The answers to 1^(1/2) and 1/0

[exfret]

Here are two things that I figured out that seem to disagree with the current consensus:

1/0 = infinite (and many other cool properties of infinite)

1^(1/2)=+/- 1

The first one is something I'm pretty sure of, while I'm near certain the 2nd one is true. I'll give my proof of the second one, because it's easier to prove:

Assuming that 1^(1/2)=sqrt(1)=1

-1=-1                         Reflexive property of equality
(-1)^1=-1                   Property of raising to the power of one (I forget what it is called)
(-1)^(1/2*2)=-1          Inverse multiplication property (1/x*x=1)
((-1)^2)^(1/2)=-1       Power of a Power (converse)
(1)^(1/2)=-1              Simplifying the square
sqrt(1)=-1                  Given/Assumption
1=-1                          Simplifying the radical

Now, the end result is not equal, meaning our assumption is false. If we use what I believe is correct, then we get:

(1)^(1/2)=-1              Simplifying the square
+/- 1=-1                    (My) Given/Assumption

And this is true, because plus or minus 1 does equal minus 1. Now, we could also define raising to the power of 1/2 as the inverse of squaring, in which case it would have to be plus or minus, because there isn't a one-to-one relation in quadratic functions. In other words, flipping an upright parabola over the line y=x will give you +/-sqrt(x) and not just sqrt(x). It makes no sense at all for it to return positive values alone. I showed this last year to my math teacher though, and she somehow found out in an Algebra I book that the Power of a Power Property only works for integer numbers, even though we used it in class for solving radical equations, quadratics, and who knows what else, so this is completely controversial. I didn't quite respond well, so she still believes in what the state of Virginia wants her to teach, but I'm hoping to eventually change her mind. Any thoughts on how I could go about it?

Well, here is the proof for the first statement (and, yes, also a bunch of neat properties about infinite):
                                                                                                                                                                                    _
Defining inifinite as a number followed by an endless stream of zeroes before the decimal point. That is, infinite = 1 0 . 0 (the decimal point and the zero after it are just for demonstration and could be removed at your discretion, and the bar is signifying an endless number of that digit following the number to the left). So, with this definition, we can create this proof:
(Just a side note: I can define the number infinite, because there aren't any properties surrounding it, except that it is not smaller than any number, which my property satisfies, which is hard to show now, so I'll do it later on. Also, we cannot just create a definition for raising to the power of 1/2 that sets it equal to square-rooting, because that would violate its property that it is the inverse of squaring.)

     _
0 . 0 = 0                                        Adding zeroes, even an endless amount of zeroes, after the decimal point will not change the value of a number
     _             _
0 . 0 1 = 0 . 0                                *** Sorry, I don't know the name of the property.
     _
0 . 0 1 = 0                                     Substitution Property of Equality
       _
( 0 . 0 1 ) ^ ( - 1 ) = ( 0 ) ^ ( - 1 )    "Operation" Property of Equality (performing an operation to each side of an equation will preserve the equality)
           _
1 / 0 . 0 1 = 1 / 0                          Property that states x^(-1)=1/x

***if you would like to see the proof for this, just ask, but I am assuming you already know because it is the foundation for turning fractions into decimals and vice versa
                                                       _
Now, we must find out what 1 / 0 . 0 1 is. So, an interesting property is that 1/0.(n)1 = 1(n).0 * 10, where n is a number of zeroes. For example, 0.0001 has 3 zeroes, so 1/0.0001=1 followed by three zeroes times ten, or 1000.0*10. We can use this right now, and since we know that our number is followed by an infinite number of zeroes, 1/it will get a number followed by an infinite number of zeroes before the decimal point (hmmm... sounds familiar) times ten.
   _
1 0 . 0 * 1 0 = 1 / 0                         Aforementioned property
                                                                                                                     _                                                      _                   _
But, wait, what about that nasty times ten in there? Hmmm... Well, since 0 . 0 1 = 0, and 10*0=0, then 1 0 * 0 . 0 1 = 0 = 0 . 0 1! <-Not factorial operator
     _                    _
0 . 0 1 = 1 0 * 0 . 0 1                      Multiplication property of zero
             _
1 / ( 0 . 0 1 * 1 0 ) = 1 / 0              Substitution Property of Equality
           _
1 / 0 . 0 1 / 1 0 = 1 / 0                  Dividing by multiplying property, or whatever it's called
   _
1 0 . 0 * 1 0 / 1 0 = 1 / 0                Aforementioned property
   _
1 0 . 0 = 1 / 0                                Inverse Property of Multiplication
   _
1 0 . 0 = infinite                             Given/Definition

infinite = 1 / 0                               Substitution Property of Equality

1/0 = infinite                                 Reflexive Property of Equality (Just to make it look nicer)

