Merdock
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Last summer my aunt and I got into an argument about math at a family reunion. She's just retired from her long career as a math professor at a university and I'm a newbie engineer. The core of the issue was the equation √(x^2)=|x|. This is because, as she put it, unless stated otherwise we just want the positive value from the square root function rather than both the positive and negative value.
Last night something triggered my OCD mind to rethink the conversation and realized that if the solution would be a complex number (combination of real and imaginary), then this presents a problem that did not occur to me previously.
I fucking love my aunt and love having these conversations with her.
As a thank you for putting up with that, here's some star trek memes
I recently started a new binge of star trek
How do these ships take so much damage and yet the bridge almost always looks pristine
I feel ya O'Brien
Cat tax: Quinn (left) and his sister Claudia. They are definitely siblings.
Doggo tax: Wolfrik enjoying some recent snow. He loves snow more than anything in the world. He had one surgery, attempting to get a second for something that was missed in the first. His spirits are up and he will be ok however things move forward from here.
gallowglacht
Soooo, is the Star Trek Adventures RPG by Modiphius any good? Seems like this thread would be a good one to ask in...
playingwithknives
Congratulations on the math, but mostly on the fur babies. Mostly b/c of the 2, my brain only produces the happy chemical for the one.
billyjonas
Given that all these are jokes, you may think #5 is as well. But no no. That’s all Garak, baby! ?
HashMaster9k
FrogBotherer
I'm quite glad that complex numbers reside in the world of things I had to learn but haven't needed to remember.
getthismanabeer
Wuz314159
gnomedeplume
So what is it?
Merdock
ReaperCDN
Heffaloo
#10 Imaginary fucks are real. In fact, that's about the only way I get to experience them.
BGravoIII
HashMaster9k
Illithidbane
"I believe that's the first totally honest thing you've ever said to me."
HashMaster9k
Merdock
Fools errand it might have been, I'm glad all the same that you took it with me.
HashMaster9k
lastmanonearthbutidroppedmyglasses
HashMaster9k
lastmanonearthbutidroppedmyglasses
HashMaster9k
vickyblr
#8 If I steal, I upvote. Take your damn +1.
vickyblr
Currently I'm high on shrooms and am just tickled pink by all of the post but I'll come back in the morning because legit wanna see the end
vickyblr
of the sqrt(i) discussion
Merdock
Enjoy your evening. I'll gladly talk about math if you talk about shrooms. I've never had the pleasure of trying them before.
DenStortalendeMester
420username69
Merdock
Imalwaysready
A starship for ants is brilliant.
USSBigBooty
#7
GreaterDog
ﻼƆUᆿ ƎHT TAHW HƆTIઘ
Illithidbane
M/AM reactor excites exotic plasma, vented through nacelles to energize coils that warp/bend spacetime as an Alcubierre drive.
USSBigBooty
My point. It's not really space magic, although IIRC, the warp field creates a null mass pocket that can be accelerated indefinitely past C.
USSBigBooty
Anyway, I know later treks were real bad about technobabble, but TNG did a good job expanding past moral/ethical into data driven ideas.
Illithidbane
Mass Effect series creates zero/negative mass ships that can zoom, but I thought Trek just bends space like Alcubierre's proposed drive.
Illithidbane
Though the Impulse engines are as described, recusing effective mass to allow fusion drives to have greater acceleration.
ApplesdaKitty
@op the square root of i is the quad root of -1. or (-1)^(1/4) no?
ApplesdaKitty
I also love to piss off my math major drinking buddy with shit like, (d/dx)*x = (d/d)*(x/x) = 1
SeriousNonsense
If f(x) = x, then (d/dx)f(x) does in fact equal 1... You were very close to the insight: (d/dx)*x = (dx/dx) = 1
Frederf
Assuming "the" (singular) is a mistake. All roots of i are fourth roots of -1. Roots of z^4=-1 are +-1/rt2 for both real & im in sum.
