The Simpsons mathematic geek received no attention

[Dark Fire]

I remember an episode of The Simpsons, where a mathematic geek received no attention of his fellow peers, so he shouted: PI IS EXACTLY THREE!
I just couldn't help laughing:D

OK, over to my recent low-leveled mathematics-discovery, regarding whether i is not equal to 1, -1 nor anything like that.

I first wrote this, to ensure I understood the principle:
9^0.5 = 3
9^0.5*9 = 3*9
27 = 27

Now, over to the subject:
i^4 = 1
1^0.25 = i
(1^0.25)*1*1*1 = (i*1*1*1) //here's the issue
1 = i

I didn't even get here earlier, I was trapped in my own confusing thinking, though when I ended up here - I saw the flaw at once..
But now, I forgot it
Anyone'd like to refresh me?
I can't use the Google calculator either, since it doesn't show flaws, it only calculates what you put in on one side, not both.

 

[Cyrus]

What, that makes no sense.$$i=\sqrt{-1}$$

1^0.25=1 not i.

 

[DaveC426913]

What exactly is the problem?

i^4=1 for i=1
i^.25=1 for i=1

So?

 

[DaveC426913]

What, that makes no sense.$$i=\sqrt{-1}$$

Not necessarily. I mean, he might be referring to i, but who says he isn't simply using the variable i?

 

[quadraphonics]

Now, over to the subject:
i^4 = 1

Right.

1^0.25 = i

i is *one* of the fourth roots of 1, the others being 1, -1 and -i.

(1^0.25)*1*1*1 = (i*1*1*1) //here's the issue

Okay, so you multiply each side by 1 three times...

1 = i

Does not follow from the previous line. The previous line just returns the same equation you had before you started: 1^0.25 = i.

I think the issue is that you're mixing roots of 1 here. There are four fourth roots of 1: 1, -1, i and -i. If you take any of these to the fourth power, you'll get one. But that does NOT mean that any product of four of these terms gives 1, nor that they're equal to one another. It seems to me that the fundamental error here is confusion of the multiplicity of roots.

 

[BryanP]

Or you can just think about it visually. I don't really get what the post is for, but just visualize a unit circle on the complex plane.

Then it's just going to alternate between 1 and -1 for each even exponent of i with i^0 = 1. This comes from Euler's formula.

 

[D H]

Now, over to the subject:
i^4 = 1
1^0.25 = i
(1^0.25)*1*1*1 = (i*1*1*1) //here's the issue
1 = i

I didn't even get here earlier, I was trapped in my own confusing thinking, though when I ended up here - I saw the flaw at once..

How many times do we have to go over this with you?

You can use the search tool to find your own posts, and thence find the answer.

The problem is that you are ignoring that 1^0.25 has four roots. There is nothing special about imaginary numbers going on here. The exact same faulty reasoning can lead you to conclude that -1=1 because (-1)^2=(1)^2. It is faulty reasoning. You can conclude $$a^r=b^r \,\Rightarrow\,a=b$$ if a and b are positive real numbers and r is real. You cannot conclude this if any of the values is complex or if a or b is negative.

 

[DaveC426913]

Question to the OP for clarity:

Is this about complex numbers and the root of -1, or not?

 

[D H]

Dave, if you look at the other threads started by the OP you will see that this is about the root of -1.

https://www.physicsforums.com/search.php?searchid=1168683

 

[Dark Fire]

Does not follow from the previous line. The previous line just returns the same equation you had before you started: 1^0.25 = i.

With real positive numbers, like I used in my first example, you can solve this:
a^0.x by multiplying the number with itself, relative to its exponent, for example:
81^0.5=9
81^0.5*81=9*81
729=729

Then with a lower exponent:
81^0.25=3
81^0.25*81*81*81=3*81*81*81
1594323 = 1594323

Therefor it would follow if it was a real positive number. <-- with a negative, it becomes complex again

I guess my point/the conclusion is that there's several 'formulas' for real numbers, that doesn't work with complex numbers...
And that just.. not surprises, but amazes me; it's interesting, since i is actually somehow a "real number" (I've been trying to figuring this out as well).
i is (-)^0.5 1 or whatever you'd like to call it.
It's not negative nor positive, until multiplied with itself, which is just an absurd idea from the beginning of, and again interesting.

