User:Pilover819/SMath
My favorite subject in math is numbers, especially Pi. It's 3.14159..... it goes on forever. It is irrational and transcendental. There are currently 12 trillion digits of pi known. It was found very recently. I know 320 digits of pi.
My second favorite number is e. It is called the Euler's number. The first ten digits are 2.7182818284... This number is like pi in that it is irrational and transcendental.
In 2004, a mathematician found out that a pandigital approximation (1+9-47*6)3285 can find 18457734525360901453873570 digits of e.
My third favorite number is the Euler-Mascheroni constant, better known as Gamma. It is 0.5772156649... This number, again, goes on forever, but we don't know yet if this number is irrational or not.
My fourth and fifth favorite numbers are the golden ratio, also known as phi or tau, and the Omega constant. The golen ratio is approximately 1.61803398874989484820... and the Omega constant is approx. 0.56714329040978387299...; those two mathematical constants are also irrational! Isn't that great?
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Pascal's triangle is an arithmetic triangle. I got interested after my dad taught me the Pascal's triangle. Here are some rows of the P.t.: As you see, the second one plus the third one is 2, thus 2 is on row 2, place 1. To see more rows of the Pascal's triangle, try the Wikipedia's article on Pascal's triangle. Here is the OEIS's site:OEIS on Pascal's Triangle
Relationship with other math systems
[edit] 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
You can use the Pascal's triangle as m-dimensional figures in an n-dimensional simplex. In a "point" (0-D simplex), there is 1 point (row 1, column 1). In a "line segment" (1-D simplex), there are 2 points (row 2, column 1) and 1 line segment (row 2, column 2). In a triangle, there are 3 points (row 3, column 1), 3 line segments (row 3, column 2), and 1 triangle (row 3, column 3), etc. Look:
As you see here[1], the shallow diagonals of the Pascal's triangle are the Fibonacci numbers. There is a section on Fibonacci numbers, go here.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Also, if you add up k rows (by that I mean from row 0 to the k-1th row), [2], you will get Mersenne numbers. For example, here is 1+1+1+1+2+1=7, which is 23-1, a Mersenne number, also a Mersenne prime.
The Cartan's triangle (no Wikipedia article), is the Pascal's triangle's OLDER BROTHER, so to speak. The number on the left is doubled everytime, while the Pascal's triangle is not. The numbers in the Cartan's triangle are related with hypercubes. Here are the first 7 rows of the triangle:
1
2 1
4 4 1
8 12 6 1
16 32 24 8 1
32 80 80 40 10 1
64 192 240 160 60 12 1
As you see, these are powers of 21 (12 if made backwards), because as you see in Row 1, there is 21. And in the second row, you see 441 which is 21². As you go on, you see two+ digit numbers on the Cartan's triangle. So then you do the same thing to it; carry it down to the number on the left. Here's a link on the Cartan's triangle: Cartan's triangle
- Note: If you have Excel, you will be able to make the Cartan's triangle bigger and bigger.
Here is OEIS's site:OEIS on Cartan's Triangle
The Fibonacci sequence is sequence where the previous two numbers are added together to make a new number. [3]The first few Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
If you want to see the 10 millionth Fibonacci number, go here.
n-nacci number
[edit]If the previous 1 number was added, it would be this: 1, 1, 1, 1, 1, 1, 1, 1, 1... so "Fibonacci numbers" were created. Let's see what Fibonacci numbers can transform into:
Tribonacci
[edit]Tribonacci numbers are where the previous 3 numbers (notice "Tri" in tribonacci) are added together to make a new number. The first few Tribonacci numbers are 1, 1, 2, 4, 7, 13, 24, etc.
Tetranacci
[edit]Tetranacci numbers are where the previous four numbers (notice "Tetra" in tetranacci) are added together to make a new number. The first couple of Tetranacci numbers are 1, 1, 2, 4, 8, 15, 29, 56, etc.
More polynacci numbers
[edit]Polynacci numbers are solved like this: _____nacci numbers are where the previous n numbers (notice _____ in _____nacci) are added to make a new number. The first couple of _____nacci numbers are, nF(n), nFn+1, nFn+2, etc. 5th order is "Pentanacci", 6th is "Hexanacci", 7th is "Heptanacci", 8th is "Octanacci", 9th is "Enneanacci" or 9th order Fibonacci numbers, 10th is "Decanacci" or 10th order Fibonacci numbers, and so forth.
Recursive relation in generalized Pascal's triangle
[edit]Let the number in the zeroth row be 1, and the first row a and b, and span it to look like a triangle. Then, if you see the sum of each diagonal, you will notice the sums form a recursive relation.
Simpleces, Hypercubes, and Cross-polytopes
[edit]These three things are multi-dimensional shapes. For example, a "heptacross" is a 7-dimensional shape, because of the prefix hepta-.
Champernowne constant in base 10
[edit]The Champernowne constant (in base ten) is a mathematical constant where all the positive whole numbers are listed in order. Any one of you can memorize most of this number, because it's simply 0.12345678910111213141516171819202122232425262728293031323334353637383940414243444546...
The Champernowne constant is transcendental. It is probably irrational, too. Since the Champernowne constant is irrational, you can't memorize all the digits.
The weird dream I had on July 30~31, 2008 about the largest known prime number
[edit]I had this dream, about the largest known prime. In my dream, I logged onto Wikipedia to see the article. But instead of 9,808,358 digits, I saw "...As of 2008 , the largest known prime contains..." and I saw a number in the 11 millions.
Also, on September 14, I had a dream about the largest known prime's exponent, and I saw a number about 40.8 million.
EDIT: It turned out that the mean of the two primes was near 40.1 million, and the 2nd prime had 11.2 million digits. It was a wrong dream. The largest known prime's exponent is in the 43 millions, and the number is nearly 13 million digits long.. but I still wonder how I dreamt the dream only a few months before the new primes were found.