Frederf
z^2=i has 2 roots (a+ai, -a-ai), z^4=-1 has 4 (a+ai, -a+ai, a-ai, -a-ai). a =1/root2.
Merdock
It is...
ApplesdaKitty
no differentiation is needed the. calc has become useless. newton and mexwell, eat me lol.
IsThereAnyUserNameThatsNotAlreadyTaken
The first one is questionable since the derivative of |x| is not continuous, while sqrt(x^2) is. So the equation is a tricky one ;)
euphoricopportunity
#1 ugh. I really should just remember the answer to this. I have forgotten most of what I learned in trigonometry.
pomax
To be fair... the square root of x² is not |x| in the slightest, which your high school math class should have taught you.
pomax
And if "my high school math class never taught me that", then maybe you live in one of the disappointing countries.
Recursiv3
Oh my God Wolfrik, that's a fuckin amazing name and they're adorable
almostalone
Nerd! Hah! J/k. I love this so much.
Zange0
The negative results may not always by physically possible, but I'd much rather include ± than be beaten to death with my own skull.
ObiHaiv
Your doggo image -- when I was scrolling, the screen stopped with his eyes and ears above. Looked like a bovine skull, and I'm thinking WTF?
Chilichunks
As someone who likely has dyscalculia I'll take your word for it.
lastmanonearthbutidroppedmyglasses
HashMaster9k
YouCantSpellSlaughterWithoutLaughter
I'm so happy that I've never been in a situation where I had to understand anything of what you just wrote.
whatspaulplayingtoday
#10 I'm sure at least one of them has experienced a negative fuck in their life.
gnomedeplume
the Vulcan might have but only gets to have any once every 7 years so will say it was at least agreeable
dunkum09
√(x^2)=|x| is ONLY true when x is a real number. if X is complex then the its different in part because |x| means something different there
lastmanonearthbutidroppedmyglasses
I understood some of those words
cheeseman93
I feel like √ means something different too, right, because typically √ is an operation that is only defined on positive real numbers?
PocketCleric
no, it IS defined for real NEGATIVE numbers also. i.e. ax+i, or a real part coupled with an imaginary part.
dunkum09
for negative real numbers x= -k; k>0 √ x = i√ k
Frederf
Root as an operation takes on the context of its domain. There is ambiguity if the range is just the positive root.
dunkum09
pretty sure its √ c = c^1/2 for Complex c. and of course c=k(e^(ni)) for some real k and n so √ c = k(e^((ni)/2))
Merdock
I believe it should be √c = √(k)(e^((ni)/2). Though I typically use c =k /_ n° so √c = √k /_ (n/2)°
dunkum09
you're right. id probably have seen that if I were writing it out or doing it in Latex or something, harder to spot here with all the parens
LtGenObvious
#1 Yes, it's an "imaginary numbers are special" scenario. Math relies heavily on context. Unless there's reason to assume otherwise, we…
LtGenObvious
…generally assume a domain of ℝ. (The presence of i in an equation—as in your second example—would be one reason to assume otherwise.)
abion47
Correct me if I'm wrong, but the equation sqrt(x^2)=|x| still works even in the case of sqrt(i) since in that case x=sqrt(i), aka -1.
Merdock
No, i^2 =-1. Sqrt(i) = (1+i)/sqrt(2). If sqrt(x^2)=|x| for complex numbers, then sqrt(i)=1. But 1^2 does not equal i
abion47
Ah right, I had that backwards for some reason.
cheeseman93
I think you're right, and the aunt should say √(x^2)=|x|, x ∈ ℝ. On the complex plane, the √ and |x| operators mean something different.
Merdock
I agree this works on the real plane, but I think if you want something equivalent on the complex plane you would need the radial sqrt eq.