@Dave
It is.
Also I didn't know there were a way to say "for i=1" ;o
Thanks, now I know that too x)
I'm not very educated, 17 years old, done primary school for 9 years, stopped with homework @ 5th grade, I'm just trying to get back :D

And D H
The issue you're addressing, is something I figured out already in the beginning, this thread is therefor not about that.

 

[Gokul43201]

With real positive numbers, like I used in my first example, you can solve this:
a^0.x by multiplying the number with itself, relative to its exponent, for example:
81^0.5=9
81^0.5*81=9*81
729=729

And what happens if I write 81^0.5=-9?

Your question has been answered in posts #5 & #7. Please read them carefully. This has nothing do do with i "being special" and has everything to do with multiple roots and principal values.

 

[HallsofIvy]

With real positive numbers, like I used in my first example, you can solve this:
a^0.x by multiplying the number with itself, relative to its exponent, for example:
81^0.5=9
81^0.5*81=9*81
729=729

Then with a lower exponent:
81^0.25=3
81^0.25*81*81*81=3*81*81*81
1594323 = 1594323

Therefor it would follow if it was a real positive number. <-- with a negative, it becomes complex again

No, it doesn't follow. In the real numbers, with x a positive real number, $x^{1/n}$ is defined as the positive real number whose nth power is x (the "principal" root). That is NOT the definition for a negative real number which has NO "principal" root.

I guess my point/the conclusion is that there's several 'formulas' for real numbers, that doesn't work with complex numbers...
And that just.. not surprises, but amazes me; it's interesting, since i is actually somehow a "real number" (I've been trying to figuring this out as well).
i is (-)^0.5 1 or whatever you'd like to call it.
It's not negative nor positive, until multiplied with itself, which is just an absurd idea from the beginning of, and again interesting.

@Dave
It is.
Also I didn't know there were a way to say "for i=1" ;o
Thanks, now I know that too x)
I'm not very educated, 17 years old, done primary school for 9 years, stopped with homework @ 5th grade, I'm just trying to get back :D

And D H
The issue you're addressing, is something I figured out already in the beginning, this thread is therefor not about that.

 

[LukeD]

Dark Fire: Be very careful. With your personality, you're at risk of becoming what most people would call a crack pot. It's not a bad thing (and it's certainly not inevitable), I'm the same way some times, but you should be aware of it. To avoid this, you need to listen to what other people are saying. Don't just post something so that you can get validity; read what people are saying and try to understand the mistakes that people are pointing out. Also, try not to immediately trust your own results (I've seen you with other similar posts where you'll write a "proof" that 1 = -1 and ask why it is the case. However, it's not. 1 = -1 is completely false.) because you can certainly be wrong. If you come up with some result that seems wrong, there's a very good chance that it is and that you've made some mistake somewhere. Even if you come up with something that is correct, it does not mean that the logic you used to arrive there is correct.

Most importantly: Don't be afraid to make mistakes, but accept that you can, have, and will continue to make them. Most people in any profession (especially science and mathematics) would be more surprised to learn that they haven't made a mistake than to find that they made several. With practice though, people learn to avoid the larger mistakes and how to spot others so that they know that their mistakes are probably small. But even then, most people know that even small mistakes can completely ruin the result. You're not yet at this point with mathematics, so try to be aware of this.

 

[Dark Fire]

Dark Fire: Be very careful. With your personality, you're at risk of becoming what most people would call a crack pot. It's not a bad thing (and it's certainly not inevitable), I'm the same way some times, but you should be aware of it. To avoid this, you need to listen to what other people are saying. Don't just post something so that you can get validity; read what people are saying and try to understand the mistakes that people are pointing out. Also, try not to immediately trust your own results (I've seen you with other similar posts where you'll write a "proof" that 1 = -1 and ask why it is the case. However, it's not. 1 = -1 is completely false.) because you can certainly be wrong. If you come up with some result that seems wrong, there's a very good chance that it is and that you've made some mistake somewhere. Even if you come up with something that is correct, it does not mean that the logic you used to arrive there is correct.