Now, this proof has some major implications. For starters, where k is any complex number (excluding infinite), k*infinite=infinite, because k*infinite=k*1/0=1/(1/k)*1/0=1/(1/k*0)=1/0, because
0*(1/k) = 0*(1/any complex number, given k != 0, because 1/0 is not a complex number according to our definition) = 0. In fact, 0*infinite is indeterminate, because since k*0=0, 0*infinite=k*0*infinite=k*0*1/0, so since k*1/k=1 (Give me a proof showing this is untrue when it comes to infinite and 0 and I'll try to disprove it or I'll offer an explanation for why 1/0 still = infinite. I think proofs "demonstrating" this exception to be "true," thus showing this notion of infinite breaks the laws of complex numbers (even though infinite isn't even a complex number in some rights) and meaning that infinite can't be 1/0, are a large part of the reason why people don't believe 1/0=infinite), 0*infinite=0*1/0=k*0*1/0=k*1=k, and since k is any complex number (it can be zero this time), infinite*0 ends up equalling indeterminate. In fact, 1/0 equalling infinite doesn't even violate either the property that k*0=0 or that 1/k*k=1, because 0*infinite=0 has an infinite number of solutions, one of them being 0, and another being 1. Moving on, we can see that infinite^l=infinite, where l is a non-infinite rational number, because infinite^l=(1/0)^l=1^l/0^l=1/0. Now to show that my definition of infinite is not smaller than any number. Well, my proof for this isn't that concrete, but once you go infinitely far out from the number line, there's no way to measure how big a number is. If you add 1 to the definition of my number, you get this:
   _
1 0 1 . 0 / 1 0

If you keep on adding one to the point that you've added an infinite number of ones, you get:
   _            _
1 0 . 0 + 1 0 . 0

Which equals the number before adding all those one. Of course this could be wrong, because we are only assuming that adding one infinitely should increase the number. The meaning behind the fact that you can't just add 1 to increase the size of the number is that you can't create a number greater than infinite. Oh, and how do we even know if infinite is positive? Well, it's both positive and negative in a way, because infinite=-1*infinite=-infinite. Strange. Well, we know that a positive over a positive is a positive. And, a negative over a negative is a positive and a negative over a positive is a negative and a positive over a negative is a negative, and then a zero over a positive is zero and a zero over a negative is zero and a zero over a zero is a positive, zero, or a negative, but then when we divide a positive or negative by zero, what do we get? We get an infinite, a now defined infinite, and a very strange one, too.

[exfret]

Congrats to anyone who was able to scroll through the whole thing!

[robly18]

Congrats to anyone who was able to scroll through the whole thing!

TL;DR

Anyway, it's well known that 1/0 is not infinity. It's simply undefined.

As for the second one, easy to prove:

1^(1/2) = x
1 = x^2

1^2 = 1
(-1)^2 = 1

Therefore, 1^(1/2) = 1 or -1

[exfret]

Congrats to anyone who was able to scroll through the whole thing!

TL;DR

What does TL;DR mean?

Anyway, it's well known that 1/0 is not infinity. It's simply undefined.

Yes, I said that. That's the reason I had to prove it wrong. Why else would I have gone to the effort to write all that down?

