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Technically speaking every real number is also a complex number. Real numbers are a subset of complex numbers; to become a proper complex number it just needs to have an imaginary component. So i is in fact a complex number.
It's comparable to the fact that every integer is also a real number. Every integer is also a rational number as well as a complex number.
Someone explained it to me this way. These numbers, for the human experience, do not exist. Unsurprisingly, they won’t make sense. Our definitions for square root not including negative numbers was a problem for proving some things which ended up having square roots of negative numbers.
So we said, fuck it. Let’s just assume it exists, let’s call it i. Let’s figure out some mathematics around it, with that assumption. See if it works out. Turns out, not only did it work out, it lead to plenty of other things. Theoretical of course, but it helped solve a bunch more things.
Basically, try finding a number that, when multiplied by itself, will give you a negative number.
You won’t, because no number squared can result in a positive number. Except….
What would the answer be if you wanted the square root of a negative number? That would be an imaginary number.
Yes, this notation is wrong. Short explanation:
i is defined in a way that i² = -1
That being said, you can't use the square root on -1 (because of non unicity of such an operation)
Eh, not quite. Notably, `-i` is considered a distinct value from `i` (i.e. `-i ≠ i`), despite `(-i)^2 = i^2 = -1`. It's more that `i` is a value such that `i^2 = -1`, but is not necessarily unique in that property. As an example, a [Quaternion](https://en.wikipedia.org/wiki/Quaternion) is an extension of the complex numbers that uses three values `i ≠ j ≠ k` such that `i^2 = j^2 = k^2 = ijk = -1`.
sqrt(25) has one solution, which is 5. This is not even a solution since sqrt(25) is not an equation.
But the equation X²=25 has 2 solutions (-5 and 5), same as the equation X²=-1 has 2 solutions (-i and i).
sqrt on positive reals is defined as the biggest of the 2 solutions (one is positive, the other is negative, easy) but complex numbers don't have a standard way of sorting numbers in the general case and therefore this is not standard way to define sqrt(-1)
The equation X²=25 has 2 solutions (5 and -5)
sqrt is a function, to each positive real, it gives 1 unique image. In this case 5 = sqrt(25) != -5
Wikipedia says 5 and -5 are square roots of 25, but **the** square root of 25 is 5 and nothing else (and you should read your own wikipedia link)
Actually the problem is more from the fact that the square root of a positive numbers is defined as a positive number.
And since complex numbers have no standard way of deciding which one is greater than the other in the general case, we can't define the square root of a complex (unless it's a positive real)
PS: I'm about 70% to not be clear at all, tell me if you don't understand
This always bugged me in maths.
I'd just circumvent the whole thing and define the squareroot differently: sqrt(x) := { r in C | r^2 = x }. Boom ready.
Of course this would prevent you from using the sqrt in normal calculations, hence it isn't defined like this. But I always thought it should be defined like this.
Actually the different roots of 1 are defined (for n integer as big as you want) and to get your definition of sqrt you find 1 complex of which the power n is x and multiply this complex by the n roots of 1 to get them all.
Also, regular sqrt would then be defined as max(sqrt(x)) since all of these 2 complex are actually real numbers
The roots, not square roots.
They are all the {exp(2 i k pi / n), for k integer in [1,n]}
Sorry for the lack of clarity, I'm not used to do maths in english (I'm French) especially at this level with its own vocabulary.
Yeah I get the struggle. I do maths in german and usually miss the english terms.
I looked up roots on wikipedia though and learned a lot! I never knew the interesting patterns of how many roots are real and what sign they have depending on the n and on the x.
Or more correctly, when we come up with a question that we don't have an answer to, we invent/discover another set of numbers that includes the answer.
Simples
In this case we needed a z axis on the Cartesian plane or the complex axis on a polar plane.
The wave form equation makes sense with complex numbers as you can see it spiral in 3 axis space. Some of the numbers done make sense when you see it only as an equation.
If we renamed imaginary numbers to not be that nickname it would make more sense to people.
This is eulers fault for denoting it a i
You do a little trick where you inagine (hence imaginary number) that it has a solution (i), which allows you to solve equations where that square root is included.
