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sunny79
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Why is it that the radius of the unit circle is 1?
Unit with radius 1Math_QED said:By definition? What do you think the "unit" in "unit circle" stands for?
Excellent responses given so far, but the question is silly.sunny79 said:Why is it that the radius of the unit circle is 1?
Because 1 is the neutral element of multiplication, which simplifies a lot of the math:sunny79 said:Why is it that the radius of the unit circle is 1?
While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.jedishrfu said:The benefits are that it provides a simplification when teaching students about the trig functions and radians. Using a 1 means the circle perimeter is now ##2\pi## radians.
Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?A.T. said:While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.
Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.sysprog said:Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?
A.T. said:Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.
Are you saying π should have been defined as circumference/radius (π=6.283...)?A.T. said:While using a circle of radius 1 provides a simplification, introducing a constant that is just half of its perimeter, makes it unnecessarily complicated again.
I think that this is a nice article on the topic: https://tauday.com/tau-manifestojedishrfu said:Sciam did a nice article on the pros and cons:
https://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/#:~:text=At its heart, pi refers,now a proponent of tau.
I am fearful though that it will become another political football as the definition of pi=3 almost did many years ago in Indiana:
https://en.wikipedia.org/wiki/Indiana_Pi_Bill
Something like this happened with the change of notation between physics and math over the spherical coordinate system. I was taught in the early 1970s ##R \theta \phi## (##\phi## for the angle with the z-axis)whereas physics usage at work was ##R \phi \theta## (##\theta## with the z-axis) . You can imagine the confusion that results in trying to understand any spherically symmetric physical systems.
sysprog said:I think that this is a nice article on the topic: https://tauday.com/tau-manifesto
In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?jedishrfu said:My only argument in favor of ##\pi## is that two pies are better than one.
Maybe two half memberships?sysprog said:In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?
How do you write the area of a triangle? The circle area can be derived from that, so it makes sense for them to have a similar form.etotheipi said:What if you want to write the area of the circle?
Based on his post, @etothei1.5pi would be more appropriate.sysprog said:In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?
##-## otherwise rendered as @etothepau ##-## why not let him have 3 memberships?Mark44 said:Based on his post, @etothei1.5pi would be more appropriate.
Isn't it true that if we are to measure the circle by use of progressive caliper measurements whereby to determine whether our diametrical measurement is maximal then we have to do that with at least two different pairs of circumferential points in order to by the intersection of the line segments between the thereby determined pairs of points find the center?gmax137 said:I didn't read the links. But it seems to me the question is, is a circle defined by its radius, or by its diameter? There is no definitive answer. If you are drawing a circle (say with a compass) then you think, "radius." If you are measuring a circle (with a ruler or calipers) then you think, "diameter."
sunny79 said:Why is it that the radius of the unit circle is 1?
I'm not sure what you're getting at here. I meant calipers like this (with parallel jaws). Squeeze and read the diameter. Much easier than measuring the radius of a given circle.sysprog said:Isn't it true that if we are to measure the circle by use of progressive caliper measurements whereby to determine whether our diametrical measurement is maximal then we have to do that with at least two different pairs of circumferential points in order to by the intersection of the line segments between the thereby determined pairs of points find the center?
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:gmax137 said:I'm not sure what you're getting at here. I meant calipers like this (with parallel jaws). Squeeze and read the diameter. Much easier than measuring the radius of a given circle.
View attachment 269823
This type of caliper would work like the Vernier caliper shown in post #29. If the jaws of this caliper are set too close, the circle wouldn't fit between the jaws. Opening the jaws to a width so that they just barely accept the circle would give the diameter.sysprog said:Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:
View attachment 270179
Isn't that still a series of trials? How do we find that we haven't exceeded the diameter? Don't we have to do repeated trials to find out exactly where "just barely" is?Mark44 said:Opening the jaws to a width so that they just barely accept the circle would give the diameter.