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Why is it that the radius of the unit circle is 1?

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Why is it that the radius of the unit circle is 1?

- #2

Math_QED

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By definition? What do you think the "unit" in "unit circle" stands for?

- #3

jedishrfu

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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.

Right triangles drawn inside the circle with their hypotenuses being the radius have sides that sin and cos measurements their acute angles.

Im sure there’s other benefits as well. Can you spot any?

https://en.wikipedia.org/wiki/Unit_circle

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Unit with radius 1By definition? What do you think the "unit" in "unit circle" stands for?

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Infrared

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DaveE

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I think this is the same as asking why does a circle with radius =13 have radius =13? They've just used different words for r=1.

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symbolipoint

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Excellent responses given so far, but the question is silly.Why is it that the radius of the unit circle is 1?

Further Thought: My hasty thinking to say, "silly". One can look at a few measurable parts of a circle. circumference, diameter, radius, area. To pick RADIUS of unit 1 allows for some ease in handling some Trigonometry.

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A.T.

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Because 1 is the neutral element of multiplication, which simplifies a lot of the math:Why is it that the radius of the unit circle is 1?

https://en.wikipedia.org/wiki/Identity_element

- #9

A.T.

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While using a circle of radius 1 provides a simplification, introducing a constant that isThe 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.

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Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?While using a circle of radius 1 provides a simplification, introducing a constant that isjust halfof its perimeter, makes it unnecessarily complicated again.

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jedishrfu

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He’s a ##(\tau)## Tau-ist.

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A.T.

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Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.Can you please elaborate further on why you think that the ##2\pi## radians is an unnecessary complication?

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etotheipi

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What if you want to write the area of the circle?Beacause of the unnecessary factor 2. It's like using a circle with radius 1/2 instead of the unit circle.

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DaveE

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Are you saying π should have been defined as circumference/radius (π=6.283...)?While using a circle of radius 1 provides a simplification, introducing a constant that isjust halfof its perimeter, makes it unnecessarily complicated again.

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jedishrfu

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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 the physics usage at work was ##R \phi \theta## (##\theta## with the z-axis) . At first, I thought my brain was losing it until I did some research and discovered I was taught the math convention.

You can imagine the confusion that results in trying to understand any spherically symmetric physical systems.

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 the physics usage at work was ##R \phi \theta## (##\theta## with the z-axis) . At first, I thought my brain was losing it until I did some research and discovered I was taught the math convention.

You can imagine the confusion that results in trying to understand any spherically symmetric physical systems.

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I think that this is a nice article on the topic: https://tauday.com/tau-manifestoSciam 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.

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jedishrfu

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True, they mention it in the Sciam article.

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jedishrfu

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My only argument in favor of ##\pi## is that two pies are better than one.

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etotheipi

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That's a really fun page; I noticed that they justified ##A = \frac{1}{2} \tau r^2## by analogy for other quadratic forms that arise in Physics. The stuff about Gaussian distributions and polar coordinates is a nice touch. Perhaps we can agree on:I think that this is a nice article on the topic: https://tauday.com/tau-manifesto

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In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?My only argument in favor of ##\pi## is that two pies are better than one.

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DaveE

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Maybe two half memberships?In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?

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DaveE

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But required to post everything twice.

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A.T.

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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.What if you want to write the area of the circle?

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Mark44

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Based on his post, @etothei1.5pi would be more appropriate.In that case what would you think about letting PF member @etotheipi have a second membership as @etothetau?

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##-## otherwise rendered as @etothepau ##-## why not let him have 3 memberships?Based on his post, @etothei1.5pi would be more appropriate.