The formula for the sum of row x in Pascal's Triangle is 2^x, and the value of each position y across in that row is xCy.
Every row beginning with a prime has only multiples of that prime in the row.
If you go down diagonally from each one in a sort-of 'knight's pattern', like in chess, and add up the numbers, you get the Fibonacci sequence.
The diagonals from the ones in the third row have triangular numbers, and if you add each adjacent one, you get square numbers.
If every odd number is colored in in an infinitely big Pascal's Triangle, you get Sierpinski's Triangle.
The contents of each row are powers of 11, with some of the contents smushed up and added together.
Stay coolio,
John
Thursday, January 30
Fractal Maker
I made an amazing fractal today! I name it....
THE Z FRACTAL!!!!!!!!
which is infinity.
Just an update, you know.
Stay coolio, John
THE Z FRACTAL!!!!!!!!
You make it by taking a line, adding lines to the end to make a Z, and repeating it with the lines you added, and et cetera. You can do it as long as the interior and exterior angles are each the same.
If the Z has base length 3 and diagonal length 5, then the formula for the entire length of the fractal is
which is infinity.
Sorry it's scratchy. I just took a partial screenshot from Wolfram-Alpha.
By the way, the best fractal maker that I found for the Mac is @
http://www.shodor.org/master/fractal/software/Snowflake.html
It's great for experimenting and other stuff. You can change the number of segments in the first iteration, the beginning of the fractal, and the configurations. You can see the fractal up to the eighth iteration–– that's pretty good, if you ask me.
Just an update, you know.
Stay coolio, John
Sunday, January 26
Population and Stuff
According to Wolfram Alpha, 31 of the world's 50 most populous cities are in China. China contains almost a fifth of the world's population.
Macau is the world's densest country, with fifty three thousand people every square mile, while Greenland is the sparsest (with one person every 14.5 square miles).
The Vatican and the Pitcairn Islands are the most Christian countries, while Somalia is the least. Morocco is the most Muslim country, and Israel is the most Jewish country.
4.3 people are born every second (on average), while 1.8 people die. This means we have a net gain of 2.5 people per second.
The world's population has quadrupled in the 20th century, and yet is has also declined in the 14th century (due to Black Death).
Adios for hoy,
John
Macau is the world's densest country, with fifty three thousand people every square mile, while Greenland is the sparsest (with one person every 14.5 square miles).
The Vatican and the Pitcairn Islands are the most Christian countries, while Somalia is the least. Morocco is the most Muslim country, and Israel is the most Jewish country.
4.3 people are born every second (on average), while 1.8 people die. This means we have a net gain of 2.5 people per second.
The world's population has quadrupled in the 20th century, and yet is has also declined in the 14th century (due to Black Death).
Adios for hoy,
John
Thursday, January 23
Paradoxes!
Well, if something just doesn't work even though everything is right, you've found a paradox.
PARADOXES-
If you have a fully booked hotel with infinite rooms and infinite people arrive to get a room, you can just move everybody (who was there beforehand) in their rooms 'n' into rooms '2n' and fit them in.
If a person says, "This statement is false," then are they lying?
If you have a perfect machine that for a half-minute is on, for a quarter minute is off, then an eighth minute is on, and et cetera, will the machine be on in two minutes exactly?
sum (n=1 through infinity) (-1)^n = ? but the equations below are the same with different answers.
(1-1+1-1...) is the same as (1+(-1+1)+(-1+1)...=1) is the same as ((1-1)+(1-1)+(1-1)....=0)
Adam owns a shop with two employees, Bob and Carl. There are two rules for the workers.
Rule 1: There must be a worker in the shop at all times.
Rule 2: If Adam is out, Bob must be out.
Let's say Carl is out. By logic, if Adam is out, then Bob must be in and out, so Carl can never leave the store. However, the fact that Carl is out leads to the fact that he can't be out.
Let's say a faraway galaxy has two 'rules of sand'.
Rule 1: One grain of sand is not a heap.
Rule 2: Adding a grain of sand to a pile that is not a heap does not make it a heap.
Let's say you have a grain of sand, and add infinity more to the grain, one by one. That would mean infinite grains of sand are not a heap, which means that there is no such thing to distinguish a heap from a pile, thereby saying that the laws that proved it are unnecessary.
