# Pi(mod 2Pi)=180 degrees?

1. Feb 16, 2006

### rtharbaugh1

I am refreshing my maths and came across this formula stated as a fact without explanation in a geometry text. How does this work? I would have thought (mod 2Pi) was the decimal portion of 2Pi but that doesn't multiply out to 180.

Thanks,

R.

2. Feb 16, 2006

### HallsofIvy

I've never seen such a thing as "pi (mod 2pi)= 180 degrees"! I suspect what is meant is simply that, since a complete circle, measured in radians, is 2pi, while measured in degrees it is 360 degrees, then pi radians is the measure of a half circle which corresponds to 180 degrees.

That's just telling how to convert from one set of units (radians) to another (degrees) and, in my opinion anyway, has nothing to do with "mod". "pi (mod 2pi)= 180 degrees" strikes me as a lot like saying "one foot (mod one yard)= 33 cm."

3. Feb 16, 2006

### rtharbaugh1

Thank you for the reply. The text is Geometry, A Comprehensive Course by Dan Pedoe, Dover, 1970. Perhaps I have misinterpreted.

On page 3 Pedoe says "We know that in Euclidean geometry, for any triangle ABC, angle BCA + angle CAB + angle ABC = Pi (mod 2Pi)"

If I recall correctly the interior angles of a triangle add up to 180 degrees, so I wrote "Pi (mod 2Pi) = 180 degrees." Did I make a mistake?

Anyway, except for using the word "Pi" instead of the Greek symbol, and using the word "angle" where Pedoe uses a symbol like the "less than" symbol "<" but with an arc drawn through it, I have now quoted exactly.

I don't get the "mod" idea. My best understanding from reading around in Wiki and Mathworld is that (mod Pi) would mean the decimal portion of Pi, in other words, Pi minus three. So (mod 2 Pi) should be about .28. Am I anywhere close to the meaning of the mod notation?

Looking on in the chapter, I don't see any further use of the mod notation. Maybe I should just ignore this sentence as some kind of aberration.

Thanks for your comment and any further idea you may have on the idea of "mod".

R

Last edited: Feb 16, 2006
4. Feb 16, 2006

### shmoe

The "mod 2Pi" means they are treating angles that differ by an integer multiple of 2Pi as the same, eg. Pi, 3Pi, 5Pi, -Pi, -99Pi all point in the same direction. This isn't saying the real numbers Pi and 3Pi are the same, rather the angles given by Pi and 3Pi radians are the same (recall a full circle is 2Pi radians).

5. Feb 17, 2006

### HallsofIvy

Yes, that's true. Although I would prefer it said specifically "Pi radians" and "(mod 2Pi radians)".

No, that's not at all the same thing. What is the same thing is
"in Euclidean geometry, for any triangle ABC, angle BCA + angle CAB + angle ABC = 180 degrees (mod 360 degrees)"

Both are saying that you can keep increasing the angle without bound- winding around and around the circle, if you like- but normally, we work within "once around the circle", 2pi radians or 360 degrees.

6. Feb 17, 2006

### rtharbaugh1

Thank you both for the replies. I am still uncertain about the idea of mod, which I have also seen in other places. I would like to feel that the next time I encounter it I will know what it means. Any direction here much appreciated.

Thanks,

Richard

7. Feb 17, 2006

### shmoe

Try looking up "Modular arithmetic". This is widely used in number theory (any elementary text will explain it), so you'll be able to find plenty of examples.

8. Feb 17, 2006

### matt grime

mod is something you use every day of the week, literally.

Suppose today is still Friday the 17th of February when you read this.

What day is nine days hence? That's a week and two days, so the day of the week is unaffected by adding seven days, so it is sunday. What date is in 3 weeks time? 3 weeks is twenty one days, so that would be the 38'th of February (bear with me), but feb only has 28 days making it ten days into March.

See, you're knocking off multiplies of something useful.

In geometry there is no reason to distinguish 180 and 540, in some sense, ie the differ by a full circle of 360 degrees.