As for the second one, easy to prove:

1^(1/2) = x
1 = x^2

1^2 = 1
(-1)^2 = 1

Therefore, 1^(1/2) = 1 or -1

I'm starting to feel like you just glanced over my post instead of really paying attention to what I wrote (which I don't really blame you for). The reason why I made this thread was because I came up with the answers to x^(1/2) and 1/0, not because I wanted them. Well, this is actually a better way to put it than the way I did in my proofs, but as I said in a very deep part of my proof:

I showed this last year to my math teacher though, and she somehow found out in an Algebra I book that the Power of a Power Property only works for integer numbers, even though we used it in class for solving radical equations, quadratics, and who knows what else, so this is completely controversial. I didn't quite respond well, so she still believes in what the state of Virginia wants her to teach, but I'm hoping to eventually change her mind. Any thoughts on how I could go about it?

So, yeah, my math teacher doesn't believe in the power of a power property for all numbers, and the state of Virginia's standards of learning wants students to be taught that x^(1/2) is just sqrt(x). Do you know of any proofs for the power of a power property in all cases? Anyways, here is a quote from Bob:

The post is not that long. You have read books pages long, so a few paragraphs should not hinder you! The post is inflated with lots of numbers and proofs. The actual content is actually very easy to handle, so do not be intimidated to read it thoroughly!

[robly18]

What does TL;DR mean?

Too Long; Didn't Read

I'm starting to feel like you just glanced over my post instead of really paying attention to what I wrote (which I don't really blame you for).

Quite

I showed this last year to my math teacher though, and she somehow found out in an Algebra I book that the Power of a Power Property only works for integer numbers, even though we used it in class for solving radical equations, quadratics, and who knows what else, so this is completely controversial. I didn't quite respond well, so she still believes in what the state of Virginia wants her to teach, but I'm hoping to eventually change her mind. Any thoughts on how I could go about it?

Well, it's a well known fact that those laws don't apply to all non-real numbers numbers.
As for your result, it's well known that every number has two square roots, one being positive and one being negative. However, a general consensus in mathematics is that in most cases, the response is the positive result. However, strictly speaking, you should always have the +/- before any roots you've made.

So, yeah, my math teacher doesn't believe in the power of a power property for all numbers, and the state of Virginia's standards of learning wants students to be taught that x^(1/2) is just sqrt(x). Do you know of any proofs for the power of a power property in all cases?

It doesn't apply to non-real numbers, but here's an easy one:

(x^2)^2 = (x*x)^2 = (x*x)(x*x) = x*x*x*x = x^4

[exfret]

What does TL;DR mean?

Too Long; Didn't Read

Oh, I guess I got that out of your post anyways. Follow the advice of Bob! (Pwease).

I showed this last year to my math teacher though, and she somehow found out in an Algebra I book that the Power of a Power Property only works for integer numbers, even though we used it in class for solving radical equations, quadratics, and who knows what else, so this is completely controversial. I didn't quite respond well, so she still believes in what the state of Virginia wants her to teach, but I'm hoping to eventually change her mind. Any thoughts on how I could go about it?

Well, it's a well known fact that those laws don't apply to all non-real numbers numbers.
As for your result, it's well known that every number has two square roots, one being positive and one being negative. However, a general consensus in mathematics is that in most cases, the response is the positive result. However, strictly speaking, you should always have the +/- before any roots you've made.

But it shouldn't ever be just the positive when raising to the power of 1/2. If you just want positive, then you should use sqrt. Oh, and why doesn't the power of a power property apply to all non-real numbers? Also, it seems that this isn't the general consensus, because this isn't what is taught.

So, yeah, my math teacher doesn't believe in the power of a power property for all numbers, and the state of Virginia's standards of learning wants students to be taught that x^(1/2) is just sqrt(x). Do you know of any proofs for the power of a power property in all cases?

It doesn't apply to non-real numbers, but here's an easy one:

(x^2)^2 = (x*x)^2 = (x*x)(x*x) = x*x*x*x = x^4

I want a proof that shows ((x^2)^(1/2))=x^(2*1/2)=x^1. This proof just shows (x^2)^2=4. Basically, I need a proof that show x^(1/2) is the inverse of x^2.

Also, any thoughts on my 1/0 proof? That's what I really want replies about.

[robly18]

Oh, I guess I got that out of your post anyways. Follow the advice of Bob! (Pwease).

I'll do it when my brain stops hurting xP

But it shouldn't ever be just the positive when raising to the power of 1/2. If you just want positive, then you should use sqrt. Oh, and why doesn't the power of a power property apply to all non-real numbers? Also, it seems that this isn't the general consensus, because this isn't what is taught.

