randomness in github commits

I just read the proof for the (a,b)=ax+by, where a,b,x,y are all integers and (a,b) is its gcd. Just like the proof for the division algorithm, this one is really awesome. The proof can be divided into two parts. The first part proves the existence of a least positive integer “l” in the set {ax+by} such that l|(ax+by). The second part proves that l=g=(a,b).

So far, I feel that the elegance of a proof lies in the way its been approached. Some are classical and you can see it coming right after you read what the initial assumptions are, but in some others the way you approach the theorem itself is so … out-of-the-box? (lack of a better word).

Lets have a look at the proof of the “divisibility algorithm”. It states that for any integers “a”,”b” there exist “q”,”r” such that “b=qa+r”, where 0<=r<a; If a does not divide b, 0<r<a.

Proof:
Consider the sequence of numbers in the AP given below

….. , b-3a, b-2a, b-a, b, b+a, b+2a, b+3a, …..

So, r; would be the smallest non-negative integer in the above sequence. Why? There are two reasons for this:-

So now we need to prove that “q” and “r” are unique. In order to do this, assume that there is another pair (q1,r1) such that (b-q1a)=r1.

So, we could have either (r1-r) or (r-r1) as less than “a” as r1<a and r<a.

(r1-r) = (b-q1a - (b-qa)) = (-q1a + qa) = a(q-q1)
Hence,
a | (r1-r)

However, according to a theorem we can say that a|b if a!=0 and a<b. In this case, we have a contradiction. Hence, (q1, r1) does not exist and we can say that (q,r) is unique.

Beautiful, isn't it? How do people come up with such stuff huh? :/

def dummy_func(x):
# statements that would
# raise the exception
….
self.assertRaises(TypeError, dummy_func, 1)

In py.test you could simply do something like:-

def test_func():
with pytest.raises(TypeError):
# statements that would
# raise the exception
….

Not a big deal, but I like this much better.

A few points :-

So if you wanna know what Im upto, check out my buzz posts.