/sci/ - Science & Math

>>9194161
I'll bite.

Imaginary numbers are not any more or less made up than numbers in general are, they are essentially multidimensional numbers where regular ("real") numbers are one dimensional numbers. In that sense one could think that -1 and 1 are the same number and of equal size, but merely in the opposite direction (see vectors). Imaginary numbers are not in the opposite direction or the original, but somewhere in between (e.g. i is the size of number one, but points 90 degrees in a different direction (where -1 points 180 degrees in a different direction). This becomes even more apparent while looking into more complicated physics.
3. There are no solutions to the two additional problems you typed out, as opposed to the first problem.

x^2=-1

To solve for x, one has to answer "which number has to be multiplied by itself in order for the result to be -1". The size of the number (or more precisely, it's absolute value) is 1, but it's direction is 90 degrees in a different direction than +1 or -1. When multiplying said number by itself, the 90 degree angle is multiplied by the power (2) to get the new angle, 180 degrees (the number is negative). Because we did not previously have a way of noting numbers that were not either positive or negative (either in the direction of choice or in the opposite direction), we began noting them as i.

1^y=0

To solve for y, one has to answer "How many times does the number one have to be multiplied by itself in order to reach 0"
Because the very definition of the number one is the multiplicative identity (That is, any number multiplied by one remains that number), no possible solution can exist. The reason for why your next problem can't be solved with a "meme number" either.

>>9194161
The meme imaginary numbers did not arise from the quadratic, or quadratic formula. There are no REAL and at the time of discovery, applicable solutions to the equation x^2=-1.

The imaginary numbers arose from the cubic formula, where multiplying, adding and what not square roots of negative numbers still gave a sensible result.

Is there a sensible result from using a number in the lower two cases? Is there an identity that uses them and still is applicable? not that I know off...

>>9194190
>There are no REAL [...] solutions to the equation X^2=-1

This depends on if you by real mean "existing" or "true" numbers or if you mean the "real numbers", which is a misleading name as "imaginary" numbers are not any more or less imaginary than regular numbers. They're simply regular numbers with a different direction.

+ means a direction that deviates from the original by 0 degrees
- means a direction that deviates from the original by 180 degrees
i means a direction that deviates from the original by (180/2)=90 degrees
-i means a direction that deviates from the original by 180+90=270 degrees

>>9194179
How can you understand that imaginary numbers exist without understanding how y such thay 1^y=0 can exist? Just like complex numbers don't satisfy all the properties that the reals do doesn't mean that they don't exist. You are free to produce new algebras to solve any equation which has no solutions, just don't expect them to obey all familiar laws.

>>9194179
So why can't we raise 1 to some new imaginary-number like number, let's say [math]n[/math] to get a negative number?

[math]1^n = -1[/math]

It seems to work out with our rules for exponents.

[math]1^n * 1^n = (1^n)^2 = 1[/math]

>>9194204

See
>Because the very definition of the number one is the multiplicative identity (That is, any number multiplied by one remains that number), no possible solution can exist.

Without you redefining the concept of "number one" or the concept of "powers" or the concept of "equivalence" or the concept of "number zero" or just generally messing with how mathemathic notation works, you can't solve 1^y=0. You can't extend existing frameworks to solve it, either. That solution requires an entire alternative framework, or as you called it, "new algebras". The quickest way to do so is probably by altering the definition of one to make a solution possible.

The key difference is that x^2=-1 <=> x=i works perfectly without altering any existing definitions or introducing new frameworks (One could argue that it alters the definition of square roots, but what we're doing is simply exploring what happens by using square roots outside their intended domain, hence extending an existing framework and as such only removing limits that are artifically imposed and do not naturally arise from the initial axioms). The concept of "power" relies on the concept of multiplication, and we can rewrite the equation as x*x=-1. Multiplication means that as many of the number being multiplied are added together as the multiplier states, and in addition, the resulting number's direction angles are added together (e.g. (-2)*(-2)=4 because the direction angle of -1 is 180, 180+180=360 which results in a direction identical to the original direction with the angle of 0).