The proof for this has to do with euler's identity, so...

e^(2*i*pi) = 1
Therefore, (e^(2*i*pi))^i = 1^i which means:
e^(2*pi*i*i) = 1
Therefore,
e^(-2pi) = 1

However, upon doing the maths(Read:google):
http://bit.ly/17oSbe9

I want a proof that shows ((x^2)^(1/2))=x^(2*1/2)=x^1. This proof just shows (x^2)^2=4. Basically, I need a proof that show x^(1/2) is the inverse of x^2.

I'll work on this later.

Also, any thoughts on my 1/0 proof? That's what I really want replies about.

I read a bit and my brain started hurting. However, from what I could tell, it relies on limits. Sadly, division by zero is a bit of an issue with limits. See, if I did the limit as h reaches zero of 1/h I'd get:

1/1 = 1
1/0.5 = 2
1/0.25 = 4
So the closer to zero the number is, the higher the result is. However, if I do this from the negatives:

1/-1 = -1
1/-0.5 = -2
1/0.25 = -4
So the closer the number is to zero, the lower it is.

So if you come in from the positives, 1/0 = a number bigger than every other
If you come in from the negatives, 1/0 = a number smaller than (inferior to?) every other

It gets more complicated. If you come in from imaginary numbers, you get even more results. So the point is, 1/0 equals infinity. It's just an undefined infinity. Not +infinity, not -infinity, not i*infinity... It's just undefined. Period.

On the other hand, 0/0 = indefinite, which is different in the sense that, while 1/0 has no answer, 0/0 has all of them. If you plot y = x/0 the result would be a vertical line at x = 0, the same way y = 0 is a horizontal line at y = 0

[exfret]

Oh, I guess I got that out of your post anyways. Follow the advice of Bob! (Pwease).

I'll do it when my brain stops hurting xP

Fair enough. It's better than never. I sometimes can't read math-y things because they make my brain hurt, too. I actually created a very good summary of what I did at the bottom of my post. Hopefully, that doesn't make your brain hurt.

But it shouldn't ever be just the positive when raising to the power of 1/2. If you just want positive, then you should use sqrt. Oh, and why doesn't the power of a power property apply to all non-real numbers? Also, it seems that this isn't the general consensus, because this isn't what is taught.

The proof for this has to do with euler's identity, so...

e^(2*i*pi) = 1
Therefore, (e^(2*i*pi))^i = 1^i which means:
e^(2*pi*i*i) = 1
Therefore,
e^(-2pi) = 1

However, upon doing the maths(Read:google):
http://bit.ly/17oSbe9

Whoever said that just because one of the solutions to e^(-tau) (tau=2pi) is that irrational number Google gives you means that 1 isn't another solution for it? If you type in 1^(1/2) to Google, you'll just get 1, but like you and I said, 1^(1/2)=+/-1, so -1 is also another viable solution. In the same way, e^(-tau) may still be equal to 1.

I want a proof that shows ((x^2)^(1/2))=x^(2*1/2)=x^1. This proof just shows (x^2)^2=4. Basically, I need a proof that show x^(1/2) is the inverse of x^2.

I'll work on this later.

I don't even know if it's possible. I mean, how would you even be able to prove that x^(1/2) is related to square roots? It must be possible, though, because we wouldn't use it to stand for square roots so much in mathematics otherwise.

Also, any thoughts on my 1/0 proof? That's what I really want replies about.

I read a bit and my brain started hurting. However, from what I could tell, it relies on limits. Sadly, division by zero is a bit of an issue with limits. See, if I did the limit as h reaches zero of 1/h I'd get:

1/1 = 1
1/0.5 = 2
1/0.25 = 4
So the closer to zero the number is, the higher the result is. However, if I do this from the negatives:

1/-1 = -1
1/-0.5 = -2
1/0.25 = -4
So the closer the number is to zero, the lower it is.

So if you come in from the positives, 1/0 = a number bigger than every other
If you come in from the negatives, 1/0 = a number smaller than (inferior to?) every other

It gets more complicated. If you come in from imaginary numbers, you get even more results. So the point is, 1/0 equals infinity. It's just an undefined infinity. Not +infinity, not -infinity, not i*infinity... It's just undefined. Period.

