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?
Best Answer
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."