19 Aug 2015, 7:59 pm
23 Aug 2015, 10:53 am
23 Aug 2015, 11:54 am
23 Aug 2015, 6:48 pm
23 Aug 2015, 7:00 pm
23 Aug 2015, 8:28 pm
Owlpic wrote:
Rudin wrote:
Transcendental numbers are a very interesting type of numbers
I find the equations or geometrical measurements that generate Transcendental numbers more interesting. I believe they reveal important physics that help understanding fundamental relationships necessary for us to solve new breakthroughs in our understanding of life and our universe.
I have worked for years to derive all the Transcendental equations by only the application of subtraction (meaning minus).
Also worked to intuitively understand geometrical measurements such as Pi*R^2= area of a circle.
Welcome to WP, Owlpic.
Sounds like you are Mathematician.
Your expertise is welcome.
23 Aug 2015, 8:35 pm
Rudin wrote:
Transcendental numbers are a very interesting type of numbers so I decided to make a thread about it.
To understand what transcendental numbers are first we'll start with the definition of natural numbers. Natural numbers are essentially counting numbers 0,1,2,3,4,.... This set is clearly infinite but it is the smallest type of infinity known as Aleph-null. It is called countable or listable which means you can make a list of them and know what the next term is. The next set of numbers we'll discuss is known as integers which are 0,1,-1,2,-2,3,-3,... and so forth. These have the same cardinal as the first one because you can map a 1 to 1 correspondence between each of the elements in both sets.
Like:
0,1,2,3,4,..
0,1,-1,2,-2...
Sure the integers are a denser set but they still have the same cardinal.
The next set of numbers we'll discuss are the rational numbers which are essential negative and positive fractions.
Q={p,q in Z: p/q}
Q is the rationals and Z is the integers, N is the natural numbers.
Using Cantor's diagonal proof we can conclude that this set has cardinality Alpeh null like the rest.
The next type of numbers we'll explore is called algebraic numbers and they are numbers that can be expressed by the solution of a non-zero polynomial equation.
Examples:
sqrt(2) is irrational meaning it cannot be expressed as a fraction however it can be expressed by the solution of a non-zero polynomial.
x^2-2=0, x=sqrt(2), sqrt(2) is algebraic
i is clearly not rational, it is complex. x^2+1=0, x=i. i is algebraic
Get the point?
It has been proven the set of algebraic numbers is listable (has cardinality Aleph null).
Transcendental numbers are numbers that cannot be expressed as the solution of a non-zero polynomial. Whose digits go on forever. There are an uncountably infinite amount of these, has a larger cardinality than the algebraics. It has cardinality Aleph 1.
This is the pinnacle of this thread.
The list of the most interesting transcendental numbers--all transcendental numbers:
Some simple ones:
π- Of course we have to add π.
e- Euler's constant, lim_{N -> infinity} (1+1/N)^N
e^a- Where a is algebraic
ln(a)- Where a is algebraic
sin(a),cos(a),tan(a), csc(a), sec(a), cot(a)- a is algebraic
More interesting ones:
0.1234567891011121314151617...
∑β^2^n where 0<|β|<1. The sum ranges from 0 to infinity by the way.
There is many more. Please post some more that I did not get to.
Also feel free to discuss potential transcendental numbers and the zeta function, of course.
To understand what transcendental numbers are first we'll start with the definition of natural numbers. Natural numbers are essentially counting numbers 0,1,2,3,4,.... This set is clearly infinite but it is the smallest type of infinity known as Aleph-null. It is called countable or listable which means you can make a list of them and know what the next term is. The next set of numbers we'll discuss is known as integers which are 0,1,-1,2,-2,3,-3,... and so forth. These have the same cardinal as the first one because you can map a 1 to 1 correspondence between each of the elements in both sets.
Like:
0,1,2,3,4,..
0,1,-1,2,-2...
Sure the integers are a denser set but they still have the same cardinal.
The next set of numbers we'll discuss are the rational numbers which are essential negative and positive fractions.
Q={p,q in Z: p/q}
Q is the rationals and Z is the integers, N is the natural numbers.
Using Cantor's diagonal proof we can conclude that this set has cardinality Alpeh null like the rest.
The next type of numbers we'll explore is called algebraic numbers and they are numbers that can be expressed by the solution of a non-zero polynomial equation.
Examples:
sqrt(2) is irrational meaning it cannot be expressed as a fraction however it can be expressed by the solution of a non-zero polynomial.
x^2-2=0, x=sqrt(2), sqrt(2) is algebraic
i is clearly not rational, it is complex. x^2+1=0, x=i. i is algebraic
Get the point?