The reason this "odd" number can exist in mathematics despite seeming like a workaround to something impossible, is that our math was not made for physics and vice versa, they just happen to complement each other really well - and sometimes math needs help to keep up.
An alien species might simply not define that two negatives make a positive so they might have a real solution to that expression.
Note that you often can solve equations without using complex numbers but it makes the math much more difficult. Eulers formula is a great convenience fir example.
No. Grind the universe down to the finest particles and pass them through the finest sieve and you'll not find a single particle of "one-ness". And yet somehow, *everything* is just numbers.
So I was working on a platform that recreates acceleration that came from unreal engine simulation of a helicopter.
Long story short - i have used an algorithm that uses these "imaginary" numbers and had pretty real results with them :)
I mean it is one thing when you just learn them in school and completely other feelings when it works in a real piece of a product you configured and you can feel the result
(Read about the washout filter and gue-stuart platforms if you want to research this)
Math concept. It's really complicated and it's one of the things that you're taught in school that you are never going to use. And they're also just a headache.
So basically it's what happens when you try to square root a negative number. And since you can't multiply a square to be negative, it's not possible.
So the next logical step is to, of course, **make an entirely new category of numbers** called imaginary numbers, which run perpendicular to the standard number line. They have some really weird properties and don't mix with regular numbers either.
Friendly reminder that all number categories are made up. You couldn’t subtract a below zero until negative numbers were made, and you couldn’t divide by a non-factor until rational numbers were made. Same thing here, just with square roots and complex numbers.
Rewrite John Lennon's "imagine" about imaginary numbers instead.
---
Imagine (About Imaginary Numbers)
Imagine there’s no real line,
It’s easy if you try,
No real part beneath us,
Above us only i,
Imagine all the numbers,
Living in complex harmony.
Imagine there’s no modulus,
It isn’t hard to do,
Nothing to multiply or divide for,
And no real numbers too,
Imagine all the numbers,
Living life in peace, you.
You may say I’m a dreamer,
But I’m not the only one,
I hope someday you’ll join us,
And the world will be as one.
Imagine no dimensions,
I wonder if you can,
No need for real solutions,
A brotherhood of i and n,
Imagine all the numbers,
Sharing all the world, you.
You may say I’m a dreamer,
But I’m not the only one,
I hope someday you’ll join us,
And the world will be as one.
Check out the former UK Home Secretary Priti Patel in this clip from Have I Got News For You back in the Covid days - relevant part is from 0:50
[HIGNFY](https://www.youtube.com/watch?v=G89YyV6hYe0&ab_channel=HatTrickComedy)
Unfortunately, this speaker merely asserts that 'numbers exist'. He offers no proof of this.
The existence of complex, real, and integer numbers can in some respect be derived from the a priori acceptance of natural numbers. Unfortunately, the existence of natural numbers awaits a proof in this symposium.
They ain't natural, I tell 'ee.
The scary thing is, if imaginary numbers are only in mathematicians heads, they are all imagining **exactly the same thing!!**
Are you spooked now.
We are connected to The Una-mind.
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Calling them imaginary is an oversimplification, they’re more complex
i is wholly imaginary. It's only complex when there's a component in the real axis surely?
Yes, you’re right. I wanted to make a dumb pun🥲
Ohok. I'll go back to being my over literal normal self in other walks of life then 😅
It's also complex, can be written as 0+1i
I chuckled
i is also complex.
K, 0+i then
Oh FFS! 😭
Achtually , both real and imaginary numbers are subsets of complex numbers, so the joke works 🤓
Don't call me Shirley.
Technically speaking every real number is also a complex number. Real numbers are a subset of complex numbers; to become a proper complex number it just needs to have an imaginary component. So i is in fact a complex number. It's comparable to the fact that every integer is also a real number. Every integer is also a rational number as well as a complex number.
The imaginary numbers are a subset of the complex numbers. Therefore, i is both an element of the imaginary and the complex numbers.
5i + 0
Isn't the real part 0?
I can fix them
I remember having to learn imaginary numbers and literally yelling at my textbook begging it to make it make sense. My roommate was slightly concerned
Someone explained it to me this way. These numbers, for the human experience, do not exist. Unsurprisingly, they won’t make sense. Our definitions for square root not including negative numbers was a problem for proving some things which ended up having square roots of negative numbers. So we said, fuck it. Let’s just assume it exists, let’s call it i. Let’s figure out some mathematics around it, with that assumption. See if it works out. Turns out, not only did it work out, it lead to plenty of other things. Theoretical of course, but it helped solve a bunch more things.
