One of Riemann's notes

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An alternative GCD algorithm

The greatest common divisor ( GCD ) of two integers a and b is usually calculated with the ( well-known ) Euclidean Algorithm. There is however an alternative algorithm which is based on an entirely different idea. Let's illustrate this idea with an example. Let a, b be integers, a >:b and (a, b) = GCD(a,b). Then the following rules can be applied recursively until a=b=GCD(a,b):
If ( a=even AND b=even) then GCD(a,b)=2*GCD(a/2,b/2)
If ( a=odd AND b=even) then GCD(a,b)=GCD(a,b/2)
If ( a=even AND b=odd) then GCD(a,b)=GCD(a/2,b)
If ( a=odd AND b=odd) then GCD(a,b)=GCD(a-b,b)

Example
(36, 27) = (27, 36/2)
(27, 18) = (27, 18/2)
(27, 9) = (27-9, 9)
(18, 9) = (18/2, 9)
(9, 9) Halt.
GCD(36,27)=9.

Compare using the Euclidean Algorithm
36 = 1 * 27 + 9
27 = 3 * 9 + 0 Halt.
GCD(36,27)=9.

However, this doesn't mean that the Euclidean Algorithm is always faster.

Calculating squares

Try ( without a calculator )

21^2 ?

37^2 ?

If you need a calculator to calculate simple squares then you may need the following simple rule.

21^2 = 441

37^2 = 1369.

Or using ( x - k ) * ( x + k ) + k^2 = x^2

21^2 = 20 * 22 + 1^2 = 440 + 1 = 441

37^2 = 34 * 40 + 3^2 = 1200 + 160 + 9 = 1369.

Cubic numbers

Create a triangle from the sequence of odd numbers s[n]=2n-1 by writing s[1] on the first row, s[2] and s[3] on the second row, the next three numbers from the sequence on the third row, ... the next k numbers on the k-th row. For example:

The vertical column contains the sums by row of the numbers in the triangle. It is easy to see that this column contains the cubic numbers.

Primes

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate. (Leonard Euler)

Roman numerals

Just noticed that the BBC uses Roman numerals for copyright dating. (c) MMVIII.

I - 1
V - 5
X - 10
L - 50
C - 100
D - 500
M - 1000

1 I
2 II
3 III
4 IV
5 V
6 VI
7 VII
8 VIII
9 IX
10 X
11 XI
12 XII
13 XIII
14 XIV
15 XV
16 XVI
17 XVII
18 XVIII
19 XIX
20 XX

My goal

I decided that I should have a goal related to my math hobby. I don't know if that is good or bad. It is my goal to get this degree in mathematics. It has been in the back of my mind for a while. It's time to come out of the closet. Having a secret goal is a sort of fear of failure I guess. Working towards a goal is more fun than having achieved a goal because once a goal has been achieved new goals turn up. Talking ( writing ) about what I am doing is the purpose of this blog anyway. ( To be continued. )

Inner product of two matrices

Let M(m,n)[R] be the vector space of m by n matrices with elements in R and scalar field R.

Let A, B, C elements in M(m,n)[R] and x,y elements in R. Define the map

f: V x V -> R by (A,B) |-> Tr(B^T * A).

Since
- (xA+yB,C) = x(A,C) + y(B,C)
- (A,B) = (B,A)
- (A,A) >= 0
(proofs are trivial)

f is an inner product.

The psychology of Charlie Eppes.

He is often shown "stimming". "Stimming" is basically self-stimulation, a repetitive sound or motion made by people who have difficulty integrating all the sensory input around them as a buffer between themselves and the sensory stimulus.

Charlie has sensory integration issues. This means he has difficulty integrating sensory input around him into a cohesive whole, something that most of us do automatically but is sometimes an exhausting task for individuals in the autism spectrum. Charlie wears enormous headphones with no music playing while figuring out a math problem.

Charlie fails to see the big picture at almost every turn and has extreme difficulty anticipating spontaneous behavior. Charlie is most comfortable with inductive reasoning, and despite his high intelligence levels, finds deductive reasoning difficult at best.

Charlie struggles to interpret social cues from others, when he picks up on them at all. Charlie rarely intuits when he annoys everyone around him, either with his single-minded preoccupation with mathematics or his inability to accurately read the nuances of social situations. He is very literal minded, often rambles past the point where anyone is listening, and in fact, sometimes DOES NOT NOTICE everyone has stopped listening.

Charlie fails to connect his concrete theorizing to abstract human reality, and in fact, doing so is deeply disturbing to him.

Charlie displays an extremely limiting and persistent social naivete that also can translate as extreme social immaturity. He is 27, doesn't date, doesn't notice that his very pretty friend is probably interested in him.

There is the compulsive and completely preoccupying nature of Charlie's obsession with discrete mathematics.

( This entry is a summary of this article on "tv makes you stupid". )

Projection Matrix

A matrix P is a projection matrix IFF P=P*P.

Henry Pitcher

92-year-old to graduate from UH with a bachelor degree in math Crossing the stage to graduate from the University of Houston, summer 2007 semester culminated a quest that Henry Pitcher, 92, began 75 years ago. Full Story