Off to the Internet, spent a couple more hours researching and trying to rearrange my formula to fit somebody elses, with no success. Finally the key to the lock presented itself, and the pieces fell into place.

Alas, preliminary, unverified results indicate that the above sum is equal to 1. Which is of absolutely no help whatsoever in determining the most pleasing shape for a superellipse. The only thing of use from all the eraser shavings was the fact that any repeated addition of a fraction of a whole will always add up to the whole. 1/3 plus 1/3 of the remaining portion plus 1/3 of that remaining portion, etc. is equal to 1. Even if it's really small. 1/million plus 1/million of the remainder plus 1/million of that remainder, etc., is still equal to 1.

This probably explains why it was no where to be found in the math text, it basically amounts to an interesting side note. It's back to the drawing board for the superellipse, but stay with me people, the superellipse survey is coming.

Comments

[tom]

Aye!!

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

http://en.wikipedia.org/wiki/Limit_%28mathematics%29

[basqueard]

Hey MathJames,

Here's a math puzzle for you. May be really basic but and you may have seen it before. You must generate all the values from 0 to 25 (and beyond if possible) using expressions with four fours. Examples:

0= 4+4-4-4 (no rocket science here)
1= (4+4)/(4+4)
2= (4*4)/(4+4)

etc... obviously there's a ton of ways to do each one, but it gets complicated and you have to come up with creative answers...

[MathJames]

I'll accept the challenge, but here are the rules i will be using. I will be using NO MORE than four 4s, if i can show a number in less i will use that. 44 is acceptable as a use of two 4s as is 4.4, though .4 is NOT, there must be an integer part. +,-,*,and / are acceptable as are ^ and 'root' when used with fours - NO SQUARE ROOTS, the two is implicit. However i will be using cascading operations, ! for factorial, @ for addition (4@ = 4+3+2+1) and ^^ for powers (4^^ = 4^3^2^1). I have not decided on modulus, but logs are allowed in conjuction with 4s, with LOG the 10 is implicit as is the e in LN and will not be used. 'i' the complex number is not allowed.

I am currently up to 17, and when I reach 25 i will post the results.

[basqueard]

I see you've gotten into it! Just for reference's sake the rules I originally saw this solved under allowed square root and logs, and assumed that since you can write these expressiones without the 2 or 10 or e they are valid.

Also, the original results ALWAYS used four fours, not up to four fours. The only cascading opertations I saw in the solution was factorial, not sum or exponents.

But I can see you're definitely on the right track, anyway and I've managed to pique your interest ;-)

[MathJames]

First, here is the traditional four 4s problem completed upto 32:

0 = 44-44
1 = 44/44
2 = 4/4 + 4/4
3 = 4-sqrt(4) + 4/4
4 = sqrt(4)+sqrt(4) * 4/4
5 = 4+sqrt(4) - 4/4
6 = 4+sqrt(4) * 4/4
7 = 4+4 - 4/4
8 = 4+4 * 4/4
9 = 4+4 + 4/4
10 = 44/4.4
11 = 4/.4 + 4/4
12 = 4+4 + sqrt(4)+sqrt(4)
13 = 4!/sqrt(4) + 4/4
14 = 4/.4 + sqrt(4)+sqrt(4)
15 = 4^sqrt(4) - 4/4
16 = 4+4+4+4
17 = 4^sqrt(4) + 4/4
18 = 4/.4 + 4+4
19 = 4!-4 - 4/4
20 = 4!-4 * 4/4
21 = 4!-4 + 4/4
22 = 4!-sqrt(4) * 4/4
23 = 4!-sqrt(4) + 4/4
24 = 4!/4 * sqrt(4)*sqrt(4)
25 = 4!+sqrt(4) - 4/4
26 = 4!+sqrt(4) * 4/4
27 = 4!+4 - 4/4
28 = 4!+4 * 4/4
29 = 4!+4 + 4/4
30 = (4+4+4)/.4
31 = 4! + (4!+4)/4
32 = (4*4) + (4*4)

[basqueard]

Niiice!! I can't remember all the original solns, but I dig the way you use division by .4 and 4.4 to get 10, nice & elegant!

[MathJames]

And here is the modified four 4s problem, with +,-,*,/,^,nth roots, !, @, ^^, log_n,; and no open decimals, sqrts, repeating decimals, or number symbols (e, pi, phi, i).