There are oh so many, and some of them were so confusing I barely looked at them. In fact, I am still confused by the Shopkeeper's Dilemma (with mah gals Adduhm, Baubbe, and Caurel).
That's all for our second post as of hoy!
Stay coolio,
John
PARADOXES-
If you have a fully booked hotel with infinite rooms and infinite people arrive to get a room, you can just move everybody (who was there beforehand) in their rooms 'n' into rooms '2n' and fit them in.
If a person says, "This statement is false," then are they lying?
If you have a perfect machine that for a half-minute is on, for a quarter minute is off, then an eighth minute is on, and et cetera, will the machine be on in two minutes exactly?
sum (n=1 through infinity) (-1)^n = ? but the equations below are the same with different answers.
(1-1+1-1...) is the same as (1+(-1+1)+(-1+1)...=1) is the same as ((1-1)+(1-1)+(1-1)....=0)
Adam owns a shop with two employees, Bob and Carl. There are two rules for the workers.
Rule 1: There must be a worker in the shop at all times.
Rule 2: If Adam is out, Bob must be out.
Let's say Carl is out. By logic, if Adam is out, then Bob must be in and out, so Carl can never leave the store. However, the fact that Carl is out leads to the fact that he can't be out.
Let's say a faraway galaxy has two 'rules of sand'.
Rule 1: One grain of sand is not a heap.
Rule 2: Adding a grain of sand to a pile that is not a heap does not make it a heap.
Let's say you have a grain of sand, and add infinity more to the grain, one by one. That would mean infinite grains of sand are not a heap, which means that there is no such thing to distinguish a heap from a pile, thereby saying that the laws that proved it are unnecessary.
There are oh so many, and some of them were so confusing I barely looked at them. In fact, I am still confused by the Shopkeeper's Dilemma (with mah gals Adduhm, Baubbe, and Caurel).
That's all for our second post as of hoy!
Stay coolio,
John
Pi and E
I didn't know that you could find pi and e with infinite series! Here are the serieses! (My preciouses)
sum(n= 1 through infinity) ( (-1)^(n+1) * (4 / (2n-1) ) ) = pi
sum(n= 1 through infinity) ( 1 / (n-1)! ) = e
Isn't it weird that the sum for e has a factorial? I never made that connection before. I thought that factorials were for combinations and permutations, not logarithms, but I guess we use exponents in probability and all that. Whatever.
Here's something weird about pi and e–– they aren't really connected to any integers, at least not by simple functions. That means they're transcendental, along with––
sum(n= 1 through infinity) (10 ^(-k!)) = 0.1100010000000000001...
e^pi
sum(n= 1 through infinity) (10 ^(-k^2))= 0.1001000010000001000000001... (my invention!)
and all of that, yada yada yada.
Personally, I think we should use 2 times pi as opposed to plain old pi. We base absolutely nothing on the diameter, and it would make radians so much more simple.
Adios for hoy,...
Stay coolio,
John
sum(n= 1 through infinity) ( (-1)^(n+1) * (4 / (2n-1) ) ) = pi
sum(n= 1 through infinity) ( 1 / (n-1)! ) = e
Isn't it weird that the sum for e has a factorial? I never made that connection before. I thought that factorials were for combinations and permutations, not logarithms, but I guess we use exponents in probability and all that. Whatever.
Here's something weird about pi and e–– they aren't really connected to any integers, at least not by simple functions. That means they're transcendental, along with––
sum(n= 1 through infinity) (10 ^(-k!)) = 0.1100010000000000001...
e^pi
sum(n= 1 through infinity) (10 ^(-k^2))= 0.1001000010000001000000001... (my invention!)
and all of that, yada yada yada.
Personally, I think we should use 2 times pi as opposed to plain old pi. We base absolutely nothing on the diameter, and it would make radians so much more simple.
Adios for hoy,...
Stay coolio,
John
Wednesday, January 22
Birthday Paradox
Happy Birthday to More-than-one-person...
A couple of years ago I saw something called the birthday paradox. It was something about the probability that any two people in a certain room with x people will have the same birthday.
