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Reading Pedoe's "Geometry: A Comprehensiveness Course" I came across the following

We know that from Euclidean geometry, for any triangle ABC,$$\sphericalangle ABC + \sphericalangle CAB + \sphericalangle BCA = \pi \quad (\text{mod}\ 2\pi) $$

I am not certain what this is supposed to mean. I know from high school geometry the sum of all three angles of a proper triangle is 180 degrees or $\pi$ radians. Is that what $\pi\ (\text{mod}\ 2\pi)$ means, 180 degrees?

Chill2Macht
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Diehardwalnut
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  • $(\mod2\pi)$ means that increasing or decreasing by $2\pi$ gives you the same result. And $\pi$ is equal to 180 degrees, so $2\pi$ is 360, and adding 360 degrees to anything doesn't change the result. It is modular arithmetic if you are interested. – Simply Beautiful Art May 21 '16 at 01:08
  • The measure of an angle, rigourously speaking, is a class of congruence of real numbers modulo $2\pi$, i.e. an element of $\mathbf R/2\pi\mathbf Z$. – Bernard May 21 '16 at 01:14
  • I meant to say multiples of $2\pi$. Its kind of like walking in circles, literally. – Simply Beautiful Art May 21 '16 at 01:15
  • So why not drop the $mod 2 \pi$ and just leave the $\pi$? Are we really interested in anything beyond $\pi$ when considering the sum of all angles of a proper triangle? – Diehardwalnut May 21 '16 at 01:38

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Modular arithmetic is a type of arithmetic where numbers "wrap around" after reaching a certain number (the modulus).

For example, a simple example is the time on a clock, this is mod 12. That is, if a clock reads 1 o'clock, then in one hour, the clock will read 2 o'clock. However, this is not unique for the passage of one hour, because the clock would also read 2 o'clock in 13 hours, or 25 hours, or 37 hours. In fact, the clock will read 2 o'clock in 1 + 12n hours, where n is any natural number. We would say that 13, 25, 37, and any number of the form 1 + 12n are congruent modulo 12. Thus, if I say 1 mod 12, I am referring to an entire class of numbers, those of the form 1 + 12n.

In the same light, in radians, our angles "wrap around" at 2π. Thus, when we say π mod 2π, we are speaking of the class of numbers which are congruent to π in modulo 2π, which are all numbers of the form π + 2πn for any natural number n.

For more information, look into the term "modular arithmetic."

Jonathan Hebert
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  • So in talking about the sum of all angles of a proper triangle, by saying they add up to $\pi mod(2 \pi)$ we are saying $\pi, 3 \pi, 5\pi ,...$ are all acceptable answers for 180 degrees or pi radians? – Diehardwalnut May 21 '16 at 01:51
  • Yes, that's right. After all, what's the difference between π and 3π radians? They represent the same thing. Your text is just acknowledging that we treat these numbers as equivalent (congruent) in the world of angles and radians. Does that make sense? – Jonathan Hebert May 21 '16 at 01:56
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It means $\pm 2k \pi$, for all $k \in \mathbb{Z}$.

I.e., the angle is unique "up to additions and subtractions of multiples of $2\pi$".

Chill2Macht
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