No new frameworks, no definition changes. The solution is very much true and real even if the resulting number does not belong in the domain of what is commonly referred to as "real numbers". This is why the "real numbers" have a misleading name; imaginary numbers are just as real as "real numbers".

>>9194211
So:

[math]1^\frac{n}{2} = i[/math]

Pretty neat desu.

>>9194211
To be honest, I completely overlooked the last equation. I was focusing more on the idea of demonstrating that the concept of "i" isn't any more made up than any other number.

The problem that we get from this is how we note directions of numbers (see >>9194196)

Technically your point IS valid as we could extend the concept of exponents in another way as well
1^n=-1 can be expressed as
1=(-1)^k

The problem is that this causes a direct conflict between two key ideas
1. Any number multiplied by one remains itself
2. When multiplication is performed, the direction angles (as deviating from the positive direction) are added up to form the new direction angle.

The problem kind of lies with the assumption that +1=1. This is a bad assumption as +1 inherently implies the positive direction, 1 simply implies scale (although at this point they're essentially the same thing).

This means that you would have to come up with a compromise along the lines of "Any number multiplied by a number the scale of one retains it's scale, but not necessarily it's direction."

>>9194232
To add to this since I forgot to mention, the compromise created here does not alter the fact that 1^y=0 is not solvable merely be extending existing concepts without redefining existing things and/or introducing new frameworks, though. (>>9194222)

>>9194161
A solution to x^2 = -1 was invented because it was useful to do so. It turned out that the complex numbers had other interesting properties and applications too, so they became mainstream. Also, quantum mechanics requires complex numbers too, so complex numbers are as "real" as the real numbers. [math] 1^z = -1 [/math] is actually true in [math] \mathbb{F}_2 [/math] (i.e., mod 2).

You can do whatever you want, but if other mathematicians don't find it interesting then they won't spend time thinking about it.

>>9194161
>meme
...still shilling this shit?
Lrn2meme fgt pls

Okay, seems like in my original post (>>9194179) I missed the point of the latter half of your post.

>>9194238
>>9194239
Are correct. If I interpret your question as "why are there no systems in which solutions exist for these equations" (I originally interpreted it as a shitpost implying that i is somehow more artificial than other numbers), then the answer is "because out of no set of non-conflicting axioms matching useful real or imaginary circumstances do such solutions logically arise". (Well 1^z=-1 does work, but the other one I haven't seen a solution for in any system.)

Those solutions may exist in some systems as >>9194238 points out, but they're not implemented in "normal math" because there are still some unsolved problems in relation to them, and as such they can't be used universally without paradoxes arising or something of the like. In a similar manner i does exist, but it's not used universally because problems may arise.

>>9194161
That's actually a very good question. Here's a better one: how come there isn't an imaginary number that solves [math]x = \frac{1}{0}[/math], or how come there isn't an imaginary number for [math]\mathbb{R}^3[/math] ? inb4 riememe sphere

>>9194161
They were invented (unavoidable) to solve

y=x^3+x^2-12x-5

Which clearly has three real roots but to get to the middle one you need complex numbers.
16th century stuff.

>>9194232
You should learn some abstract algebra. You can absolutely create a new set of rules which has a solution to 1^y=0 without changing any of the old ones. The real number 1 is not defined as "the multiplicative identity" in any concrete construction of the reals. Even if you just say its the totally ordered uncountable field with the least upper bound property you can still extend the operations however you want to obtain something bigger. After all 1^(1/2) isn't 1 even though 1 times anything is 1.

In short, you put the phrase "new algebra" in scare quotes, so I know you haven't seen any of this before and have no idea what you're talking about.

>>9194290
I've got unpublished stuff on division by zero that I've given talks on. You can absolutely do that, but without a solid algebra background you wont be able to handle it very well. Your multiplication can't be associative, for one.

The second thing is well-studied, look up the process that goes reals, complexes, quaternions, octonions, sedenions, and you will find the relevant discussion.

