What is a Euclidean Number

For the discussion of math. Duh.

Moderators: gmalivuk, Moderators General, Prelates

jewish_scientist
Posts: 649
Joined: Fri Feb 07, 2014 3:15 pm UTC

What is a Euclidean Number

Postby jewish_scientist » Wed Oct 25, 2017 5:54 pm UTC

http://mathworld.wolfram.com/EuclideanNumber.html wrote:A Euclidean number is a number which can be obtained by repeatedly solving the quadratic equation. Euclidean numbers, together with the rational numbers, can be constructed using classical geometric constructions.


Why is the underlined word there?

DavidSh
Posts: 67
Joined: Thu Feb 25, 2016 6:09 pm UTC

Re: What is a Euclidean Number

Postby DavidSh » Wed Oct 25, 2017 7:55 pm UTC

I think they mean the following vaguely algorithmic process:

Start with S= the set of all natural numbers (include 0). These can all be constructed with compass and ruler, given a unit length.
Repeat forever:
Pick a, b, and c from S.
Let x be a solution to ax^2 + bx + c = 0. (Found using the quadratic formula).
x is also constructable.
Insert x into S.
End Repeat.

All numbers ever in S will be constructable. I think these are what they are calling Euclidean numbers.
They were being vague with what the starting set is. The set of non-negative integers is sufficient to generate the negative integers (If a=0 and b=1, x= -c), and then all rational numbers (If a=0, and b is not 0, then x= -c/b).

User avatar
Thesh
Made to Fuck Dinosaurs
Posts: 5496
Joined: Tue Jan 12, 2010 1:55 am UTC
Location: Colorado

Re: What is a Euclidean Number

Postby Thesh » Wed Oct 25, 2017 10:35 pm UTC

Honesty replaced by greed, they gave us the reason to fight and bleed
They try to torch our faith and hope, spit at our presence and detest our goals

User avatar
Eebster the Great
Posts: 2738
Joined: Mon Nov 10, 2008 12:58 am UTC

Re: What is a Euclidean Number

Postby Eebster the Great » Thu Oct 26, 2017 7:59 pm UTC

These are usually called the Constructible numbers. They include all numbers which can be defined using only integers, addition, multiplication, division, and square roots. So for instance, (21/2-1)1/2 is constructible, but 21/3 is not, since that requires taking a cube root. This is provably equivalent both to the algebraic definition above (solving quadratic equations) and to the traditional Euclidean method of construction.

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5211
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Somerville, MA
Contact:

Re: What is a Euclidean Number

Postby doogly » Thu Oct 26, 2017 8:06 pm UTC

And maybe to cut to the point of your question - the "repeated" is there because to be a nice algebraic definition, if there is some operation we're talking about, we really want the set to be include repeated actions of those operations. It is because geometry and algebra are very close friends, despite the high school experience of throwing the barrier of freshman summer up in between them.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

User avatar
PM 2Ring
Posts: 3619
Joined: Mon Jan 26, 2009 3:19 pm UTC
Location: Mid north coast, NSW, Australia

Re: What is a Euclidean Number

Postby PM 2Ring » Thu Oct 26, 2017 8:29 pm UTC

They forgot to mention that the number of repetitions is supposed to be finite. Of course there are infinite sequences whose limit is a constructible number, but in general that's not the case. If you permit infinite sequences then any real number can be constructed.

I was going to say something about it being tricky to do ruler and compass constructions for quadratics with complex solutions, but I guess that's not really a problem if you use the Argand diagram.

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5211
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Somerville, MA
Contact:

Re: What is a Euclidean Number

Postby doogly » Fri Oct 27, 2017 12:02 pm UTC

I mean I carry an imaginary ruler with me everywhere I go, are you not likewise prepared?
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

User avatar
measure
Posts: 100
Joined: Sat Apr 04, 2015 4:31 pm UTC
Location: Time-traveling kayak

Re: What is a Euclidean Number

Postby measure » Fri Oct 27, 2017 3:38 pm UTC

doogly wrote:I mean I carry an imaginary ruler with me everywhere I go, are you not likewise prepared?

Isn't that just two rulers stapled together at right angles?

User avatar
doogly
Dr. The Juggernaut of Touching Himself
Posts: 5211
Joined: Mon Oct 23, 2006 2:31 am UTC
Location: Somerville, MA
Contact:

Re: What is a Euclidean Number

Postby doogly » Fri Oct 27, 2017 4:05 pm UTC

That sounds like a pretty mundane imagination.
LE4dGOLEM: What's a Doug?
Noc: A larval Doogly. They grow the tail and stinger upon reaching adulthood.

Keep waggling your butt brows Brothers.
Or; Is that your eye butthairs?

Derek
Posts: 2145
Joined: Wed Aug 18, 2010 4:15 am UTC

Re: What is a Euclidean Number

Postby Derek » Thu Nov 02, 2017 7:04 pm UTC

This reminds me of a mathematical website/game I've been playing recently, www.euclidea.xyz, which has you solve geometric problems using Euclidean constructions. The problems get pretty difficult and I haven't finished it yet.

MachEps
Posts: 2
Joined: Fri Nov 03, 2017 5:13 pm UTC

Re: What is a Euclidean Number

Postby MachEps » Fri Nov 03, 2017 11:52 pm UTC

Derek wrote:This reminds me of a mathematical website/game I've been playing recently, http://www.euclidea.xyz, which has you solve geometric problems using Euclidean constructions. The problems get pretty difficult and I haven't finished it yet.

Wow it really has been some time since I had geometry. I am also getting stuck, but a fun thing to tinker with in the bus.

User avatar
jestingrabbit
Factoids are just Datas that haven't grown up yet
Posts: 5963
Joined: Tue Nov 28, 2006 9:50 pm UTC
Location: Sydney

Re: What is a Euclidean Number

Postby jestingrabbit » Sun Nov 05, 2017 1:33 pm UTC

I genuinely loved that euclidea game for a while, but then it got to be a bit of a "solve this equation" thing in my head, and I lost interest.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

User avatar
Xanthir
My HERO!!!
Posts: 5210
Joined: Tue Feb 20, 2007 12:49 am UTC
Location: The Googleplex
Contact:

Re: What is a Euclidean Number

Postby Xanthir » Mon Nov 06, 2017 7:38 pm UTC

Yeah, near the end it starts explicitly depending on you knowing some funky geometry proofs on your own; some of them are *not* reasonable to figure out on your own.

But the rest of the game is great!
(defun fibs (n &optional (a 1) (b 1)) (take n (unfold '+ a b)))

jewish_scientist
Posts: 649
Joined: Fri Feb 07, 2014 3:15 pm UTC

Re: What is a Euclidean Number

Postby jewish_scientist » Wed Nov 08, 2017 4:14 pm UTC

This is so much fun! I seriously wish that this is what we did in geometry class.


Return to “Mathematics”

Who is online

Users browsing this forum: No registered users and 8 guests