A probability

Aug 2018
118
7
România
Hello all,

What is the probability that a square constructed with the line and the compass has the side represented by a transcendental number?

All the best,

Integrator
 

romsek

Math Team
Sep 2015
2,679
1,480
USA
The compass can be set to any width. With no pre-conditions at all, the set of sides represented by non-transcendental numbers has measure 0.

Thus the probability that the side will be represented by a transcendental number is 1.
 
Aug 2018
118
7
România
The compass can be set to any width. With no pre-conditions at all, the set of sides represented by non-transcendental numbers has measure 0.

Thus the probability that the side will be represented by a transcendental number is 1.
The line is not graded!
 

romsek

Math Team
Sep 2015
2,679
1,480
USA
If the line is not graded , then about what the compass setting we can talk?:rolleyes:
From what I just watched you first use the compass w/o caring about it's setting to establish a line perpendicular to a line you first drew with the straight edge.

You then set the compass to control the size of the square you want to draw and draw a circle with the intersection of the above mentioned lines as the center.

You then connect the points where this circle intersects those lines to draw your square.

So the setting of the compass in step 2 determines what the length of the side of the square will be.

The probability of selecting a compass width such that a side is non-transcendental is zero for the reason I mentioned in the earlier post.

I still have no idea what you mean by a graded line.
 
Jun 2019
439
236
USA
Remember that length is not unitless, so you have to pick a unit in order to assign a number. If you measure the side of the square in terms of multiples of the distance between the compass points, then there is a 100 % chance it will be a transcendental number. Other than that, I agree with romsek. A randomly chosen value has 100 % chance of being transcendental.
 
Aug 2018
118
7
România
Remember that length is not unitless, so you have to pick a unit in order to assign a number. If you measure the side of the square in terms of multiples of the distance between the compass points, then there is a 100 % chance it will be a transcendental number. Other than that, I agree with romsek. A randomly chosen value has 100 % chance of being transcendental.
Hello,

If we take as a unit of measure even the length of the opening of the compass and as the line has no gradations , then I think the required probability is equal with 0.Is my reasoning correct? Thank you very much!

All the best,

Integrator
 
Jun 2019
439
236
USA
Let's let the distance between the points of the compass be $l$.

Naturally, since we're talking about transcendental numbers, we're working in a world where this distance can be defined (measurable or not) to infinite precision. We're also assuming $l$ is fixed (in a realistic compass, it would change by probably microns during use -- even more if there's a hinge that's allowed to slip slightly).

To put a number on $l$, we need a unit of length. Maybe $l=7.43892~45635~41113~81... ~cm$. Assuming this is a randomly-chosen length with infinite precision, there is a 100 % chance the numeric value (the number of centimetres) is transcendental. We could also express it as $l=2.92871~04580~87052~68... ~in.$. This number will also be transcendental. However, we could also define a new unit, one compasslength (cpl), such that $1~cpl \equiv l$. 1 is algebraic.

I presume any type of square you can construct with a compass distance $l$ will have sides proportional to $l$, with the constant of proportionality being algebraic. Thus, the length of a side divided by any standard unit is pretty much guaranteed to be transcendental (in our mathematically perfect world). However, you can get around this by defining new units.