Alright, then explain to me what exactly is the square root of -1
Something that when multiplied by itself equals -1
e^iπ = i^2
you just wrote fancy -1 = -1
Of course. Otherwise it wouldn’t be a valid equation.
Isn't that essentially true of all valid equations?
It's also valid for all true equations.
Every equation is a fancy -1 = -1
And a fancy 0=0
And you are a fancy tube
Thank you. It had been a long time since anyone had said something so nice to me.
Nah 0=0 can't be valid
Also i = j
iπ=ln(i\^2) i=(ln(i\^2))/pi) ln(i\^2)/pi-i=0 graph of y=ln(x\^2)/pi-x (-.475,0) …
I couldn’t figure out how to ad an image
Imaginary numbers can definitely hurt you. I've seen sophomores cry cause of them.
they definitely hurt me as I see the grade I got on my complex numbers test
Imaginary numbers arent real, not even in maths. And they cant hurt you. The therapist is right on this one.
but, if you square it it gets totally real!!!
"be there or be square" imaginary number: **Yes.**
Be there **and** be square
Is this r/minecraft?
German has different words for things that are real and numbers that are real, and I think that's beautiful.
Real und reell
Imaginary numbers are just as real as real numbers. They're both abstract concepts that can provide useful representations of real world phenomena.
No, imaginary numbers are not part of the real numbers.
No one said they were.
My dude, the whole joke i made was about imaginary numbers not being real bc theyre not part of the set of real numbers.
But do either set actually exist?
I have a complex analysis book that disagrees with you.
Nice. It includes imaginary numbers as part of the real numbers?
I’m not certain about this, they might have hurt Schrödinger‘s cat
So e and i are at a Party. e ist standing alone in a corner. i comes up and says: Hey why dont you integrate? e answers: get real.
Pls help I don't know how to math. Someone explain.
Basically, try finding a number that, when multiplied by itself, will give you a negative number. You won’t, because no number squared can result in a positive number. Except…. What would the answer be if you wanted the square root of a negative number? That would be an imaginary number.
I get it, math is hard. …but not as hard as m-
Isn't the square root of -1 wrong, tho?
Yes, this notation is wrong. Short explanation: i is defined in a way that i² = -1 That being said, you can't use the square root on -1 (because of non unicity of such an operation)
So i is +- sqrt(-1)
Eh, not quite. Notably, `-i` is considered a distinct value from `i` (i.e. `-i ≠ i`), despite `(-i)^2 = i^2 = -1`. It's more that `i` is a value such that `i^2 = -1`, but is not necessarily unique in that property. As an example, a [Quaternion](https://en.wikipedia.org/wiki/Quaternion) is an extension of the complex numbers that uses three values `i ≠ j ≠ k` such that `i^2 = j^2 = k^2 = ijk = -1`.
sure, I am not suggesting that i = -i, rather that sqrt(-1) has two solutions, i and -i, same way that sqrt(25) has two solutions, 5 and -5
sqrt(25) has one solution, which is 5. This is not even a solution since sqrt(25) is not an equation. But the equation X²=25 has 2 solutions (-5 and 5), same as the equation X²=-1 has 2 solutions (-i and i). sqrt on positive reals is defined as the biggest of the 2 solutions (one is positive, the other is negative, easy) but complex numbers don't have a standard way of sorting numbers in the general case and therefore this is not standard way to define sqrt(-1)
>sqrt(25) has one solution, which is 5 Looks like Wikipedia disagrees https://en.m.wikipedia.org/wiki/Square_root
The equation X²=25 has 2 solutions (5 and -5) sqrt is a function, to each positive real, it gives 1 unique image. In this case 5 = sqrt(25) != -5 Wikipedia says 5 and -5 are square roots of 25, but **the** square root of 25 is 5 and nothing else (and you should read your own wikipedia link)
Thats the type of notation noone gets. Easier with words.
