My simple proof of x^0=1 part 2 (axioms)

[why Fenix]

I was looking for explanation why x^0=1.
thread
https://www.physicsforums.com/threads/my-simple-proof-of-x-0-1.172073/
is locked and i did't found solution in it from axioms. People using exp(x) and log(x) and x^a-a=x^ax^-a as given.
If you have x^a-a=x^ax^-a for a∈ℤ and x∈ℕ₊ then there is no question... (thank to M5 below...)But from axioms ... maybe something like this?...:Starting from "Principles of mathematical analysis. Walter Rudin" page 5 and 6.
For any field F we get axioms:
...
"(M3) Multiplication is commutative: xy=yx for all x,y ∈ F"
...
"(M5) If x ∈ F and x ≠ 0 then there exists an element 1/x ∈ F such that
x⋅(1/x) =1." Let assume (Rudin does not specify that) that we call this element x^-1.

On next page (6) Rudin in Remark 1.13 states:
"One usually writes (in any field) ...x², x³... "
"in place of ... xx, xxx, ..."

Let assume (Rudin does not specify that) that x=x¹.

From (M5) we get
1=x⋅(1/x) =x¹x^-1
so
1=x¹x^-1 ⋅ 1 = x¹x^-1 ⋅ x¹x^-1 = (from M3) = x¹⋅x¹⋅x^-1⋅x^-1=(from remark 1.13)=
x²⋅x^-1⋅x^-1.

But we got for x² from (M5)
1=x²(x²)^-1
we get (not proving that if ab=c and ad=c then b=d)
x^-1⋅x^-1=(x²)^-1=(from now on)=x^-2 let's call this (equation 1). Now we know what x^-n is.

(from remark 1.13) where ∏_kX=X⋅X⋅...⋅X k-times
X^A+B=∏_A+BX=∏_AX⋅∏_BX=X^A⋅X^B
so
X^A+B=X^A⋅X^B and A,B ∈ℤ base on (equation 1)

If A=1 and B=-1 we get from above
X^A+B=X^A⋅X^B=X¹⋅X^-1=(from M5 only for X≠0)=1
and
X^A+B=X^(1)+(-1)=X⁰

we get X⁰=1for X≠0 because of (M5) (we don't know what 0^-1 is...)
the end ...
and now just for fun :)
0⁰=(1-1)^1-1=(1-1)¹(1-1)^-1=(0)¹(1-1)^-1=0⋅(1-1)^-1=0/(1-1)¹=0/(0)¹=0/0
so
0⁰=0/0

 

[Mark44]

I was looking for explanation why x^0=1.
thread
https://www.physicsforums.com/threads/my-simple-proof-of-x-0-1.172073/
is locked and i did't found solution in it from axioms. People using exp(x) and log(x) and x^a-a=x^ax^-a as given.
If you have x^a-a=x^ax^-a for a∈ℤ and x∈ℕ₊ then there is no question... (thank to M5 below...)But from axioms ... maybe something like this?...:Starting from "Principles of mathematical analysis. Walter Rudin" page 5 and 6.
For any field F we get axioms:
...
"(M3) Multiplication is commutative: xy=yx for all x,y ∈ F"
...
"(M5) If x ∈ F and x ≠ 0 then there exists an element 1/x ∈ F such that
x⋅(1/x) =1." Let assume (Rudin does not specify that) that we call this element x^-1.

On next page (6) Rudin in Remark 1.13 states:
"One usually writes (in any field) ...x², x³... "
"in place of ... xx, xxx, ..."

Let assume (Rudin does not specify that) that x=x¹.

From (M5) we get
1=x⋅(1/x) =x¹x^-1
so
1=x¹x^-1 ⋅ 1 = x¹x^-1 ⋅ x¹x^-1 = (from M3) = x¹⋅x¹⋅x^-1⋅x^-1=(from remark 1.13)=
x²⋅x^-1⋅x^-1.

But we got for x² from (M5)
1=x²(x²)^-1
we get (not proving that if ab=c and ad=c then b=d)
x^-1⋅x^-1=(x²)^-1=(from now on)=x^-2 let's call this (equation 1). Now we know what x^-n is.

