Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 230

Which is larger

Which is larger,

or

and how

Last edited by tony123 (2008-07-18 06:55:25)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Which is larger

No, they’re not.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Which is larger

2040 is less than a billionth larger than the computation.
source: a c program I wrote once.

Here's the irrational number to 200 digits, if you want more, just ask.
2039
. (decimal point here)

99999 99990 42712 92398 86785 04689 50535 08021 04757 16285
50913 62457 66368 90051 21473 00492 80558 35449 46853 12843
52155 84652 98204 70426 64166 14492 92266 20637 07858 28301
93203 63850 87528 78826 86043 86197 27273 43510 51071 44710

Last edited by John E. Franklin (2008-07-18 08:25:37)

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igloo myrtilles fourmis

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Which is larger

Here are some more digits.
2039
. (decimal point here)

99999 99990 42712 92398 86785 04689 50535 08021 04757 16285
50913 62457 66368 90051 21473 00492 80558 35449 46853 12843
52155 84652 98204 70426 64166 14492 92266 20637 07858 28301
93203 63850 87528 78826 86043 86197 27273 43510 51071 44710
31147 52846 94678 69112 67220 26003 62428 33592 34557 54119
97655 39369 37505 56020 77950 46023 19711 35935 79378 55601
07894 89721 71851 94511 03254 10955 93966 78762 21522 10741
07643 88051 88425 96575 41076 07466 77171 92192 98665 93620
25831 50559 04797 49997 82998 65837 79008 51639 15303 42265
82857 61575 49321 82783 30846 65500 61774 77085 57232 69470
63089 85192 25341 07372 34741 96896 61090 91819 49507 18368
96216 91523 20995 52459 12351 18914 39701 82474 11939 37328
19638 50047 48577 02514 13164 33281 82526 35280 16004 11740
78903 89778 74745 36099 66643 13810 28413 09008 80686 90680
22382 95355 36582 42763 25511 23049 26855 32540 64936 04736
55691 82407 87785 48082 40331 81278 85310 53448 23553 04292
85649 00676 36549 89970 66877 09116 36314 75795 71246 43431
95978 79994 92949 76209 10078 22491 34543 23118 84124 60665
63198 60830 84320 26022 59390 67680 42741 64741 20058 89814
16549 61669 81463 13278 60790 49116 30061 91474 39873 62828
98354 77131 29212 57588 00046 39235 47157 24724 69812 63634
06061 73636 74040 36105 89634 59997 32287 62359 95813 45115
20026 48785 49693 07500 17335 93050 09636 66170 28363 62118
19362 04460 98348 15518 05032 53378 94802 64335 64283 44760
24223 59873 95316 04432 42415 64507 35275 42416 59500 34749
55065 00893 00956 79712 48792 51563 91234 72747 11988 20068
59719 06041 33352 46396 55985 79956 36254 17684 78802 79177
09747 81463 69948 45984 23436 04418 08134 90243 41062 46706
19593 81917 22497 96831 56940 70523 86183 75038 37834 32774
28867 64762 62793 51080 07211 38137 14947 40843 06073 95767
47160 93744 43919 04539 34769 24388 58124 80452 97296 39173
20437 74540 17687 04327 70787 67358 43564 20245 41415 82818
76887 42728 31372 52575 66523 90869 15668 46625 26956 86811
27571 38477 11634 74007 72834 20522 58482 66164 07904 54148
72881 08677 36757 19795 82223 40069 94461 32632 52716 52680
96522 79197 83183 01712 74609 80026 65167 69937 85766 65831
94585 88084 95870 00216 80412 36022 12835 57683 18997 59558
97219 11197 63870 76204 31469 23260 15117 58954 11559 21620
98587 30434 41458 43577 82292 45314 96854 38880 65228 39963
42124 74325 75983 46368 71403 69624 51946 02350 06619 16904
the next digit after this is 5, so round up if want or ask for more digits.

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igloo myrtilles fourmis

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Which is larger

Well, yes. 2040 is larger, but only just. Obviously the trick to proving this other than by using a program or calculator is to come up with a convergent series of some sort like

where the RHS is equal to 2040 and the LHS is equal to the other number for a suitable n. That’s my idea anyway.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Which is larger

I haven't thoroughly checked my work, but:

Then:

Solve for sqrt(d), and compare.

Last edited by Ricky (2008-07-18 08:48:18)

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Which is larger

Wow!  Check this out:

By my previous result, we have:

Now:

Squaring both sides:

And this method completely generalizes!  Integers never lie!  Algebra > Analysis.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Which is larger

is not rational.

And this method completely generalizes!  Integers never lie!  Algebra > Analysis.

Amen!

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Which is larger

I double checked the ratio of 16912742401:16912742400 against 2040:(my long number)
and it almost worked out, so I don't know if I have a bug or
if these ratios are not supposed to be the same??
Here's some data:
FillVariable(2L,"<7");
square_root_calculation(4L,2L,200,1225L/*max repeats*/);
//PrintVariable(4L);
FillVariable(3L,"<759");
MultiplyVariables(5L, 3L, 4L);//759 * 4L
FillVariable(6L,"<254");
square_root_calculation(7L,6L,200,1225L/*max repeats*/);
FillVariable(8L,"<2");
MultiplyVariables(9L,8L,7L);
AddVariables(11L,9L,5L);
PrintVariable(11L);
FillVariable(12L,"<16912742401");
MultiplyVariables(13L,12L,11L);
FillVariable(14L,"<2040");
DivideVariables(15L,13L,14L);
PrintVariable(15L);

Here is variable #: 15   It is a positive number.
00000 00000 00000 00000 <16912 74240 0
. (decimal point here)

99206 35540 58634 32009 32448 40434 42506 69051 52954 78720
81208 76588 44852 71777 67658 63327 58341 69275 47453 80852
94786 73695 15207 98523 34633 94226 02875 78715 98097 99893
13252 18687 41989 49842 13398 00845 50011 30759 70539 84191
15898 00189 40436 32062 41824 12253 35439 46157 99766 46031
81132 66815 56519 42825 76442 51104 24261 93318 93859 72196
87753 24752 20961 63000 03834 66060 63138 69395 17041 62313
21695 68973 65421 01454 54509 48163 83720 54729 71543 03381

(note the point 99206 after the decimal, weird how it isn't closer like 16912742400.999999 or something
I don't know what's going on here??
Woops, actually, the number that was supposed to come up was 16912742400.000000, we
are way off somehow unless the ratio is not linear due to
squaring or something else in Ricky's algebra, I just don't know?

Last edited by John E. Franklin (2008-07-18 09:25:18)

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Which is larger

is not rational.

I meant the equals sign just to take the place of a comparator.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Which is larger

I squared the ratio of the two 16.9 billion integers hoping they would
diverge further apart, but unfortunately, then didn't budge much,
so that didn't seem to work.  Maybe I made a mistake.?
Or maybe the 2 numbers don't have a simple coorespondance
to each other.  Any ideas Ricky?

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Which is larger

Alternate question:

Given positive integers a, b, c, and d, is it possible to find:

With b or d (or both) not perfect squares, such that the expression above results in an integer?  What about a rational?

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Which is larger

John, I'm really not even sure what it is you're trying to do.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."