Why does the radius of a unit circle need to be 1?

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Discussion Overview

The discussion revolves around the question of why the radius of the unit circle is defined as 1. Participants explore the implications of this definition in mathematics, particularly in trigonometry, and consider historical and pedagogical perspectives.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that the term "unit" in "unit circle" indicates a radius of 1 by definition.
  • Others argue that defining the unit circle with a radius of 1 simplifies teaching trigonometric functions and radians, as it allows for straightforward calculations of circumference and arc lengths.
  • A participant questions the necessity of the unit circle's radius being 1, comparing it to circles with different radii and suggesting that it is merely a convention.
  • Some participants mention that using a radius of 1 makes the math simpler, particularly regarding the relationship between angles and arc lengths in radians.
  • There are discussions about the historical context, with references to earlier conventions in trigonometry that used different radii, such as 60.
  • One participant expresses concern that the discussion around the definitions of pi and tau could become contentious, drawing parallels to historical debates in mathematics.
  • Several participants engage in light-hearted banter regarding the definitions of pi and tau, suggesting alternative membership names based on these constants.
  • Some participants raise questions about whether circles are defined by their radius or diameter, indicating a lack of consensus on this point.

Areas of Agreement / Disagreement

Participants generally agree that the definition of the unit circle as having a radius of 1 is a matter of convenience, but there is no consensus on the necessity or implications of this definition. Multiple competing views remain regarding the historical context and the relevance of different radii in mathematical discussions.

Contextual Notes

Some discussions touch on the historical use of different radii in trigonometry and the potential confusion arising from varying conventions in mathematics and physics. There are unresolved questions about the definitions and implications of pi and tau in relation to the unit circle.

  • #31
@hutchphd! Still here...just terribly busy.
 
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  • #32
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
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:

1601392725965.png
 
  • #33
sysprog said:
Oh, heck, I thought of regular (not necessarily with the spring and screw-wheel apparatus) outside calipers like this:

View attachment 270179
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.
I'm assuming we have an object with circular cross section here, although you could also do the measurement reasonably well for a circle drawn on paper. Put one jaw at any point on the circumference, and adjust the caliper opening so that the other jaw intersects a single point as the caliper is rotated through an arc.
 
  • #34
I'm sorry, my post #26 was a distraction, intending to address the distraction that started around post #9, debating radius vs. diameter, and pi vs. 2*pi.
 
  • #35
Mark44 said:
Opening the jaws to a width so that they just barely accept the circle would give the diameter.
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?
 
  • #36
Mark44 said:
Put one jaw at any point on the circumference, and adjust the caliper opening so that the other jaw intersects a single point as the caliper is rotated through an arc.
This method makes good sense to me.
 
  • #37
of course the assumption that caliper measure equals ruler measure is equivalent to the Side angle side axiom of euclidean geometry.

Pardon me, I am entranced by a thread that goes on this long, in answer to OP's question of "why is 1 = 1?"
 
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