>>9194161
What the fuck did you just fucking say about imaginary numbers, you little bitch? I’ll have you know I stopped caring about math when I was introduced to the concept of imaginary numbers, and I’ve been involved in numerous secret raids on Al-Gebra, and I have over 300 crocks of shit. I am trained in equations that can only be solved by inventing numbers that can't exist and I’m the top math deity in the entire US academic forces. You are nothing to me but fucking wrong. I will wipe you the fuck out with math the flaws of which have never been seen before on this Earth, mark my fucking words. You think you can get away with saying that shit to me over the Internet? Think again, fucker. As we speak I am contacting my secret network of algebra solutions across the USA and your IP is being traced right now so you better say "the correct answer is whatever the correct answer is", maggot. The math that says the pathetic little thing transcribed to words. You’re fucking dead, kid. I can be anywhere, anytime, and I can mark you wrong in over seven hundred ways, and that’s just if you write it down in english instead of ancient math runes. Not only am I extensively trained in unarmed combat, but I have access to the entire arsenal of the United States Logical Math Corps and I will use numbers that never lie to their full extent to wipe your miserable ass off the face of the continent, you little shit. If only you could have known what unholy flaws your little “clever” human construct was about to bring down upon you, maybe you would have held your fucking tongue. But you couldn’t, you didn’t, and now you’re paying the price, you goddamn idiot. I will shit complex numbers all over you and you will drown in it. You’re fucking dead, kiddo.

>>9195770
>1^(1/2) isn't 1

que

>>9195821
There are two complex roots, so maybe it would have been better to say it's not just one. The complex logarithm is not actually an operation since it's multivalued so exponentiation is multivalued too.

>>9194161
You can absolutely solve this, anon, for nonzero a

1^y=a

This is obviously equal to: e^(y(ln 1))=a

And, as we know, ln z is a infinite valued function for all complex z but 0. Ln 1 =0 is just one of the branches, but in general ln 1 = 2pi*k*i

Now, for k=/=0, set y= (ln a)/(2pi*k*i). Problem solved.

>>9196211
Just one last thing: it should say set-valued map, point-to-set map, multi-valued map, multimap, correspondence or carrier; but not function, of course.

In the strict sense, a well-defined function associates one, and only one, output to any particular input. The term "multivalued function" is, therefore, a misnomer because functions are single-valued.

>>9194161
the complex numbers are characterized by three properties

1. they obey the same algebraic rules like R (like a(b+c) = ab + ac etc.)
2. they contain R as a sub-structure
3. the equation x^2 = -1 has solution in C

if we were to "make up new meme numbers", those are quite reasonable demands. but if 3. is replaced with 1^y = 0, there's no structure which satisfies all three conditions. you can invent your own algebra where your equation has a solution, but it won't be related to R in any nice way.

oh, and 1^y = -1 is totally solvable in C.

>>9196232
C doesn't obey all the same identities as R. You can't pull things out from under a radical, for one.

sorry for the late responce but the answer you are looking for is epsilon.

https://en.wikipedia.org/wiki/Epsilon_numbers_(mathematics)

so yes we have invented another imaginary number to explain these cases

>>9194161
y=-inf
z=1/2

you moron

>>9197742
Your first answer is plain wrong (literally, 1^n for all n is one)
Second one is only a single case. All correct answers are given by:
>z=1/(2n) where n is a natural number (excluding 0)

>>9197776
>literally, 1^n for all n is one
Wrong.

>>9194179
Then explain me why we can't multiply 1 through a n-dimensional wormhole that loops back on itself in an infinite number line and reach 0 that way. Just name this new wormhole creating number t and define it in a way that makes 1^y=0 feasible.

>>9200673

>>9197776
>literally, 1^n for all n is one
if it is a fraction, depends on the branch cut you choose

>>9194179
I'm sure I'm a brainlet and all that but I found this post interesting. Thank you.

>>9200930
You found it interesting because it was written by a brainlet who thinks they know what they're talking about.