What makes the problem a paradox is that the actual chance is a lot higher than you'd think. In fact, in a room with 23 people, it's 50-50. That's cray-cray when you think about it: of 365 days, two on the same day?
The graph below tells us the chance (y) for the number of people (x).
Nothing much to say hoy...
Stay coolio,
John
A couple of years ago I saw something called the birthday paradox. It was something about the probability that any two people in a certain room with x people will have the same birthday.
What makes the problem a paradox is that the actual chance is a lot higher than you'd think. In fact, in a room with 23 people, it's 50-50. That's cray-cray when you think about it: of 365 days, two on the same day?
The graph below tells us the chance (y) for the number of people (x).
Nothing much to say hoy...
Stay coolio,
John
To Infinity and Beyond
Infinity is like, the weirdest thing ever. I think sometimes that it is obvious, but what it does is change whole numbers like one into zero.
1/0=infinity
1=infinity*0
1=0
???????
Whatever 1=0 is supposed to mean, it does not fit the laws of algebra. If I see something like above on the Algebra 2 midterm today, I might throw up infinity gallons.
Personally, I don't really care yet. How about we just forget this conversation until I get to high school? Well, whatever–– here's what I think
The thing about the equation above is that you can't divide one into zero pieces. If you get no pieces at all when dividing something, you either have zero matter whatsoever or just infinitely small pieces. But no matter what, you always get 1=0 as an answer, unless 0/0=1 by multiplicative identity. This is our only (remotely) plausible answer, so that's my theory (as infinity has no properties because it doesn't exist in algebra).
1/0=0
1/(0*0)=1?
1/0=1?
Well, it seems that my theory has a (differing) counterpart. Adios, infinity! I'll be back much later...
If you fold a piece of paper in half infinity times and unfold it, you get the Dragon Curve (or a tiny version because infinite material is used in such a small shape). I tried it myself and it worked.
Well, it seems that I am about to suffer Spanish midterms, so that's it for hoy.
Stay coolio,
John
1/0=infinity
1=infinity*0
1=0
???????
Whatever 1=0 is supposed to mean, it does not fit the laws of algebra. If I see something like above on the Algebra 2 midterm today, I might throw up infinity gallons.
Personally, I don't really care yet. How about we just forget this conversation until I get to high school? Well, whatever–– here's what I think
The thing about the equation above is that you can't divide one into zero pieces. If you get no pieces at all when dividing something, you either have zero matter whatsoever or just infinitely small pieces. But no matter what, you always get 1=0 as an answer, unless 0/0=1 by multiplicative identity. This is our only (remotely) plausible answer, so that's my theory (as infinity has no properties because it doesn't exist in algebra).
1/0=0
1/(0*0)=1?
1/0=1?
Well, it seems that my theory has a (differing) counterpart. Adios, infinity! I'll be back much later...
According to this graph, I am unofficially wrong, which means the graph is glitching. LOL
Well, it seems that I am about to suffer Spanish midterms, so that's it for hoy.
Stay coolio,
John
Tuesday, January 21
Infinite Series
Something I just noticed–– linear and quadratic infinite series almost always total to infinity or negative infinity. When I say linear, I mean in the form (an+b) or in standard form for quadratics. Anyway, it depends on the coefficient of the highest degree of n–- every time.
There are a lot of good math You-Tubers: Vihart and Numberphile are two of my particular favorites, Vihart in particular being particularly kid-friendly. Her "Doodling in Math Class" videos are some of my favorites.
In one of her videos, she argues with the plural of series. I think it should be 'serieses', as the plural of proper nouns is (Not to be one of those Gollumses). However, some singular nouns ending in one -s double the 's' and add -es instead. However, those nouns have a hard 's' as in 'cross' rather than the soft one in 'has'... well, frankly, I don't care too much.
We learned something cool in math class the other day. In this Remainder Theorem, we can say that if we do synthetic division with P(x) and a, then the remainder is always P(a). COOL. There's a proof at http://www.purplemath.com/modules/remaindr.htm that's really cool, btw.
Other polynomial things--
There are a lot of good math You-Tubers: Vihart and Numberphile are two of my particular favorites, Vihart in particular being particularly kid-friendly. Her "Doodling in Math Class" videos are some of my favorites.