Most importantly: Don't be afraid to make mistakes, but accept that you can, have, and will continue to make them. Most people in any profession (especially science and mathematics) would be more surprised to learn that they haven't made a mistake than to find that they made several. With practice though, people learn to avoid the larger mistakes and how to spot others so that they know that their mistakes are probably small. But even then, most people know that even small mistakes can completely ruin the result. You're not yet at this point with mathematics, so try to be aware of this.

This is deeper than that I'm not aware of "denial"/to ignore others statements.
"nth power of blabla" <-- wtf..? power of = exponent, n = real number?
I'm living in a very emotional place, with emotional people, I'm trying hard not to burst out in anger or sadness, and instead of going mad, and then possibly come with shallow, angry statements, I just ignore whatever I need to, to not burst out~

I need a clearcut, easy-English statement that explains not the obvious, but the complex - the part of the equation I don't get, or I'm having trouble with~
I've probably written this a 100 times, but I live not in an English-speaking country, also I never payed attention to school; what seem to be a possibly great vocabulary (or not) is nothing but what I've learned the last year through Googling and games, I could barely write and say simple words such as: yes, no, bye, what - a couple of years ago.
While my life's gotten just worse and worse, I've gotten less and less space of learning, and I've ended up barely Googling anything, though I'm still philosophizing sometimes, and solving math puzzles, since it steals no energy for me to philosophize, rather than figuring out, for example how a Buffer Overflow actually works by source (not just the idea, but in practice).

I'm just trying to focus at something else, since there's no way I can solve my problems IRL, and I just have to keep acting shallow, a "crack pot".

Thanks for your concerns, though.

Oh, and another thing: I also got bad experiences about people that sighs @ me all the time, mom, teachers, I guess we 'all' experience that, but since I've never felt hope - I quit acting out, I guess.
Rather just change the subject than experiencing a million sighs, like they're truly my superior, and I'm worthless.

Enough about me now >_<
Thanks for replies.

 

[matt grime]

Now, over to the subject:
i^4 = 1
1^0.25 = i

Well, if you are asking 'where have I gone wrong', then the answer is there. (The line you wrote after this also makes no sense.)

There is nothing to justify that this assertion is true; it isn't as it happens since the principal (not principle) root is taken to be the positive real root anyway.

 

[cristo]

This is deeper than that I'm not aware of "denial"/to ignore others statements.
"nth power of blabla" <-- wtf..? power of = exponent, n = real number?
I'm living in a very emotional place, with emotional people, I'm trying hard not to burst out in anger or sadness, and instead of going mad, and then possibly come with shallow, angry statements, I just ignore whatever I need to, to not burst out~

So what are you trying to say? You have been given the answer to your problems several times in this thread. Are you just intending on ignoring them?

 

[Dark Fire]

@up
1^0.25=i is wrong?
The line after is what I'm using to make a number with an exponent lower than 1 to become 1, though with complex numbers, as you can see, doesn't work.
If you watch the first line (9^0.5=3)-thingy, u can find the method~

 

[Dark Fire]

So what are you trying to say? You have been given the answer to your problems several times in this thread. Are you just intending on ignoring them?

If not written in a clearcut, satisfying way, then yes.

Sorry about double post

 

[D H]

So what are you trying to say? You have been given the answer to your problems several times in this thread. Are you just intending on ignoring them?

What's even frustrating is that he has raised this same issue in three other threads, has been answered multiple times in each and every one of those threads, and has ignored all of those answers. Sigh.

 

[matt grime]

@up
1^0.25=i is wrong?

Yes.

The line after is what I'm using to make a number with an exponent lower than 1 to become 1, though with complex numbers, as you can see, doesn't work.
If you watch the first line (9^0.5=3)-thingy, u can find the method~

And it doesn't make any sense. At best you're claiming (by adding powers in the exponent on the left hand side) that

1^{3.25}=i

Since you're arbitrarily choosing roots at will there is no logical reason to suppose that this equality holds, or even makes sense. It is, in fact, equivalent to you asserting that since 1 and -1 square to the same number, then 1=-1. As is frequently the case, light can be shed on the problem without recourse to the overly complicated ideas initially used. The infamous James Whatisisname (honestly can't remember) was a prince of this on sci.math with his "over-interpretation of galois theory".

 

[D H]

I'm sorry I made fun of you.

