Posted on 12/13/2014 6:32:47 PM PST by SunkenCiv
Addition and subtraction would be cumbersome operations.
Common Core will have us there before you know it.
Or, alternately, ::1
http://en.wikipedia.org/wiki/Collatz_conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem;[1][2] the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud),[3][4] or as wondrous numbers.[5]
Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO[6]) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.[7]
Here are the first few Collatz number series...
2 1
3 10 5 16 8 4 2 1
4 2 1
5 16 8 4 2 1
6 3 10 5 16 8 4 2 1
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
8 4 2 1
9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
10 5 16 8 4 2 1
11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
12 6 3 10 5 16 8 4 2 1
13 40 20 10 5 16 8 4 2 1
14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
16 8 4 2 1
17 52 26 13 40 20 10 5 16 8 4 2 1
18 9 28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
20 10 5 16 8 4 2 1
21 64 32 16 8 4 2 1
22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
24 12 6 3 10 5 16 8 4 2 1
25 76 38 19 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
26 13 40 20 10 5 16 8 4 2 1
27 82 41 124 62 31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
28 14 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
30 15 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
31 94 47 142 71 214 107 322 161 484 242 121 364 182 91 274 137 412 206 103 310 155 466 233 700 350 175 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
And now, a biggie...
1424322332 712161166 356080583 1068241750 534120875 1602362626 801181313 2403543940 1201771970 600885985 1802657956 901328978 450664489 1351993468 675996734 337998367 1013995102 506997551 1520992654 760496327 2281488982 1140744491 3422233474 1711116737 5133350212 2566675106 1283337553 3850012660 1925006330 962503165 2887509496 1443754748 721877374 360938687 1082816062 541408031 1624224094 812112047 2436336142 1218168071 3654504214 1827252107 5481756322 2740878161 8222634484 4111317242 2055658621 6166975864 3083487932 1541743966 770871983 2312615950 1156307975 3468923926 1734461963 5203385890 2601692945 7805078836 3902539418 1951269709 5853809128 2926904564 1463452282 731726141 2195178424 1097589212 548794606 274397303 823191910 411595955 1234787866 617393933 1852181800 926090900 463045450 231522725 694568176 347284088 173642044 86821022 43410511 130231534 65115767 195347302 97673651 293020954 146510477 439531432 219765716 109882858 54941429 164824288 82412144 41206072 20603036 10301518 5150759 15452278 7726139 23178418 11589209 34767628 17383814 8691907 26075722 13037861 39113584 19556792 9778396 4889198 2444599 7333798 3666899 11000698 5500349 16501048 8250524 4125262 2062631 6187894 3093947 9281842 4640921 13922764 6961382 3480691 10442074 5221037 15663112 7831556 3915778 1957889 5873668 2936834 1468417 4405252 2202626 1101313 3303940 1651970 825985 2477956 1238978 619489 1858468 929234 464617 1393852 696926 348463 1045390 522695 1568086 784043 2352130 1176065 3528196 1764098 882049 2646148 1323074 661537 1984612 992306 496153 1488460 744230 372115 1116346 558173 1674520 837260 418630 209315 627946 313973 941920 470960 235480 117740 58870 29435 88306 44153 132460 66230 33115 99346 49673 149020 74510 37255 111766 55883 167650 83825 251476 125738 62869 188608 94304 47152 23576 11788 5894 2947 8842 4421 13264 6632 3316 1658 829 2488 1244 622 311 934 467 1402 701 2104 1052 526 263 790 395 1186 593 1780 890 445 1336 668 334 167 502 251 754 377 1132 566 283 850 425 1276 638 319 958 479 1438 719 2158 1079 3238 1619 4858 2429 7288 3644 1822 911 2734 1367 4102 2051 6154 3077 9232 4616 2308 1154 577 1732 866 433 1300 650 325 976 488 244 122 61 184 92 46 23 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1
Starting with the initial value of 1424322332, it takes 302 steps to reach unity.
I have a script that will do these all day long :)
2 takes 1 step, and 2 is the highest number reached.
3 takes 7 steps, and 16 is the highest number reached.
4 takes 2 steps, and 4 is the highest number reached.
5 takes 5 steps, and 16 is the highest number reached.
6 takes 8 steps, and 16 is the highest number reached.
7 takes 16 steps, and 52 is the highest number reached.
8 takes 3 steps, and 8 is the highest number reached.
9 takes 19 steps, and 52 is the highest number reached.
10 takes 6 steps, and 16 is the highest number reached.
11 takes 14 steps, and 52 is the highest number reached.