It has been proven the set of algebraic numbers is listable (has cardinality Aleph null).
Transcendental numbers are numbers that cannot be expressed as the solution of a non-zero polynomial. Whose digits go on forever. There are an uncountably infinite amount of these, has a larger cardinality than the algebraics. It has cardinality Aleph 1.
This is the pinnacle of this thread.
The list of the most interesting transcendental numbers--all transcendental numbers:
Some simple ones:
π- Of course we have to add π.
e- Euler's constant, lim_{N -> infinity} (1+1/N)^N
e^a- Where a is algebraic
ln(a)- Where a is algebraic
sin(a),cos(a),tan(a), csc(a), sec(a), cot(a)- a is algebraic
More interesting ones:
0.1234567891011121314151617...
∑β^2^n where 0<|β|<1. The sum ranges from 0 to infinity by the way.
There is many more. Please post some more that I did not get to.
Also feel free to discuss potential transcendental numbers and the zeta function, of course.
Hey Rudin.
Never lose your willingness to share your knowledge with others, it is a very nice trait.
Ik you are young, but you are obviously brilliant.
A lot of great minds don't like to share their knowledge, but you do.
I appreciate that trait and wanted to compliment you on it.
warm regards,
slave
24 Aug 2015, 7:02 am
slave wrote:
Rudin wrote:
Transcendental numbers are a very interesting type of numbers so I decided to make a thread about it.
To understand what transcendental numbers are first we'll start with the definition of natural numbers. Natural numbers are essentially counting numbers 0,1,2,3,4,.... This set is clearly infinite but it is the smallest type of infinity known as Aleph-null. It is called countable or listable which means you can make a list of them and know what the next term is. The next set of numbers we'll discuss is known as integers which are 0,1,-1,2,-2,3,-3,... and so forth. These have the same cardinal as the first one because you can map a 1 to 1 correspondence between each of the elements in both sets.
Like:
0,1,2,3,4,..
0,1,-1,2,-2...
Sure the integers are a denser set but they still have the same cardinal.
The next set of numbers we'll discuss are the rational numbers which are essential negative and positive fractions.
Q={p,q in Z: p/q}
Q is the rationals and Z is the integers, N is the natural numbers.
Using Cantor's diagonal proof we can conclude that this set has cardinality Alpeh null like the rest.
The next type of numbers we'll explore is called algebraic numbers and they are numbers that can be expressed by the solution of a non-zero polynomial equation.
Examples:
sqrt(2) is irrational meaning it cannot be expressed as a fraction however it can be expressed by the solution of a non-zero polynomial.
x^2-2=0, x=sqrt(2), sqrt(2) is algebraic
i is clearly not rational, it is complex. x^2+1=0, x=i. i is algebraic
Get the point?
It has been proven the set of algebraic numbers is listable (has cardinality Aleph null).
Transcendental numbers are numbers that cannot be expressed as the solution of a non-zero polynomial. Whose digits go on forever. There are an uncountably infinite amount of these, has a larger cardinality than the algebraics. It has cardinality Aleph 1.
This is the pinnacle of this thread.
The list of the most interesting transcendental numbers--all transcendental numbers:
Some simple ones:
π- Of course we have to add π.
e- Euler's constant, lim_{N -> infinity} (1+1/N)^N
e^a- Where a is algebraic
ln(a)- Where a is algebraic
sin(a),cos(a),tan(a), csc(a), sec(a), cot(a)- a is algebraic
More interesting ones:
0.1234567891011121314151617...
∑β^2^n where 0<|β|<1. The sum ranges from 0 to infinity by the way.
There is many more. Please post some more that I did not get to.
Also feel free to discuss potential transcendental numbers and the zeta function, of course.
To understand what transcendental numbers are first we'll start with the definition of natural numbers. Natural numbers are essentially counting numbers 0,1,2,3,4,.... This set is clearly infinite but it is the smallest type of infinity known as Aleph-null. It is called countable or listable which means you can make a list of them and know what the next term is. The next set of numbers we'll discuss is known as integers which are 0,1,-1,2,-2,3,-3,... and so forth. These have the same cardinal as the first one because you can map a 1 to 1 correspondence between each of the elements in both sets.
Like:
0,1,2,3,4,..
0,1,-1,2,-2...
Sure the integers are a denser set but they still have the same cardinal.
The next set of numbers we'll discuss are the rational numbers which are essential negative and positive fractions.
Q={p,q in Z: p/q}
Q is the rationals and Z is the integers, N is the natural numbers.
