My Math Forum Definition of the angle and pi???

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 February 28th, 2019, 04:07 AM #1 Senior Member   Joined: Oct 2015 From: Greece Posts: 137 Thanks: 8 Definition of the angle and pi??? As a University student, I'm re-reading calculus in order to get a full understanding of what is going on and how everything was actually invented (I just love to know how things work). By reading calculus 1 of Thomas and Finney I found out for the first time that the angle in radians is being defined as followed: $\displaystyle \theta = \frac{s}{r}$ where s is the arc of the circle and r the radius. For a unit circle: $\displaystyle \theta = s$ I can not understand how they came up with this division. I can not visualize how: $\displaystyle \theta \cdot r = s$ will actually give me the length of the arc.
 February 28th, 2019, 05:58 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,978 Thanks: 2229 It's $\theta \cdot r$ that gives the arc length, as the arc length is proportional to both the angle and the radius. Thanks from topsquark
 February 28th, 2019, 06:01 AM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,685 Thanks: 2666 Math Focus: Mainly analysis and algebra There are $2\pi$ radians in a circle. What is the circumference of the circle of radius $r$? What fraction of this circumference does the arc of angle $\theta$ have? And therefore, what is the length of the arc of angle $\theta$ and radius $r$? PS: what you read in current textbooks is not how this stuff was invented - that was generally incredibly messy, with inaccuracies and gaps that were only corrected and filled in later. Textbooks give a sanitised and logical overview of how one might invent this stuff if one knew in advance where one was trying to get to. Thanks from topsquark
February 28th, 2019, 08:41 AM   #4
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Quote:
 Originally Posted by v8archie PS: what you read in current textbooks is not how this stuff was invented - that was generally incredibly messy, with inaccuracies and gaps that were only corrected and filled in later. Textbooks give a sanitised and logical overview of how one might invent this stuff if one knew in advance where one was trying to get to.
This is actually so true! But this makes me sad because I really want to understand how things were invented.

So in other words, I also need to know and understand how pi was invented in order to understand this formula which connects the angle with the circle's radius and arc?

By the way, can I find any books or other sources that explain mathematics to a more detailed level, including history? How these people exactly planned everything and invented those things. For example, how did they calculate pi? How did they think of calculating it and why in the first place? What is the mathematical proof (with steps)?

February 28th, 2019, 10:15 AM   #5
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Quote:
 Originally Posted by babaliaris So in other words, I also need to know and understand how pi was invented in order to understand this formula which connects the angle with the circle's radius and arc?
Not really, it's sufficient to know the formula for the circumference of a circle.

Quote:
 Originally Posted by babaliaris For example, how did they calculate pi? How did they think of calculating it and why in the first place? What is the mathematical proof (with steps)?
There are many ways of calculating PI. The first was this: https://www.pbs.org/wgbh/nova/physic...mating-pi.html

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