(from remark 1.13) where ∏_kX=X⋅X⋅...⋅X k-times
X^A+B=∏_A+BX=∏_AX⋅∏_BX=X^A⋅X^B
so
X^A+B=X^A⋅X^B and A,B ∈ℤ base on (equation 1)

If A=1 and B=-1 we get from above
X^A+B=X^A⋅X^B=X¹⋅X^-1=(from M5 only for X≠0)=1
and
X^A+B=X^(1)+(-1)=X⁰

we get X⁰=1for X≠0 because of (M5) (we don't know that 0^-1 is...)
the end ...
and now just for fun :)
0⁰=(1-1)^1-1=(1-1)¹(1-1)^-1

No, this isn't valid, but maybe you don't seriously mean it. If you do mean it, though, in the second factor you have essentially 0^-1. M5 applies only if x ≠ 0.

=(0)¹(1-1)^-1=0⋅(1-1)^-1=0/(1-1)¹=0/(0)¹=0/0
so
0⁰=0/0

 

[why Fenix]

...don't seriously mean it...

Yes. Proof is finished on "the end...".
I was just trying too write something funny in the end of the post... :)

But now I wonder.
Let us assume that we are in Riemann sphere or just ℂ∪{∞} and (M5) is given for (X=0) by 1/0=∞.
Of course ∞=z*∞=z*1/0=z/0.
and now we will get 0⁰=0/0 (of course we don't know what 0/0 is but is equal to 0⁰ from axioms ).

 

[Mark44]

But now I wonder.
Let us assume that we are in Riemann sphere or just ℂ∪{∞} and (M5) is given for (X=0) by 1/0=∞.

From the Wiki article on the extended real numbers, http://en.wikipedia.org/wiki/Extended_real_number_line :
"The expression 1/0 is not defined either as +∞ or −∞..."

Of course ∞=z*∞=z*1/0=z/0.
and now we will get 0⁰=0/0 (of course we don't know what 0/0 is but is equal to 0⁰ from axioms ).

 

[jbriggs444]

Let us assume that we are in Riemann sphere or just ℂ∪{∞} and (M5) is given for (X=0) by 1/0=∞.
Of course ∞=z*∞=z*1/0=z/0.
and now we will get 0⁰=0/0 (of course we don't know what 0/0 is but is equal to 0⁰ from axioms ).

In the Riemann sphere, 0/0 is left undefined for a reason. If you define it and if you allow yourself to do algebra with it, you are begging for trouble.

From the Wiki article on the extended real numbers, http://en.wikipedia.org/wiki/Extended_real_number_line :
"The expression 1/0 is not defined either as +∞ or −∞..."

In the two point compactification of the real number line, 1/0 is left undefined. In the one point compactification of the complex plane, it is defined. However... http://en.wikipedia.org/wiki/Riemann_sphere: "The quotients 0/0 and ∞/∞ are left undefined."

 

[why Fenix]

From the Wiki article on the extended real numbers, http://en.wikipedia.org/wiki/Extended_real_number_line :
"The expression 1/0 is not defined either as +∞ or −∞..."

As I said. We are not in real numbers or extended real numbers.

But in ℂ∪{∞} or something like Riemann sphere.
Here http://en.wikipedia.org/wiki/Riemann_sphere you can find "...The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0 = ∞ well-behaved..."

There is
+∞∉ ℂ∪{∞}
-∞∉ ℂ∪{∞}
∞∈ ℂ∪{∞}

 

[micromass]

But neither $0/0$ or $0^0$ are defined on the Riemann sphere

 

[why Fenix]

But neither $0/0$ or $0^0$ are defined on the Riemann sphere

When you start from axioms and wonder about x⁰=1 you can show that x⁰=1 for x≠0 because of (M5). (As I did in my first post above.)
So when you extend (M5) with (1/0)=∞, like it is done in Riemann sphere, you should get what x⁰ for x=0 is.

Thanks to (M5) with (1/0)=∞
for x=0 we get from (M5) 0*(1/0)=1
and from (1/0)=∞ and above we get 0*∞=1.
and because 0 is inverse to ∞ so we will put 1/∞=0
now from (M5) we get ∞(1/∞)=1.

Is this a field?? What will break ?? Which axiom ?? We need just one or we will get 0*(1/0)=0/0=1.