You are very frustrating to deal with. You don't seem to listen and you don't seem to learn, because you keep rehashing the same arguments after having been explicitly told these arguments are faulty and why they are faulty. Instead you try to blame it on us for not being "clear cut". We have been very patient and very clear cut, from the moment that you first asked why the product of two negative numbers is positive.

This warning was right on target:

Dark Fire: Be very careful. With your personality, you're at risk of becoming what most people would call a crack pot.

You have already seen that arbitrarily choosing roots will lead to erroneous results. The simple thing to do is to stop doing that. Don't keep making the same mistakes over and over. Everyone makes mistakes. It is very important to recognize that you too make mistakes. Learn to recognize them, learn to avoid them, and learn from them.

 

[dx]

Dark Fire,

Do you agree that $$1^{1/4} = 1, i, -1, -i$$? That is, are you aware that the operation of taking the fourth root is not single valued?

 

[matt grime]

Basically, I've seen at least 4-5 people agreeing, or seem to be agreeing on that 1^0.25=i, because i^4=1, but whatever.

Then they (and you) are incorrect. It is universally accepted that the symbol 1^{0.25} is 1. This is called 'choosing the principal branch'. This is required to make raising to the power 1/4 a function. This in no way contradicts the fact that there are 4 complex numbers that when raised to the 4th power give 1.

There are, as far as I can tell, no faulty conclusions presented to you: you cannot choose anyone of the fourth roots of a number at will and presume everything you say will be true. It simply, and demonstrably, doesn't work like that.

 

[quadraphonics]

I guess my point/the conclusion is that there's several 'formulas' for real numbers, that doesn't work with complex numbers...
And that just.. not surprises, but amazes me;

Why is that surprising (let alone amazing)? Complex (and imaginery) numbers are *defined* precisely to give different results under power functions than real numbers do, and all of the equations you've considered are power functions.

it's interesting, since i is actually somehow a "real number" (I've been trying to figuring this out as well).

Err... i is emphatically not a real number. So you can stop working on figuring that one out.

Also, as someone pointed out, the issues with power-function relations occur with negative roots as well; the issue really has nothing to do with imaginery numbers as such.

 

[matt grime]

Darkfire, a function should take one input (1, say) and return *one* answer. It is thus completely incorrect to say that 1^1/4 is a *set* of four possible numbers. You cannot claim something is a function and treat it as such when you allow it to return multiple answers for one input.

The convention is to define 1^0.25 as 1. It is called choosing a principal branch because you are choosing one of the 4 possible complex numbers that satisfy x^4=1. You must (and do not) make your choices consistent, and accept the consequences of making those choices.

 

[dx]

DX
My question is: WHHHHHHHHHYYYYYYYYYYYY.
This is what I'm talking about: I want the deep explanation, not the obvious conclusion.

I'm not sure what you mean by the deep explanation.

If you raise 1 to the 4th power, you get one, if you raise i to the fourth power, you get 1, if you raise -1 to the 4th power, you get 1, if you raise -i to the fourth power, you get 1. That is why all of these numbers (1, i, -1, -i) are fourth roots of 1.

If by deep explanation you mean something that you can see, then here it is:

Multiplying by 1 is the same as not doing anything. So if you multiply by 1 four times, then it's no different than multiplying by 1.

Multiplying by i is the same as rotating by 90 degrees in the complex plane, so if you multiply by i four times, you get back to where you were. So its the same as multiplying by 1.

Multiplying by -1 is rotating by 180 degrees, and multiplying by -i is rotating by 270 degrees. Do either of them four times, and you will end up at the same place, so its the same as multiplying by 1.

If you don't know what the complex plane is, read a book.

 

[Dark Fire]

I'm not sure what you mean by the deep explanation.

Something satisfying for a philosopher, not a mathematician.

Multiplying by 1 is the same as not doing anything. So if you multiply by 1 four times, then it's no different than multiplying by 1.

Me multiplying both sides with 1, was to make 1^0.25 into 1^1, by multiply 1 with itself 4 times, because 0.25*4=1.
My method obviously gave the wrong output for a complex number, which was the whole god damn point.

If you don't know what the complex plane is, read a book.

Buy me a book & give me the concentration that's required, and I'll read it.