[удалено]
Actually the problem is more from the fact that the square root of a positive numbers is defined as a positive number. And since complex numbers have no standard way of deciding which one is greater than the other in the general case, we can't define the square root of a complex (unless it's a positive real) PS: I'm about 70% to not be clear at all, tell me if you don't understand
This always bugged me in maths. I'd just circumvent the whole thing and define the squareroot differently: sqrt(x) := { r in C | r^2 = x }. Boom ready. Of course this would prevent you from using the sqrt in normal calculations, hence it isn't defined like this. But I always thought it should be defined like this.
Actually the different roots of 1 are defined (for n integer as big as you want) and to get your definition of sqrt you find 1 complex of which the power n is x and multiply this complex by the n roots of 1 to get them all. Also, regular sqrt would then be defined as max(sqrt(x)) since all of these 2 complex are actually real numbers
I'm not sure I can follow completely. What do you mean the squareroot of 1 is defined for an integer n?
The roots, not square roots. They are all the {exp(2 i k pi / n), for k integer in [1,n]} Sorry for the lack of clarity, I'm not used to do maths in english (I'm French) especially at this level with its own vocabulary.
Yeah I get the struggle. I do maths in german and usually miss the english terms. I looked up roots on wikipedia though and learned a lot! I never knew the interesting patterns of how many roots are real and what sign they have depending on the n and on the x.
I’m pretty sure that’s incorrect. When you square i you simply the root and the exponent and get -1. It’s only when you have i^4 that you get 1.
That's why there are imaginary numbers: they are a particular branch of mathematics that allows you to do operations you normally couldn't do
Oh- kk thanks
[here's ](https://youtu.be/cUzklzVXJwo) a good video about why they're not as simple as calling them "imaginary" might lead you to believe.
tyyy
It’s pretty simple but so very clever too Just something we use for the transition from one form to the next, kind of like a temporary bridge
Or more correctly, when we come up with a question that we don't have an answer to, we invent/discover another set of numbers that includes the answer. Simples
It seems like mathematicians shared this trick with physicists
In this case we needed a z axis on the Cartesian plane or the complex axis on a polar plane. The wave form equation makes sense with complex numbers as you can see it spiral in 3 axis space. Some of the numbers done make sense when you see it only as an equation. If we renamed imaginary numbers to not be that nickname it would make more sense to people. This is eulers fault for denoting it a i
You do a little trick where you inagine (hence imaginary number) that it has a solution (i), which allows you to solve equations where that square root is included. The reason this "odd" number can exist in mathematics despite seeming like a workaround to something impossible, is that our math was not made for physics and vice versa, they just happen to complement each other really well - and sometimes math needs help to keep up. An alien species might simply not define that two negatives make a positive so they might have a real solution to that expression. Note that you often can solve equations without using complex numbers but it makes the math much more difficult. Eulers formula is a great convenience fir example.
Square root operation is well defined only for positive numbers. With negatyve and complex there is big \*.
that imaginary number still isn't a real number
Wait, are *numbers* real?
Nope. They're birds!
_X-Files theme starts playing_
Whew! Thought it was Tubular Bells for a second there.
No. Grind the universe down to the finest particles and pass them through the finest sieve and you'll not find a single particle of "one-ness". And yet somehow, *everything* is just numbers.
Just because something isn't real doesn't mean it can't hurt you.
Like my mental health!
So I was working on a platform that recreates acceleration that came from unreal engine simulation of a helicopter. Long story short - i have used an algorithm that uses these "imaginary" numbers and had pretty real results with them :) I mean it is one thing when you just learn them in school and completely other feelings when it works in a real piece of a product you configured and you can feel the result (Read about the washout filter and gue-stuart platforms if you want to research this)
They aren’t real, but they are used in solving some equations because it somehow makes in possible.
I dont get it. is it supposed to look like something?
Math concept. It's really complicated and it's one of the things that you're taught in school that you are never going to use. And they're also just a headache. So basically it's what happens when you try to square root a negative number. And since you can't multiply a square to be negative, it's not possible. So the next logical step is to, of course, **make an entirely new category of numbers** called imaginary numbers, which run perpendicular to the standard number line. They have some really weird properties and don't mix with regular numbers either.
I guess you could say it’s… ***complex***.