In one of her videos, she argues with the plural of series. I think it should be 'serieses', as the plural of proper nouns is (Not to be one of those Gollumses). However, some singular nouns ending in one -s double the 's' and add -es instead. However, those nouns have a hard 's' as in 'cross' rather than the soft one in 'has'... well, frankly, I don't care too much.
We learned something cool in math class the other day. In this Remainder Theorem, we can say that if we do synthetic division with P(x) and a, then the remainder is always P(a). COOL. There's a proof at http://www.purplemath.com/modules/remaindr.htm that's really cool, btw.
Other polynomial things--
- The graphs of even-degreed polynomials always head for the same infinity while odds head towards opposites. However, the pattern skips over 0 because that's just what zero does. (Just kidding, it creates a horizontal line because anything to the zeroth degree is a constant)
- If a polynomial has an imaginary or even irrational root, then that root's conjugate is also a root. Also, a polynomial's degree is equal to its amount of roots (not including multiplicity).
- A polynomial to the nth degree has a graph with (n-1) curves.
So yeah, that's it for hoy.
Stay coolio...
John
Stay coolio...
John
Monday, January 20
Fractals!!!
It seems that if you add up all the terms in the series
1-(1/2)+(1/4)-(1/8)... and so on,
you get two thirds, which is quite odd because powers of two have nothing to do with 3.
Anyway, I've always liked fractals because of their endless continuity.
One especially cool fractallish pattern is the Dragon Curve, which is not only cool for its name but also for the methods it uses. Basically, you take a vertical line segment, rotate it 45 degrees counterclockwise, and then add another copy rotated 90 degrees clockwise to it. Another, more simple way to find it in nature is to take a piece of paper, fold it in half, fold it in half again, fold it half again, and et cetera. If you look down the edge you fold, it slowly becomes the Dragon Curve. A cool Youtube animation thing by a guy that calls himself Numberphile is really good.
The most basic fractal, fortunately triangular, is the Sierpinski Triangle. If you take a regular triangle, fit in it the largest regular triangle of the upside-down orientation, fit a similar triangle in the remaining space not occupied by the second, and et cetera, you get the Sierpinski Triangle.
My favorite name, however, is the Minkowski Sausage. I mean, it doesn't look anything like a sausage, which makes it so stupidly funny! However, it's really a simple fractal with a simple iteration.
The most basic type of fractal is the base-motif fractal. In each iteration, or stage of the fractal, each part of the previous stage of the fractal that fits the description of the base is replaced by a certain motif. In other words, you replace each base with a motif, producing fractals such as Sierpinski's Triangle, the Minkowski Sausage, and the Koch snowflake.
Well, that's pretty much it for hoy.
Stay coolio...
John
1-(1/2)+(1/4)-(1/8)... and so on,
you get two thirds, which is quite odd because powers of two have nothing to do with 3.
Anyway, I've always liked fractals because of their endless continuity.
One especially cool fractallish pattern is the Dragon Curve, which is not only cool for its name but also for the methods it uses. Basically, you take a vertical line segment, rotate it 45 degrees counterclockwise, and then add another copy rotated 90 degrees clockwise to it. Another, more simple way to find it in nature is to take a piece of paper, fold it in half, fold it in half again, fold it half again, and et cetera. If you look down the edge you fold, it slowly becomes the Dragon Curve. A cool Youtube animation thing by a guy that calls himself Numberphile is really good.
The most basic fractal, fortunately triangular, is the Sierpinski Triangle. If you take a regular triangle, fit in it the largest regular triangle of the upside-down orientation, fit a similar triangle in the remaining space not occupied by the second, and et cetera, you get the Sierpinski Triangle.
My favorite name, however, is the Minkowski Sausage. I mean, it doesn't look anything like a sausage, which makes it so stupidly funny! However, it's really a simple fractal with a simple iteration.
The most basic type of fractal is the base-motif fractal. In each iteration, or stage of the fractal, each part of the previous stage of the fractal that fits the description of the base is replaced by a certain motif. In other words, you replace each base with a motif, producing fractals such as Sierpinski's Triangle, the Minkowski Sausage, and the Koch snowflake.
Well, that's pretty much it for hoy.
Stay coolio...
John
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