Here's the really-hurt-your-head-part: Just because it's smaller and bigger than every number doesn't mean it doesn't exist. Basically, infinite is both the smallest and largest number. There isn't really a +infinite, or a -infinite, because all those are the same as infinite (It's like how 0 is neither positive or negative). At least, this is what I think. Also, my proof is more about what actually happens at 0 and at infinite, not the limit, because limits can be very deceiving. Even so, I haven't seen any place where limits lie, they're just easy to interpret the wrong way. That's why I didn't really use limits. Instead, I came up with a number followed by an endless stream of zeroes after the decimal point, then I placed a 1 at the end (which is both possible and allowed). This number is equal to zero (if you want the proof, you can ask, but I am assuming that you already agree with this), so when I divide 1 by it, I get 1/0. Now, I saw the pattern that 1/0.1=10, 1/0.01=100, 1/0.001=1000, and so on, so when I have 1/0, that's the same as 1 divided by a number composed of the digit 0 followed by an endless stream of zeroes after the decimal point and then a one, so I get ten times one followed by an infinite number of zeroes before the decimal point, which is my definition of infinite. And, yes, I can define infinite this way, because this definition of infinite follows infinite's sole property: it is larger than any other number (I'm actually having a little trouble making a concrete proof of this). In a way, I'm taking advantage of the ambiguity of infinite, but since it isn't really defined as a number, it's okay to just define it yourself.

So, I guess what I really did in my proof was to define infinite, which has been left undefined all along. That is, until now.

[robly18]

The problem is, if you define infinite as:
Infinity > x for any x
Then you get a problem. You still can't state 1/0 = infinity. That would mean 1/0 > x which appears to be true because, upon multiplying both sides by zero, you get 1 > 0x which is indeed true.
However, take this into account:
0 = -0
Therefore, 1/0 = 1/(-0), meaning 1/0 = -(1/0). Divide both sides by 1/0 and you get 1 = -1 which is obviously not true.
Or, if you want another angle of approach, add 1/0 to both sides, getting 2/0 = 0. Divide both sides by two and you get 1/0 = 0.

My point is, the statement 1/0 as a whole is just plain bogus. One cannot solve for 1/0 because a whole batch of problems arise from that division. But I disgress.

Moving back to my original statement, you can't state 1/0 as infinity, because while 1/'0 > x in every case, seeing as 0 = -0 the statement -(1/0) > x is also true in every case. After doing a bit of work, you get to the conclusion 1/0 < -x which leads to 1/0 being inferior to a number, so it can therefore not be infinity because infinity is superior to every number.

Therefore, 1/0 is not infinity, the latter being defined as superior to any number.

[exfret]

Moving back to my original statement, you can't state 1/0 as infinity, because while 1/0 > x in every case, seeing as 0 = -0 the statement -(1/0) > x is also true in every case. After doing a bit of work, you get to the conclusion 1/0 < -x which leads to 1/0 being inferior to a number, so it can therefore not be infinity because infinity is superior to every number.

Therefore, 1/0 is not infinity, the latter being defined as superior to any number.

Proving that 1/0 is superior to and inferior to every number at the same time doesn't make it not a number. It's like saying i isn't a number because you can't compare it in value to other numbers to see if it's less than or greater than them. Another analogy would be saying 1^(1/2) isn't a number because it returns a value less than 0 (-1) and a value greater than 0 (1). Actually, this leads me to something I overlooked. 1/0 may have multiple values. Before, I thought this meant there was just infinite, no positive infinite or negative infinite, but 1/0 may return multiple values of infinite. So, there still may be a 2infinite and a -infinite. Of course, I don't really see any way to prove whether there is only one infinite or if there are two. Of course, since infinite is larger than any number, 2infinite would have to equal infinite, so that means that there could only be two possible infinites. One would be negative, the smallest number, and the other positive, the largest number. 1/0 could be either of these numbers, just like 1^(1/2) is either 1 or -1, or 1/0 could be an infinite that just happens to be the smallest and greatest number at the same time. Either way, it doesn't change my results, except that I may have to place a +/- symbol before the word infinite when I say 1/0=infinite.