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**PatternMan****Member**- Registered: 2014-03-08
- Posts: 174

I have seen people using:

Even number = 2n

Odd numbr = 2n +1 or 2n -1

Multiple of 3 = 3n

consecutive = n -1, n, n+1 ....

cosecutive odd = 2n + 1, 2n + 3, 2n + 5...

consecutive ^2 = n^2, (n+1)^2, (n+2)^2

I can see that this notation stems from defining sequences but do any of these expressions require proof(if so where are they?) or are they so self evident that they are just taken to be true?

I have seen people use 2n +1 or 2n -1 to prove that something is odd.

61 = (30 * 2)+1

-15 = (-8 * 2) +1

211 = (105 * 2) + 1

The numbers equations above show that in each of the instances, 2 times a number + 1 will give you an odd number. I think those equations are verifications of the expression but not proof according to my understanding of how they prove conjectures in mathematics.

"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Yes, 2n + 1 or 2n - 1, provided n is an integer will always give an odd number. They are almost self evident and using any of the ones you mentioned is not going to cause any argument.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**PatternMan****Member**- Registered: 2014-03-08
- Posts: 174

bobbym wrote:

Yes, 2n + 1 or 2n - 1, provided n is an integer will always give an odd number. They are almost self evident and using any of the ones you mentioned is not going to cause any argument.

Gracias. I can't get answers to questions like this from textbooks.

"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

The chance that any answer I give is better than a textbook is about one chance in 1239. The probability that my answer if correct and is not in a textbook is one in 67 million. The reasons I am sometimes better than a textbook are:

1) I do not rip

2) You do not have to carry me around.

3) You will never spill hot liquids on me.

4) You will never leave me on a bus.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,531

^

Patternman, when you learn about fields or even before that, you'll prove basic things which you considered axioms using 9 statements which are actually axioms over the field of rational numbers. For now, just proving the theorems is what most people focus on. If you want to prove the "axioms" which aren't really axioms, then read the first three chapters of Principles of Mathematics by Oakley and Allendoerfer.

*Last edited by ShivamS (2014-03-21 08:17:16)*

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

ShivamS wrote:

Patternman, when you learn about fields or even before that, you'll prove basic things which you considered axioms using 7 statements which are actually axioms over the field of rational numbers. For now, just proving the theorems is what most people focus on. If you want to prove the "axioms" which aren't really axioms, then read the first three chapters of Principles of Mathematics by Oakley and Allendoerfer.

7 statements? What set of axioms are you using? The usual for fields are:

(1) Addition is commutative.

(2) Addition is associative.

(3) There exists an additive identity 0.

(4) Existance of additive inverses (opposites).

(5) Multiplication is commutative.

(6) Multiplication is associative.

(7) There exists a multiplicative identity 1.

(8) Existance of multiplicative inverses (except for 0).

(9) Distributivity of multiplication over addition.

Those specify an arbitrary field. To get the rational or real numbers in particular, you need to add more.

And they *are* axioms, or definitions (the two concepts are really the same). Oakley and Allendoerfer may build a *model* of the rational or real numbers and prove that these axioms hold for their model. But commonly in math we simply consider such things as primative elements defined by axioms, rather than tying them to a particular model. Models are used to guarantee that our axioms are not contradictory, and they sometimes can provide good insight into the objects of study, but we are studying the objects themselves, not ways to construct them.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,531

I meant 9.

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**eigenguy****Member**- Registered: 2014-03-18
- Posts: 78

Okay. There are ways to combine some of the axioms into one, but the result is usually some odd statement that is hard to make sense of until you manage to derive the normal axioms from them. So normally, we go with the easily comprehended axioms, instead of the fewest possible.

I was just curious if you were refering to some such reduction.

"Having thus refreshed ourselves in the oasis of a proof, we now turn again into the desert of definitions." - Bröcker & Jänich

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**ShivamS****Member**- Registered: 2011-02-07
- Posts: 3,531

I meant proving things like a/b/(c/d) = ad/bc or that a * 0 = 0

For example:

Prove a*0=0

0 = 0 + 0

a * 0 = (0+0) * a

a * 0 = a * 0

Subtract the equations:

0 = a * 0

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**PatternMan****Member**- Registered: 2014-03-08
- Posts: 174

Why can say 2n = even and 2n + 1 is odd. That is based on experimentation right? There is no proof for this as far as I know. But from this you can prove if a number divdes by 2 it's even etc. Well how is it different from having a formula for PI that only works up to a certain number? I haven't heard of any logic that shows 2n is even in all cases. This seems like an axiom to me.

"School conditions you to reject your own judgement and experiences. The facts are in the textbook. Memorize and follow the rules. What they don't tell you is the people that discovered the facts and wrote the textbooks are people like you and me."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**Maburo****Member**- From: Alberta, Canada
- Registered: 2013-01-08
- Posts: 286

The fact that 2n is even is based on the definition of an even number. An even number is any number divisible by 2, and 2n is always divisible by 2. An odd number is any number which is not divisible by 2, and 2n±1 is never divisible by 2.

"Pure mathematics is, in its way, the poetry of logical ideas."

-Albert Einstein

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**PatternMan****Member**- Registered: 2014-03-08
- Posts: 174

Okay Just wanted to check on here because I'm trying to prove some of these thereoms on my own using 2n and 2n±1

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,380

Hi;

Good luck and have fun.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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