0 = 4-4
1 = 4/4
2 = (4+4)/4 or log_4(4*4)
3 = 4 - 4/4
4 = 4
5 = 4 + 4/4
6 = 4!/4
7 = (4!+4)/4
8 = 4+4
9 = 4@ - 4/4
10 = 4@
11 = 4@ + 4/4
12 = 4+4+4
13 = 4@+4 - 4/4
14 = 4@+4
15 = (4 + 4/4)@
16 = 4*4
17 = (4!/4)@ - 4
18 = 4! - 4!/4
19 = 4!-4 - 4/4
20 = 4!-4
21 = (4!/4)@
22 = 4! - (4+4)/4
23 = 4! - 4/4
24 = 4!
25 = 4! + 4/4
26 = 4! + (4+4)/4
27 = 4!+4 - 4/4
28 = 4!+4
29 = 4!+4 + 4/4
30 = 4! + 4!/4
31 = (4@)@ - 4!
32 = (4+4)*4
33 = 4!+4@ - 4/4
34 = 4! + 4@
35 = 4!+4@ + 4/4
36 = (4+4)@
37 = (4+4)@ + 4/4
38 = 4! + 4@ + 4
39 = 4*4@ - 4/4
40 = 4*4@
41 = 4*4@ + 4/4
42 = 4!+4@ + (4+4)
43 = 44 - 4/4
44 = 44
45 = (4@)@ - 4@
46 = (4+4)@ + 4@
47 = (4@)@ - (4+4)
48 = 4! + 4!
49 = (4@)@ + (4!/4)
50 = (4@)@ - 4 - 4/4
51 = (4@)@ - 4
52 = (4@)@ - 4 + 4/4
53 = (4@)@ - (4+4)/4
54 = (4@)@ - 4/4
55 = (4@)@
56 = (4@)@ + 4/4
57 = (4@)@ + (4+4)/4
58 = (4@)@ + 4 - 4/4
59 = (4@)@ + 4
60 = (4@)@ + 4 + 4/4
61 = (4@)@ + (4!/4)
62 = (4@)@ + (4!+4)/4
63 = (4@)@ + (4+4)
64 = (4^4)/4
65 = (4@)@ + 4@
66 = (4@ + 4/4)@
67 = (4@)@ + (4+4+4)
68 = (4^4)/4 +4
69 = (4@)@ + 4@ + 4
70 = (4@)@ + (4 + 4/4)@
71 = (4@)@ + (4*4)
72 = (4@)@ + (4!/4)@ - 4
73 = (4@)@ + 4! - (4!/4)
74 = (4^4)/4 + 4@
75 = (4@)@ + 4! - 4
76 = (4@)@ + (4!/4)@
77 = (4*4)@ - (4@)@ - 4
78 = (4+4+4)@
79 = (4@)@ + 4!
80 = (4@)@ + 4! + 4/4
81 = (4*4)@ - (4@)@
82 = (4+4+4)@ + 4
83 = (4@)@ + 4! + 4
84 = (4!/4)@ * 4
85 = (4@)@ + 4! + 4!/4
86 = (4@)@ + 4@ + (4!/4)@
87 = (4@)@ + 4! + (4+4)
88 = (4^4)/4 + 4!
89 = (4@)@ + 4! + 4@
90 = 4@*4@ - 4@
91 = (4@+4 + 4/4)@
92 = 4!*4 - 4
93 = (4@)@ + 4! + 4@+4
94 = (4@)@ + (4@)@ - (4*4)
95 = 4!*4 - 4/4
96 = 4!*4
97 = 4!*4 + 4/4
98 = unknown
99 = 4@*4@ - 4/4
100 = 4@*4@

I had decided to go until i couldn't find an answer or 100, which ever came first. Interestingly that turned out to be nearly the same. I don't know an answer for 98, if anybody can think of one, please let me know.

[MathJames]

Thanks, basqueard. By the way does your real name contain the letter X?

[MathJames]

Here's a quick and dirty variation: each of the numbers upto my age using the four digits of my birth year. :)

1974
0 = (7*1) - sqrt(9) - 4
1 = (7+4) - (1+9)
2 = (1+9+4) / 7
3 = (1*9) / (7-4)
4 = (9+7) / (4*1)
5 = ((7*sqrt(4))+1) / sqrt(9)
6 = (7-1) * (sqrt(9) - sqrt(4))
7 = sqrt(9)*7 / (sqrt(4)-1)
8 = 4! / (7 - sqrt(9) - 1)
9 = (7+4+1) - sqrt(9)
10 = (7+sqrt(9)) * (sqrt(4)-1)
11 = sqrt(9)+sqrt(4)+7-1
12 = (7+sqrt(9)) + (sqrt(4)*1)
13 = 7+sqrt(9)+1+sqrt(4)
14 = sqrt(9) + 7 + 4*1
15 = 1 + sqrt(9) + 7 + 4
16 = ((9*7) + 1) / 4
17 = 91 - 74
18 = 9 * (7+1) / 4
19 = 1 + 9 + 7 + sqrt(4)
20 = (9+7+4)^1
21 = 1+9+7+4
22 = (7*sqrt(9)) + (sqrt(4)-1)
23 = 94 - 71
24 = (7*sqrt(9)) + 4 - 1
25 = (7*sqrt(9)*1) + 4
26 = (7*sqrt(9)) + 4 + 1
27 = 9 * (7-4) * 1
28 = 7*sqrt(4)*(sqrt(9)-1)
29 = 4! + 7 - 3 + 1
30 = (7-1) * (sqrt(9)+sqrt(4))
31 = sqrt(9)! * 4 + (7^1)