12 takes 9 steps, and 16 is the highest number reached.
13 takes 9 steps, and 40 is the highest number reached.
14 takes 17 steps, and 52 is the highest number reached.
15 takes 17 steps, and 160 is the highest number reached.
16 takes 4 steps, and 16 is the highest number reached.
17 takes 12 steps, and 52 is the highest number reached.
18 takes 20 steps, and 52 is the highest number reached.
19 takes 20 steps, and 88 is the highest number reached.
20 takes 7 steps, and 20 is the highest number reached.
21 takes 7 steps, and 64 is the highest number reached.
22 takes 15 steps, and 52 is the highest number reached.
23 takes 15 steps, and 160 is the highest number reached.
24 takes 10 steps, and 24 is the highest number reached.
25 takes 23 steps, and 88 is the highest number reached.
26 takes 10 steps, and 40 is the highest number reached.
27 takes 111 steps, and 9232 is the highest number reached.
28 takes 18 steps, and 52 is the highest number reached.
29 takes 18 steps, and 88 is the highest number reached.
30 takes 18 steps, and 160 is the highest number reached.
31 takes 106 steps, and 9232 is the highest number reached.
32 takes 5 steps, and 32 is the highest number reached.
33 takes 26 steps, and 100 is the highest number reached.
34 takes 13 steps, and 52 is the highest number reached.
35 takes 13 steps, and 160 is the highest number reached.
36 takes 21 steps, and 52 is the highest number reached.
37 takes 21 steps, and 112 is the highest number reached.
38 takes 21 steps, and 88 is the highest number reached.
39 takes 34 steps, and 304 is the highest number reached.
40 takes 8 steps, and 40 is the highest number reached.
41 takes 109 steps, and 9232 is the highest number reached.
42 takes 8 steps, and 64 is the highest number reached.
43 takes 29 steps, and 196 is the highest number reached.
44 takes 16 steps, and 52 is the highest number reached.
45 takes 16 steps, and 136 is the highest number reached.
46 takes 16 steps, and 160 is the highest number reached.
47 takes 104 steps, and 9232 is the highest number reached.
48 takes 11 steps, and 48 is the highest number reached.
49 takes 24 steps, and 148 is the highest number reached.
50 takes 24 steps, and 88 is the highest number reached.
The number 9232 shows up as the highest number reached a lot.
It means that numbers obviously exist, whether someone (e.g., oblomov) believes it or not. :’)
Update on the old saying:
There’s no place like 127.0.0.1
Divide by zero is essentially infinity in the limit.
If they’re trying to dump us to the bottom, I suspect that we’re already there.
.
Physical processes do generate numeric scaling.
If this were not so, we would have no hope of designing anything to respond to natural conditions.
.
Dividing by zero is an excellent way to wipe out a day’s work on a digital device.
.
Is 127.0.0.1 in Kentucky? Is it old?
Exactly.
However, the models we generate, and the numbers themselves, are human representations of physical reality.
A representation just needs to be accurate enough to enable decisions.
It’ll just stop.
I used one of those a long time ago.
In essence, all they do is add, or subtract, but do it multiple times automatically.
.
If humans invent something, it exists.
The question is, whether number (in the abstract, as we understand it) exists outside of human cognition.
The development of number systems reveals more about the cross-cultural similarity of human beings, than it does the underlying structure of physical reality.
Lol. Sounds like you are speaking from experience.
Learned that way back when I was taking the first few classes in Fortran.
.
So I got to know many geniuses along the way.
My mind could only take me just so far, before it couldn't "grasp" higher concepts, like quaternions or octornions.
But some math people could.
One guy, a friend who was bound for a PhD used to say: "(boop), all we are doing is "discovering" things that already exist. We can noodle out mathematical concepts and pat ourselves on the back for "finding" them, but that's about it."
So the answer is YES, math exists outside of the human mind.
We can do our best to ATTEMPT an understanding of its existence.
Maybe take a swipe at trying to bring it down to our level.
But in the end, no matter how smart we think we are, mathematical concepts in 2014 are scratching the surface.
I asked my professor in functions of a complex variable: "What's really the POINT of this? Fine and dandy to know all of this, but why?"
The answer: "Because we are setting the foundation for the physics of 300 years from now."
Makes sense.
We use the math from hundreds of years ago for our physics today.
Mathematics does exist outside of the human mind. But not numbers. Numbers are a convention we use to do math.
Goedel showed that the idea of computation does not by necessity imply the axiomatic system. The axiomatic system we use (Peano postulates plus a few others) may have a basis in observed reality. But it is nonetheless convention and is not strictly formalizable.
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