Anything Math
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06-02-2017, 05:14 AM
RE: Anything Math
You said anything math.



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06-02-2017, 09:38 AM
RE: Anything Math
(05-02-2017 12:36 AM)Kernel Sohcahtoa Wrote:  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.

One of my favorite proofs is the proof that the square root of 2 is irrational (your examples above provide much of the preliminary work).

Another is Euclid's proof that there are an infinite quantity of prime numbers (alternative statement of the same thing: there is no largest prime).

These two proofs are well known and easily found, so I will not give the details.

But here's another favorite: For any integer n, no matter how large, there exists a sequence of n consecutive non-prime integers. Furthermore, it is easy to construct such a sequence.

(n+1)!+2 is divisible by 2; (n+1)!+3 is divisible by 3; ... (n+1)!+(n+1) is divisible by (n+1), and voila! We have n consecutive non-prime integers!
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10-02-2017, 10:25 PM
RE: Anything Math
On this day in math in 1996, history will show that the machine overlords first realized Skynet would rule the world. So did we. Are you Sarah Connor?

Deep Blue v. Kasparov

#sigh
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10-02-2017, 10:37 PM
RE: Anything Math
When I was in high school, I hated algebra, but when we got to geometry for some reason it just clicked with me. I loved doing proofs. We were tasked with taking an exam for the state and I had the highest score ever given for the test. But if you asked me anything about geometry now, the only thing I could possibly remember is the Pythagorean Theorem.

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15-02-2017, 11:27 PM
RE: Anything Math
(10-02-2017 10:37 PM)WillHopp Wrote:  When I was in high school, I hated algebra, but when we got to geometry for some reason it just clicked with me. I loved doing proofs. We were tasked with taking an exam for the state and I had the highest score ever given for the test. But if you asked me anything about geometry now, the only thing I could possibly remember is the Pythagorean Theorem.

This is very common, in my experience on both sides of the situation- student and teacher. Some people just "get" one more than the other. What do you think would have happened had you taken the Algebraic Geometry class? Big Grin Yes, that's a "real" thing. https://en.wikipedia.org/wiki/Algebraic_geometry
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15-02-2017, 11:37 PM
RE: Anything Math
(06-02-2017 09:38 AM)Grasshopper Wrote:  
(05-02-2017 12:36 AM)Kernel Sohcahtoa Wrote:  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.

One of my favorite proofs is the proof that the square root of 2 is irrational (your examples above provide much of the preliminary work).

Another is Euclid's proof that there are an infinite quantity of prime numbers (alternative statement of the same thing: there is no largest prime).

These two proofs are well known and easily found, so I will not give the details.

But here's another favorite: For any integer n, no matter how large, there exists a sequence of n consecutive non-prime integers. Furthermore, it is easy to construct such a sequence.

(n+1)!+2 is divisible by 2; (n+1)!+3 is divisible by 3; ... (n+1)!+(n+1) is divisible by (n+1), and voila! We have n consecutive non-prime integers!

A long time ago, I had a friend who mentioned the 3-4-5 right triangle as a way to verify that one has a right angle when building something. He further stated that this was the only triangle with integer sides that was a right triangle. I hated to burst his bubble, but I had to tell him that there is an infinite number of such triangles. He and his friend looked at me like I was from Mars, or something! Laugh out load Hell, we were just a bunch of guys standing around drinking beer, poolside. How would they know that I had a degree in mathematical physics? Though my friend had a degree in civil engineering.
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15-02-2017, 11:56 PM
RE: Anything Math
If you follow the Fibonacci Sequence you approach the Golden Mean. Smartass

“I am quite sure now that often, very often, in matters concerning religion and politics a man’s reasoning powers are not above the monkey’s.”~Mark Twain
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16-02-2017, 12:58 AM
RE: Anything Math
If you're up for a fun challenge, download Euclidea to your phone. It is the fucking shit. It's all geometry puzzles.

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(06-02-2014 03:47 PM)Momsurroundedbyboys Wrote:  And I'm giving myself a conclusion again from all the facepalming.
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22-02-2017, 12:28 AM
RE: Anything Math
I need help on working out internal rates of return. The formula isn’t helping me (hurts my head).

How do I calculate the following.

Initial Investment $357 Jan 1
Subsequent investments $21 Feb 21
$22 Oct 7
Final value $450 Dec 31

IRR Defined
The IRR is the interest rate, also called the discount rate, that is required to bring the net present value (NPV) to zero. That is, the interest rate that would result in the present value of the capital investment, or cash outflow, being equal to the value of the total returns over time, or cash inflow.

NPV= ∑ {Period Cash Flow / (1+R)^T} - Initial Investment

where R is the interest rate and T is the number of time periods. IRR is calculated using the NPV formula by solving for R if the NPV equals zero.


http://www.investopedia.com/ask/answers/...z4ZOQ700rP

“I am quite sure now that often, very often, in matters concerning religion and politics a man’s reasoning powers are not above the monkey’s.”~Mark Twain
“Ocean: A body of water occupying about two-thirds of a world made for man - who has no gills.”~ Ambrose Bierce
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22-02-2017, 02:48 AM (This post was last modified: 22-02-2017 04:17 AM by Robvalue.)
RE: Anything Math
Is the value of T increasing during the sum, 1, 2, 3... I imagine so? Or is it fixed as being the total number?

And I see the added cash is staggered, so would T be in months giving lots of zero terms...? I've read the article but it doesn't seem to give a lot of detail.

I have a website here which discusses the issues and terminology surrounding religion and atheism. It's hopefully user friendly to all.
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