Double post again, sorry.

 

[dx]

Something satisfying for a philosopher, not a mathematician.

Unfortunately, we're talking about mathematics. All explanations of mathematical results are mathematical explanations. You may want to speculate on the philosophical aspects of mathematics, but to do that you have to understand the mathematics first.

 

[BryanP]

Buy me a book & give me the concentration that's required, and I'll read it.

I'm not sure how much mathematics background you have but you can read about the complex plane for free. You can google it.

Complex Plane Overview
http://en.wikipedia.org/wiki/Complex_plane

Euler's Formula (might come to your interest) Overview
http://en.wikipedia.org/wiki/Euler's_formula

 

[LukeD]

DarkFire: I'm sorry, I did not realize that you didn't understand some of the messages. If you don't, try to say so, because otherwise people often assume that you're ignoring them for some other reason.
Also, congratulations on learning English as well as you have. I've yet to learn a foreign language beyond simple phrases (though I intend to some day)

I'll try to explain some of the things that you seem to be confused about. If you get lost somewhere (or don't understand something because of the language or not knowing some technical details, etc) let me know.

--

The "definition" of the Real Number is a little difficult to describe. They're usually either defined by a list of properties that they have or by a mathematical structure that has those properties. One way of defining the real numbers is by saying that it is a decimal number (such as 2.3141... and so on) where the number of digits goes on infinitely (so there is no last one). They might have some sort of pattern, but they probably do not.

Now note that by this definition, the number -1 has no square root in the real numbers because any time you square any decimal number, you get a positive number. Therefore, the number i cannot be a real number (by this definition), because when squared, it gives you a negative number.
i also cannot be a real number by any other definition of the real numbers because every definition of the real numbers has the property that any real number squared is positive.

Therefore, any property that holds for real numbers, but not necessarily for all of the complex numbers, might not hold for i.

Note that the real numbers have a very specific definition. Not everything that is considered a "number" is a real number.

--

You asked why some people say that 1^(1/4) = i is valid and others say that it is not but that everyone agrees that for some reason saying 1^(1/4) = i and 1^(1/4) = 1, so i=1 is not true.

For the real numbers (remember the definition that we used above), $x^{1/n}$, where n is an integer, is usually assigned a value called the principle value. If x is positive, then $x^{1/n}$ is the unique positive number y such that $y^n = x$. If x is negative and n is odd, then $x^{1/n}$ is the unique negative number y such that $y^n=x$. If x is negative and n is even, then $x^{1/n}$ is $i(-x)^{1/n}$

However, this is all convention. In some sense, we can assign any number to $x^{1/n}$ as long as it satisfies $y^n=x$. This is why some people say that there are 4 values to $1^{1/4}$. However, if one does this, then certain properties that are given to exponents no longer hold, so you can't assume that any of the properties that you've been expecting to work for exponents are true.

Also, you cannot then assign multiple values to $x^{1/n}$ at once because then you are not using equality correctly. There are a lot of reasons for this, and if you want, someone can explain them.

--

Feel free to comment on this explanation.

 

[Dark Fire]

If x is positive, then $x^{1/n}$ is the unique positive number y such that $y^n = x$. If x is negative and n is odd, then $x^{1/n}$ is the unique negative number y such that $y^n=x$. If x is negative and n is even, then $x^{1/n}$ is $i(-x)^{1/n}$

This is usually where I fall out.
Too complicated relative to what I find interesting and relevant at the moment (and usually in general)
I've had a problem with this at school and in other situations as well.
I'm somehow not able to participate in other peoples thinking within specific subjects.
I guess this is one of my social lacks.
I'd probably understand it if I sat down; putting my mind into it, but I'll never be able to do that, since I rather sit down, putting my mind into Perl if I one day had some spare energy (this, because it's easier to earn money on some Perl skills, than some general math, and I'm in deep need of money without having to walk outside, or basically do something I'm interested in while earning money)
OK, this went OT again.
Point is that I got a problem understanding it "this way".
It's easier if you add examples of numbers to make it practical, in addition.

Thanks for your understanding, patience and everything else :)

 

[Integral]

Something satisfying for a philosopher, not a mathematician.

...

Perhaps you ought to go find some philiosophers to talk to then.

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