31.10.05
bumper sticker
91
if you want to remember the prime numbers up to 100:
- 1 isn't a prime, unless you reckon it is, in which case it might be.
- the even numbers end in 0, 2 ,4 ,6 or 8 and aren't primes (except 2)
- multiples of three have digits which sum to a multiple of 3 and aren't primes (except 3)
- odd multiples of 5 end in 5 and aren't primes (except 5)
- multiples of 11 have two digits the same and aren't primes (except 11)
- 49 isn't a prime = 7*7. most people know that.
- 91 isn't a prime = 7*13. most people don't know that.
if you remember the primes up to 100 you can factorise (and test for primality) numbers up to 10,201 (= 101*101)
happy weierstrass day
his 190th birthday.
27.10.05
definitions
the reason noone has come up with a good definition of what mathematics is because all the people who know what it is are not interested in playing word games.
they have games of their own.
they have games of their own.
26.10.05
'chebyshev'
"At least five different transliterations of this name exist in the literature."
A very useful footnote in a damn useful book Probability, by John Lamperti
A very useful footnote in a damn useful book Probability, by John Lamperti
25.10.05
i knew that
John Dvorak, moaning about photo editing software, hits the nail on the head:
"Microsoft has a tendency to combine powerful tools with the dumbed-down "What do you want to do?" interface more than any single company."
23.10.05
a poem my printer wrote me
AQKM!
SEU!
QAGEQSOUECU=AUUOAQKM!
SEU!
SESOMUUIOO=711AQKM!
EOUES!
MAOGUAGE=QCM
SEU!
QAGEQSOUECU=AUUOAQKM!
SEU!
SESOMUUIOO=711AQKM!
EOUES!
MAOGUAGE=QCM
13.10.05
scanner
My co-worker from a long time ago on the phone, against a background of prime numbers
11.10.05
'tyranny'
i wonder..
did god tell dubya that there were weapons of mass destruction in iraq?
or did he tell him to um, 'pretend' that there were?
did god tell dubya that there were weapons of mass destruction in iraq?
or did he tell him to um, 'pretend' that there were?
10.10.05
A computer company with sense
From the BBC Micro User Guide:
"After that, do by all means press every button in sight on the computer - you can't do it any harm at all."
This gave me so much confidence as a 10-yr-old.
And from the Advanced User Guide:
"The final paragraph in this introduction must be a word of apology to those programmers engaged in the task of software protection. Many of the details contained in this book will give those intent on pirating software inspiration to circumvent their protection techniques. On the other hand these same details may also give the software protectors inspiration. In the end no software protection is complete. Any protection technique relies on the fact that the person trying to break the protection has a threshold at which he decides that the effort and resources required are greater than the reward. For some this threshold is higher than others but for these people the reward is often the victory in the intellectual battle with the programmer of the protection method. For those intent on denying the software producer his income, one hopes that this threshold is somewhat lower."
"After that, do by all means press every button in sight on the computer - you can't do it any harm at all."
This gave me so much confidence as a 10-yr-old.
And from the Advanced User Guide:
"The final paragraph in this introduction must be a word of apology to those programmers engaged in the task of software protection. Many of the details contained in this book will give those intent on pirating software inspiration to circumvent their protection techniques. On the other hand these same details may also give the software protectors inspiration. In the end no software protection is complete. Any protection technique relies on the fact that the person trying to break the protection has a threshold at which he decides that the effort and resources required are greater than the reward. For some this threshold is higher than others but for these people the reward is often the victory in the intellectual battle with the programmer of the protection method. For those intent on denying the software producer his income, one hopes that this threshold is somewhat lower."
9.10.05
The number of ways to tie a necktie using n+2 turns
1, 1, 3, 5, 11, 21, 43, 85, 171...
The Jacobsthal sequence.
I came across this sequence, or originally, double the sequence when I was trying to solve a problem involving counting binary subintervals of the [0, 1] interval. This way of working it out is to add on to each succesive power of 2 the amount by which the previous term falls short of it. Start with 1.
eg.
Here we first subtract the previous (n-1th) term from the 2n-1, then add the result we get back on to 2n-1 to get the nth term. The number we add back on turns out to be the last but one term, illustrating another property of the sequence; that 2 successive terms sum to a power of 2.
An easier way of working it out is to start with 0, double it and then add one (to get 1), double that and _subtract_ 1, (= 1) double and add 1 again (1*2 + 1 = 3), double and subtract 1 (3*2 - 1 = 5) and so on.
If you do something similar, but alternate simple doubling with either doubling and then adding 1, or doubling and subtracting 1, you get the similar sequences 1, 2, 5, 10, 21, 42, 85, 170 (alternately doubling, then doubling and adding 1) and 1, 2, 3, 6, 11, 22, 43, 86, 171 (doubling with doubling and subtracting 1). In the first case every other number is 1 smaller than our original sequence, in the second one every other number is 1 bigger.
Sloane's entry A001045 gives a different recursive definition; a(n) = a(n-1) + 2.a(n-2) (essentially the same as the 1st method above) and lists a huge number of places this sequence occurs, including the one in the title, but doesn't seem to specifically point out that it is the closest integer to 2n/3.
The Jacobsthal sequence.
I came across this sequence, or originally, double the sequence when I was trying to solve a problem involving counting binary subintervals of the [0, 1] interval. This way of working it out is to add on to each succesive power of 2 the amount by which the previous term falls short of it. Start with 1.
eg.
