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#1 2016-09-15 02:29:47

Eulero
Member
Registered: 2016-09-14
Posts: 9

An interesting equation

Hi,
Today I want to propose you a beautiful problem:
Solve the equation:
mant{x^(-1)}=mant{x}=mant{x^2}
Where mant{x} is the mantissa function

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#2 2016-09-15 03:15:41

phrontister
Real Member
From: The Land of Tomorrow
Registered: 2009-07-12
Posts: 4,593

Re: An interesting equation

Hi Eulero;


"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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#3 2016-09-15 03:16:32

Eulero
Member
Registered: 2016-09-14
Posts: 9

Re: An interesting equation

It's a solution but not the only one

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#4 2016-09-15 06:59:46

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Hi;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2016-09-15 07:44:26

Eulero
Member
Registered: 2016-09-14
Posts: 9

Re: An interesting equation

Good job!
But the fourth solution you proposed is wrong as you can see here. The exercise exalts one of the most curious proprieties of golden ratio : it's the only not integer number whose himself and reciprocal and square have the same fractionary part. When you have time would you post the execution?

Last edited by Eulero (2016-09-15 07:46:15)

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#6 2016-09-15 09:05:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Hi;

From the link you gave, mant is the positive fractional part .

So:

f[x_] := Abs[FractionalPart[x]];
n = -(8/5); 
FindRoot[f[1/x] == f[x] && f[1/y] == f[y^2] && f[z] == f[z^2], {{x, n}, {y, n}, {z, n}}, WorkingPrecision -> 50]

yields a root at

{x -> -1.6180339887498948482045868343656381177203091798058,
y -> -1.6180339887498948482045868343656381177203091798058,
z -> -1.6180339887498948482045868343656381177203091798058}

which is

When you have time would you post the execution?

My solution is a numerical one and I will post it when I can.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2016-09-15 18:54:02

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,128
Website

Re: An interesting equation

Last edited by zetafunc (2016-09-16 01:52:00)

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#8 2016-09-15 20:57:02

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: An interesting equation

When x=-1.618033989
floor(x)= -2
mant(x)=-1.618033989-(-2)=0.381966011
So

does not form a solution.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#9 2016-09-16 01:29:51

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Are you guys using the absolute value as the page describes?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#10 2016-09-16 01:35:00

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,128
Website

Re: An interesting equation

I interpreted "positive fractional part" as being
(no absolute values). I guess "negative fractional part" would be
I think the error could be coming from the fact that you defined
as
when perhaps it should not have the absolute values (unless I have misunderstood the OP's definition of mant(x) from the Wolfram article).

Last edited by zetafunc (2016-09-16 01:35:22)

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#11 2016-09-16 01:36:55

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Hi;

The page he gave http://mathworld.wolfram.com/Mantissa.html describes it like that.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#12 2016-09-16 01:43:15

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,128
Website

Re: An interesting equation

Are you sure? I am interpreting it as how they've written it down, i.e.
so that, for instance, the positive fractional part of 5.2 is 0.2, whilst its negative fractional part is 0.8. Their description does not include the word "of" between "positive fractional part" and the formula they give -- in other words, I think they're just calling that formula "positive fractional part", rather than saying to take the absolute value of that formula. That formula is how the mantissa is usually defined (although in for example analytic number theory nobody uses the word "mantissa", they just write the formula out).

Last edited by zetafunc (2016-09-16 01:44:43)

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#13 2016-09-16 01:46:53

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Hi;

Nope, I am not sure. I will  agree then that - phi is not a solution.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#14 2016-09-16 03:19:46

Eulero
Member
Registered: 2016-09-14
Posts: 9

Re: An interesting equation

Good job zetafunc!

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#15 2016-09-16 03:43:12

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: An interesting equation

Hi Bobbym,
If you recall characteristic and mantissa of log to the base 10 in logarithm tables, the concept becomes clear.


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#16 2016-09-16 04:06:33

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Hi;

If you recall characteristic and mantissa of log to the base 10 in logarithm tables, the concept becomes clear.

I do not agree. Eric's page is ambiguous to me in this case.

I agree with Knuth:

Knuth wrote:

For instance Knuth adopts the third representation 0.12345 × 10+3 in the example above, and calls 0.12345 the fraction part of the number; he adds:[7] "[...] it is an abuse of terminology to call the fraction part a mantissa, since this concept has quite a different meaning in connection with logarithms [...]".

But as this is the interpretation that the OP wanted I can only blame myself for not asking him to clear up the confusion in how I see that page.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#17 2016-09-16 04:29:18

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,128
Website

Re: An interesting equation

It might have been confusing because the Wolfram page says the phrase "positive fractional part" immediately before the formula, which is somewhat tautological (if the author had just written the formula, that would have been made clearer to the reader). But the lack of the word "of" between that phrase and the formula makes it technically unambiguous, I think. (But easy to misinterpret.)

It is made slightly worse by the fact that its counterpart "negative fractional part" is never explicitly defined on the website -- at least, I cannot seem to find it. But I'm assuming that if it does exist, it would have a definition something like the one I gave in post #10.

Last edited by zetafunc (2016-09-16 04:34:23)

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#18 2016-09-16 08:46:16

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: An interesting equation

Yes, but even the alphabet is confusing to a bumpkin.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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