Using Cantor's diagonal proof we can conclude that this set has cardinality Alpeh null like the rest.
The next type of numbers we'll explore is called algebraic numbers and they are numbers that can be expressed by the solution of a non-zero polynomial equation.
Examples:
sqrt(2) is irrational meaning it cannot be expressed as a fraction however it can be expressed by the solution of a non-zero polynomial.
x^2-2=0, x=sqrt(2), sqrt(2) is algebraic
i is clearly not rational, it is complex. x^2+1=0, x=i. i is algebraic
Get the point?
It has been proven the set of algebraic numbers is listable (has cardinality Aleph null).
Transcendental numbers are numbers that cannot be expressed as the solution of a non-zero polynomial. Whose digits go on forever. There are an uncountably infinite amount of these, has a larger cardinality than the algebraics. It has cardinality Aleph 1.
This is the pinnacle of this thread.
The list of the most interesting transcendental numbers--all transcendental numbers:
Some simple ones:
π- Of course we have to add π.
e- Euler's constant, lim_{N -> infinity} (1+1/N)^N
e^a- Where a is algebraic
ln(a)- Where a is algebraic
sin(a),cos(a),tan(a), csc(a), sec(a), cot(a)- a is algebraic
More interesting ones:
0.1234567891011121314151617...
∑β^2^n where 0<|β|<1. The sum ranges from 0 to infinity by the way.
There is many more. Please post some more that I did not get to.
Also feel free to discuss potential transcendental numbers and the zeta function, of course.
Hey Rudin.
Never lose your willingness to share your knowledge with others, it is a very nice trait.
Ik you are young, but you are obviously brilliant.
A lot of great minds don't like to share their knowledge, but you do.
I appreciate that trait and wanted to compliment you on it.
warm regards,
slave
Thank you
The things are too interesting not to share, although sometimes I share too much...
24 Aug 2015, 11:45 am
Rudin wrote:
Listable is a synonym for countable, no? If it's not, my apologies.
No apologies necessary!
I guess they are synonyms in a general sense. But 'countable' is the canonical term when referring to sets. The only usage of 'listable' I'm familiar with is as an attribute of functions in mathematical programming, e.g. Mathematica.
25 Aug 2015, 8:44 pm
25 Aug 2015, 9:02 pm
27 Aug 2015, 12:34 pm
naturalplastic wrote:
What's the difference between a "transcendental number", and an "irrational" number?
also think 'ratio' when you think of irrational numbers as oppose to irrational as defined as being unreasonable
an irrational number is any real number than cannot be expressed as a RATIO of integers
hope that adds to Rudin's good explanation
27 Aug 2015, 2:52 pm
slave wrote:
naturalplastic wrote:
What's the difference between a "transcendental number", and an "irrational" number?
also think 'ratio' when you think of irrational numbers as oppose to irrational as defined as being unreasonable
an irrational number is any real number than cannot be expressed as a RATIO of integers
hope that adds to Rudin's good explanation
Slave, I need to ask you something.
Do you think my signature is too long and making threads longer? Should I shorten it?
27 Aug 2015, 9:12 pm
Rudin wrote:
slave wrote:
naturalplastic wrote:
Slave, I need to ask you something.
Do you think my signature is too long and making threads longer? Should I shorten it?
Do you think my signature is too long and making threads longer? Should I shorten it?
I am not Slave, but..
Yes, and Yes, and pretty please with sugar on top!
27 Aug 2015, 9:29 pm
naturalplastic wrote:
Rudin wrote:
slave wrote:
naturalplastic wrote:
Slave, I need to ask you something.
Do you think my signature is too long and making threads longer? Should I shorten it?
Do you think my signature is too long and making threads longer? Should I shorten it?
I am not Slave, but..
Yes, and Yes, and pretty please with sugar on top!
Signature has been modified. Was it really that bad? Why did someone not say anything? Did you guys lie to me?
01 Sep 2015, 8:00 pm
Rudin wrote:
slave wrote:
naturalplastic wrote:
What's the difference between a "transcendental number", and an "irrational" number?
also think 'ratio' when you think of irrational numbers as oppose to irrational as defined as being unreasonable
an irrational number is any real number than cannot be expressed as a RATIO of integers
hope that adds to Rudin's good explanation
Slave, I need to ask you something.
Do you think my signature is too long and making threads longer? Should I shorten it?
Sry, Rudin, I just read this right now.
I would have suggested removing the double spacing to attenuate it a little, but it wasn't bothering me.
Mine is longer than most users so I can hardly criticize anyone else.
I see that you have altered it, so my response is effectively moot.