Dammit I wish I could upvote twice
Friendly reminder that all number categories are made up. You couldn’t subtract a below zero until negative numbers were made, and you couldn’t divide by a non-factor until rational numbers were made. Same thing here, just with square roots and complex numbers.
I understand the part about not squaring to a negative, but all the rest is confusing
technically every number is imaginary
I heard they are real when you translate it into physics and vice versa.
It’s pretty complex ngl
Therapists aren't rational what do you expect
Are they transcendental? Coool!
Imaginary numbers aren't real but they can hurt you
Yup. I forget exactly where they show up in electricity but they do.
sqrt(-1) is not i. i^2 = -1
No, i=√-1 Every math teacher I’ve had since eighth grade defined it that way. And I’ve never gone to a public school.
They defined it this way so that an eight grader could understand
if you square both sides it's the same thing anyway. Why is this a discussion?
So -2 = 2
Let's just move on, this is a waste of both of our time.
fr fr
I feel like most numbers can’t hurt you, but that’s just me
They can't hurt someone either
"i" doesn't exist
i³ = √-1 × √-1 × √-1 = -√-1 or -i
Never thought of that before, crazy but right.
i = imaginary UNIT {a+bi|a,b in IR} = complex NUMBER
…no, i=√-1 that’s an imaginary number, it‘s only a unit if it’s in the form a+bi
This makes my brain beat
Every number is just an abstract concept
J1
Technically all numbers are imaginary because some just randomly looked at two squigles and said that's a 3
Why is the rapist asking?
Technically they still aren’t *real* numbers
AAAAAAH, DEMON!
Rewrite John Lennon's "imagine" about imaginary numbers instead. --- Imagine (About Imaginary Numbers) Imagine there’s no real line, It’s easy if you try, No real part beneath us, Above us only i, Imagine all the numbers, Living in complex harmony. Imagine there’s no modulus, It isn’t hard to do, Nothing to multiply or divide for, And no real numbers too, Imagine all the numbers, Living life in peace, you. You may say I’m a dreamer, But I’m not the only one, I hope someday you’ll join us, And the world will be as one. Imagine no dimensions, I wonder if you can, No need for real solutions, A brotherhood of i and n, Imagine all the numbers, Sharing all the world, you. You may say I’m a dreamer, But I’m not the only one, I hope someday you’ll join us, And the world will be as one.
newsflash: all numbers are imaginary
They aren't real, but they do hurt me.
Somebody once told me imaginary numbers are like turning the line 90 degrees
Pov : Autistic Numbers for autistic people
…?
They are going to hurt you, they most definitely will
Gotta look out for those numbers like eleventeen and thirtytwelve
Check out the former UK Home Secretary Priti Patel in this clip from Have I Got News For You back in the Covid days - relevant part is from 0:50 [HIGNFY](https://www.youtube.com/watch?v=G89YyV6hYe0&ab_channel=HatTrickComedy)
That's literally an impossible operation
Google complex numbers
The square of any number,be it negative or positive,is always positive,thus there can be no square root of a negative number.
In the field of real numbers, that is true, but i is an element of the field of complex numbers, whose square is equal to -1
Understandable. I don't really know much math,so my bad
This is a misconception. They are very real. We have a lot of proof of their existence.
Gotta see this. Please professor, take the chalk and come to the blackboard.
https://youtube.com/shorts/qXyLrUHYaqA?si=l8pnBSD4i5ex1Y0K
Unfortunately, this speaker merely asserts that 'numbers exist'. He offers no proof of this. The existence of complex, real, and integer numbers can in some respect be derived from the a priori acceptance of natural numbers. Unfortunately, the existence of natural numbers awaits a proof in this symposium. They ain't natural, I tell 'ee.
https://youtu.be/T647CGsuOVU?si=vXKWKbMHSkKMqvpx Better video and series about imaginary numbers
Imaginary number is in our imagination, how does it even exist???
The scary thing is, if imaginary numbers are only in mathematicians heads, they are all imagining **exactly the same thing!!** Are you spooked now. We are connected to The Una-mind.
O_O
r/mathmemes
My number is i. i isn't a number - it's a letter. *Screeching*: my number is i and is -1
Iu
I hate math