1 1 - 1 = 0 2 - 1 = 1 4 - 3 = 1 8 - 5 = 3 16 - 11 = 5 | 1 + 0 = 1 2 + 1 = 3 4 + 1 = 5 8 + 3 = 11 16 + 5 = 21 |
An easier way of working it out is to start with 0, double it and then add one (to get 1), double that and _subtract_ 1, (= 1) double and add 1 again (1*2 + 1 = 3), double and subtract 1 (3*2 - 1 = 5) and so on.
If you do something similar, but alternate simple doubling with either doubling and then adding 1, or doubling and subtracting 1, you get the similar sequences 1, 2, 5, 10, 21, 42, 85, 170 (alternately doubling, then doubling and adding 1) and 1, 2, 3, 6, 11, 22, 43, 86, 171 (doubling with doubling and subtracting 1). In the first case every other number is 1 smaller than our original sequence, in the second one every other number is 1 bigger.
Sloane's entry A001045 gives a different recursive definition; a(n) = a(n-1) + 2.a(n-2) (essentially the same as the 1st method above) and lists a huge number of places this sequence occurs, including the one in the title, but doesn't seem to specifically point out that it is the closest integer to 2n/3.
8.10.05
algebraic number theory
my professor started this course by saying "the problem with language is that it's not associative." meaning the name of the course could be read as (algebraic number) theory, the theory of algebraic numbers, or algebraic (number theory), number theory using algebra. both of these make some kind of sense mathematically.
i'm glad he said this because it's something i've often worried about, my favourite example being Third World War.
i'm glad he said this because it's something i've often worried about, my favourite example being Third World War.
6.10.05
a number sequence
one of three integer sequences i have come across this summer:
1, 3, 7, 13, 19, 27, 39, 49..
These are the numbers generated by Flavius Josephus's sieve.
Sloane's A000960
Start with the integers
1, 2, 3, 4, 5, 6, 7,.. etc.
Cross out every 2nd number, ie the even numbers, leaving the odd numbers
1, 3, 5, 7, 9, 11, 13...
Go through these from the start and cross out every 3rd number, leaving
1, 3, 7, 9, 13...
Then cross out every 4th number, then every 5th number remaining after the 4th stage, and so on.
The numbers you have left form the sequence.
You will have some left; if you look at the last line of numbers above, 1, 3 and 7 will never be crossed out. They are the first 3 numbers, and we're only going to cross out the 4th, 5th, 6th.. numbers at any later stage. If you look at the next stage, 9 will be crossed out and 13 will become the 4th number. Then 13 will stay forever, since only numbers after the 4th one will be crossed out later.
In practice, you can't write out every integer to start with, since there are infinitely many. Nor can you go through infinitely many times, crossing out the nth number for each n: 2 ,3, 4, 5... But this is normal; we can only calculate and write down finitely many numbers from any infinite sequence.
Suppose you want to find all numbers from the sequence less than 1,000. Write out the numbers up to 1,000. Go through them taking out every 2nd number, every 3rd number... Eventually you will be taking out say, the 50th number, but there will be less than fifty numbers left. When this happens you are done. As explained above, the numbers you still have will remain in the final sequence, no matter how many steps you carry on for.
A 'sieve' is a procedure like this one to construct a set of numbers by repeatedly removing certain sets you don't want. The original sieve was Eristophanes' sieve, which generates the prime numbers. I came across this one when i misunderstood a definition of the 'lucky numbers', which are found using a very similar sieving method (Sloane's A000959).
1, 3, 7, 13, 19, 27, 39, 49..
These are the numbers generated by Flavius Josephus's sieve.
Sloane's A000960
Start with the integers
1, 2, 3, 4, 5, 6, 7,.. etc.
Cross out every 2nd number, ie the even numbers, leaving the odd numbers
1, 3, 5, 7, 9, 11, 13...
Go through these from the start and cross out every 3rd number, leaving
1, 3, 7, 9, 13...
Then cross out every 4th number, then every 5th number remaining after the 4th stage, and so on.
The numbers you have left form the sequence.
You will have some left; if you look at the last line of numbers above, 1, 3 and 7 will never be crossed out. They are the first 3 numbers, and we're only going to cross out the 4th, 5th, 6th.. numbers at any later stage. If you look at the next stage, 9 will be crossed out and 13 will become the 4th number. Then 13 will stay forever, since only numbers after the 4th one will be crossed out later.
In practice, you can't write out every integer to start with, since there are infinitely many. Nor can you go through infinitely many times, crossing out the nth number for each n: 2 ,3, 4, 5... But this is normal; we can only calculate and write down finitely many numbers from any infinite sequence.
Suppose you want to find all numbers from the sequence less than 1,000. Write out the numbers up to 1,000. Go through them taking out every 2nd number, every 3rd number... Eventually you will be taking out say, the 50th number, but there will be less than fifty numbers left. When this happens you are done. As explained above, the numbers you still have will remain in the final sequence, no matter how many steps you carry on for.
A 'sieve' is a procedure like this one to construct a set of numbers by repeatedly removing certain sets you don't want. The original sieve was Eristophanes' sieve, which generates the prime numbers. I came across this one when i misunderstood a definition of the 'lucky numbers', which are found using a very similar sieving method (Sloane's A000959).
2.10.05
me and björk
