## What is a Euclidean Number

For the discussion of math. Duh.

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jewish_scientist
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### What is a Euclidean Number

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?
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DavidSh
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### Re: What is a Euclidean Number

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).

Thesh
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### Re: What is a Euclidean Number

Summum ius, summa iniuria.

Eebster the Great
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### Re: What is a Euclidean Number

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.

doogly
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### Re: What is a Euclidean Number

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.
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PM 2Ring
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### Re: What is a Euclidean Number

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.

doogly
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### Re: What is a Euclidean Number

I mean I carry an imaginary ruler with me everywhere I go, are you not likewise prepared?
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measure
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### Re: What is a Euclidean Number

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?

doogly
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### Re: What is a Euclidean Number

That sounds like a pretty mundane imagination.
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Derek
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### Re: What is a Euclidean Number

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
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### Re: What is a Euclidean Number

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.

jestingrabbit
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### Re: What is a Euclidean Number

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.
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Xanthir
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### Re: What is a Euclidean Number

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
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### Re: What is a Euclidean Number

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