Good luck and have fun.

]]>For example:

Prove a*0=0

0 = 0 + 0

a * 0 = (0+0) * a

a * 0 = a * 0

Subtract the equations:

0 = a * 0]]>

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

]]>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.

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.

]]>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.

]]>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.

]]>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.

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