List of axioms from http://math.elinkage.net/showthread.php?tid=102
A field is a set F with two operations addition and multiplication, satisfying axioms (A),(M) and (D):
(A) Axioms for addition
(A1) Addition is closed in F: (x,y∈F)⇒(x+y∈F)
(A2) Addition is commutative: x+y=y+x(∀x,y∈F)
(A3) Addition is associative: (x+y)+z=x+(y+z)(∀x,y,z∈F)
(A4) F has additive unit: ∃0∈F∀x∈F0+x=x
(A5) F has additive inverse:∀x∈F∃−x∈F(x+(−x)=0)

(M) Axioms for multiplication
(M1) multiplicationis closed in F: (x,y∈F)⇒(xy∈F)
(M2) multiplication is commutative: xy=yx(∀x,y∈F)
(M3) multiplication is associative: (xy)z=x(yz)(∀x,y,z∈F)
(M4) F has multiplicative unit: ∃1∈F∀x∈F1x=x
(M5) F has multiplicative inverse:∀x∈F∃1/x∈F(x(1/x)=1)

(D) The distributive law: x(y+z)=xy+xz(∀x,y,z∈F)

 

[micromass]

The Riemann sphere is not a field because the multiplication is not always defined. Specifically, something like $0\cdot \infty$ is not defined.

It shouldn't be a field. And $\frac{0}{0}=1$ shouldn't be allowed. Indeed, because if it is, then

$$1 = \frac{0}{0} = \frac{2\cdot 0}{0} = 2\frac{0}{0} = 2$$

and thus $1=2$, which should definitely be false.

 

[why Fenix]

$$1 = \frac{0}{0} = \frac{2\cdot 0}{0} = 2\frac{0}{0} = 2$$
and thus $1=2$, which should definitely be false.

yes, you are right.
1 = 0/0 = ((1+1) * 0)/0 = (1+1)*(0/0) = (1+1) and would be 0=1.
and we got zero ring...
http://en.wikipedia.org/wiki/Zero_ringBut main thing in this post is the proof that x^0=1. for x≠0. In any field. Is it good?

 

[micromass]

yes, you are right.
1 = 0/0 = ((1+1) * 0)/0 = (1+1)*(0/0) = (1+1) and would be 0=1.
and we got zero ring...
http://en.wikipedia.org/wiki/Zero_ringBut main thing in this post is the proof that x^0=1. for x≠0. In any field. Is it good?

That $x^0 = 1$ is essentially a definition. While your proof is good it requires:

1) That $x^0$ exists
2) That $x^{-1}$ exists
3) That the usual law $x^{a+b} = x^a x^b$ is satisfied

 

[why Fenix]

That $x^0 = 1$ is essentially a definition. While your proof is good it requires:

1) That $x^0$ exists
2) That $x^{-1}$ exists
3) That the usual law $x^{a+b} = x^a x^b$ is satisfied

Ad.2. For x≠0 we get from (M5) that $x^{-1}$ exists.
Ad.1. I think i will get it from 3)
Ad.3. Is given for positive natural a,b, in Remark 1.13. For negative a,b we can get it from remark 1.13 and (equation1 --- x^-1x^-1=x^-2
extended to (k-times)
x^-1x^-1⋅...⋅x^-1 =x^-k).

Now we can take integers "a" positive, "b" negative.
And knowing that x^a and x^b well defined ( as remark )
we get that x^a+b for integers "a" positive, "b" negative is equal to x^a⋅x^b ...Extention of the definision?? Remark ?? Wishful thinking? Or just notation? Let's go with Extention of Remark 1.13.From one side if a+b>0 then we say that x^a+b=x^c for some positive c.
other side if a+b<0 then we say that x^a+b=x^c for some negative c.
so we need 1)?

for b=-a and (M5) we would get x^a+b=x^a+(-a)=x^ax^(-a)=x^a(1/x^a)=(M5)=1
and we get x^a+(-a)=x⁰ so exists 1) ??
It does not look professionally.

So in any field we have to add to axioms remark about notation like this?
a∈ℤ.
x^a=x^a+1/x
x⁰=1
or
a∈ℤ.
x^a=x^a+1/x
x¹=x