Anything Math
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05-02-2017, 12:36 AM (This post was last modified: 05-02-2017 12:45 AM by Kernel Sohcahtoa.)
Anything Math
IMO, at its core, mathematics is a very beautiful and awesome subject; however, I'd like to learn how other people see mathematics and what intrigues them about it. Hence, the purpose of this thread is for members to share anything math-related that they find interesting, cool, beautiful, etc.

In order to kick-start this thread, I will post some proofs for even and odd integers. Now, we intuitively know that an integer is even or odd, but, IMO, it is still cool to see how this fits into a broader framework via definitions and proofs.

* I'm posting the proofs in spoiler tags, so that people can attempt the proofs for themselves if they wish.


Definitions


Definition of an even integer: An integer n is even if n=2a for some integer a

Definition of an odd integer: An integer n is odd if n=2a+1 for some integer a

Prove the following statements

1. If x is even, then x^2 is even

Proof

Suppose x is even. Then via the definition of an even number, x=2a for some integer a. Now, x^2= (2a)^2= 4a^2= 2(2a^2)= 2b for some integer b equals 2a^2. Therefore, by the definition of an even number, x^2 is even.


2. If x^2 is even, then x is even

Note


It's somewhat tricky to start our proof with x^2, because the definitions posted above are for integers with an exponent of one (x^1=x). As a result, in order to make use of the definitions above, we will prove the contrapositive of the statement, as this will allow us to begin our proof with x. Recall that via logic and truth tables, a conditional statement has the form "if p then q". Now, the contrapositive of "if p then q" is "if not q, then not p." Now, since the contrapositive of a statement has an equivalent truth value to the statement itself, then proving the contrapositive of a statement is the same as proving the statement itself.

Proof

Suppose it is not the case that x is even. Then this means that x is odd. Via the definition of an odd number, x=2a+1 for some integer a. Now, x^2= (2a+1)^2 (remember foil from algebra)= 4a^2 + 4a+1= 2(2a^2+2a) + 1= 2b+1 for some integer b equals 2a^2 + 2a. Thus, by the definition of an odd number, x^2 is odd. Therefore, it is not the case that x^2 is even.


3. If x is odd then x^2 is odd

Proof

Suppose that x is odd (from this point, we can make use of our work from proof two). Via the definition of an odd number, x=2a+1 for some integer a. Now, x^2= (2a+1)^2 (remember foil from algebra)= 4a^2 + 4a+1= 2(2a^2+2a) + 1= 2b+1 for some integer b equals 2a^2 + 2a. Hence, via the definition of an odd number, x^2 is odd.


4. If x^2 is odd then x is odd

Proof

We will prove the contrapositive of this statement. Suppose that it is not the case that x is odd. Then x is even (from this point we can make use of our work from proof one). Then via the definition of an even number, x=2a for some integer a. Now, x^2= (2a)^2= 4a^2= 2(2a^2)= 2b for some integer b equals 2a^2. Thus, by the definition of an even number, x^2 is even. Therefore, it is not the case that x^2 is odd.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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05-02-2017, 12:43 AM
RE: Anything Math
Cantor's diagonalization method is always fun at parties. Big Grin

There is only one really serious philosophical question, and that is suicide. -Camus
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05-02-2017, 01:02 AM (This post was last modified: 05-02-2017 08:32 AM by Kernel Sohcahtoa.)
RE: Anything Math
(05-02-2017 12:43 AM)GirlyMan Wrote:  Cantor's diagonalization method is always fun at parties. Big Grin

Agreed. IMO, I found this to be very fascinating and mind boggling when I read about it in the Book of Proof by Richard Hammack. In particular, I remember that Cantor's diagonal method proves that there are no surjective functions from ℕ to ℝ, which means that there are no bijective functions. Consequently, the cardinality of ℕ does not equal the cardinality of ℝ, as ℕ is a countable infinity while ℝ is uncountable; thus, ℕ and ℝ are two different types of infinity.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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05-02-2017, 05:31 AM
RE: Anything Math
[Image: ra3l78.png] [Image: 1pv78y.png]

Made these two fuckers yesterday Tongue

Julia sets. Parameters taken from the wikipedia article

We'll love you just the way you are
If you're perfect -- Alanis Morissette
(06-02-2014 03:47 PM)Momsurroundedbyboys Wrote:  And I'm giving myself a conclusion again from all the facepalming.
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05-02-2017, 03:08 PM
RE: Anything Math
(05-02-2017 01:02 AM)Kernel Sohcahtoa Wrote:  
(05-02-2017 12:43 AM)GirlyMan Wrote:  Cantor's diagonalization method is always fun at parties. Big Grin

Agreed. IMO, I found this to be very fascinating and mind boggling when I read about it in the Book of Proof by Richard Hammack. In particular, I remember that Cantor's diagonal method proves that there are no surjective functions from ℕ to ℝ, which means that there are no bijective functions. Consequently, the cardinality of ℕ does not equal the cardinality of ℝ, as ℕ is a countable infinity while ℝ is uncountable; thus, ℕ and ℝ are two different types of infinity.

Or you could just say, whoa look at that, there are more real numbers between 0 and 1 than we can count. Well fuck me sideways, how can that be? Big Grin

There is only one really serious philosophical question, and that is suicide. -Camus
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05-02-2017, 03:29 PM
RE: Anything Math
One of my life goals is to understand math.

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05-02-2017, 03:31 PM
RE: Anything Math
(05-02-2017 03:29 PM)GenesisNemesis Wrote:  One of my life goals is to understand math.

That is also a goal of mine. I'm still very far from achieving it. Live long and prosper, sir.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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05-02-2017, 04:19 PM (This post was last modified: 05-02-2017 04:29 PM by GirlyMan.)
RE: Anything Math
Not to politicize this thread or anything but to politicize this thread, Cuban/Zuckerberg 2020! Thumbsup

[Image: cuban_zpszmojffiq.png]

To Live Your Best Life, Do Mathematics

There is only one really serious philosophical question, and that is suicide. -Camus
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05-02-2017, 04:33 PM
RE: Anything Math
(05-02-2017 03:29 PM)GenesisNemesis Wrote:  One of my life goals is to understand math.

She's a cruel cruel mistress taunting me with PDEs and Lagrangians and homomorphic manifolds and ZFC and .... Weeping

There is only one really serious philosophical question, and that is suicide. -Camus
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05-02-2017, 05:01 PM (This post was last modified: 06-02-2017 07:16 AM by Kernel Sohcahtoa.)
RE: Anything Math
(05-02-2017 04:19 PM)GirlyMan Wrote:  Not to politicize this thread or anything but to politicize this thread, Cuban/Zuckerberg 2020! Thumbsup

[Image: cuban_zpszmojffiq.png]

To Live Your Best Life, Do Mathematics

Those were some marvelous words, GirlyMan. Thank you for posting this. Live long and prosper.

"I'm fearful when I see people substituting fear for reason." Klaatu, from The Day The Earth Stood Still (1951)
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