Merdock
√(x^2 /_ 2y°) = |x| /_ y°
dreikommavierzehn
https://en.wikipedia.org/wiki/Absolute_value#Complex_numbers
Merdock
That's interesting, I had always thought of abs as strictly distance from 0. Thank you!
FINNSTAR7
The problem is you're treating i as a number, not a dimension. As NineLongWall eluded to, |x| is actually shorthand for "the distance of x".
FINNSTAR7
The most common measure of distance is the Euclidean distance, which is the square root of the sum of the squares of each component value of
FINNSTAR7
x. That is to say, if x is multidimensional it will have component values associated with each of its dimensions that you sum the squares of
FINNSTAR7
and take the square root of. So, for a complex number, we see |c| = sqrt(r² + i²), r being the value of the real component, and i being the
FINNSTAR7
value of the imaginary component. In your discussion, you are asking about the complex number 0 + 1i. Plugging into the formula, |0 + 1i| =
FINNSTAR7
sqrt(0² + 1²), which is simply 1. Note that it is 1², and not i². i is simply a label, the dimension, not an actual value. 1 is the value.
[deleted]
[deleted]
4vie
I know Frakes was tall, but I'd have imagine Word to be the "Absolute Unit" here....
4vie
Worf*
Somnipresent
Word.
whatbuttondoipush
Reminds me of that Russian version of return to treasure island cartoon
combatwombat0
Yeah it's a reference to dr livesey's phonk walk meme
gnomedeplume
if you search Livesy Walk combined with any fandom you can think of you'll probably get a result
DrMarioSThompson
What are they looking at on the ceiling?
whatbuttondoipush
They struttin'
auspaco
Sqrt(i) = (1+i)/Sqrt(2).
SeriousNonsense
Indeed, whereas the solution to x^2=i could be that or -(1+i)/sqrt(2).
Icantgoogle
Dont forget -(1+i)/√2
ipointoutrepeats
Oo yeah Sqrt all over my Imgur, daddy
FranticRed
This comment made me laugh entirely too much.
dwarfinathonginacaskinthewaterwoooo
there are always two kinds of people
NineLongWall
And one of them is always something like a dwarf in a thong in a cask in the water
QuiteHugeMegaPlusUltra
Merdock
Exactly, and the absolute value of that answer would be 1, which would make no sense as 1^2 does not equal i
QuiteHugeMegaPlusUltra
1) The absolute value of that is |√i | = |(1+i)/√2| = 1 indeed. You can also see it as |√i | = √((√i)²). I did this by simply following your
QuiteHugeMegaPlusUltra
2) equation √(x²) = |x|. Where x = √i. So yeah, 1² ≠ i but that is totally unrelated here. So, what you are looking for is 1² = |√i|² = 1,
QuiteHugeMegaPlusUltra
3) Which is true. Well perhaps I'm not getting at all where is the problem. Please explain the entire reasoning behind the "paradox".
Merdock
There is no doubt that |√(i)|=1, it's that √(i)≠1, showing a counter-example to √(x^2)=|x|.
SeriousNonsense
Instead of sqrt(x^2) = | x |, which isn't always true in C. A general statement would be sqrt(|x^2|)=|x|.
wintermute93
What part of that do you think doesn't make sense?
WellHelloTh3r3
I think a central issue here might be that we're abusing the idea of absolute value. What you're talking about is the magnitude
WellHelloTh3r3
It seems the abs(x) that your aunt is referring to is just |x| = {x, for x >= 0; -x, for x < 0}, which is only the same as taking the
WellHelloTh3r3
Magnitude if x is real, since real numbers only lie on 1 axis, the number line. On the complex plane, things are different.
WellHelloTh3r3
If we're to take sqrt(x²) = |x| as meaning that sqrt() outputs the positive root (not the magnitude of the root) as standard, then
NineLongWall
The absolute value of a complex number is its distance from 0 on the complex plane, |(1+i)/sqrt(2)| = 1
squillis
It seems lot of the issue is i is being equated to x and not x^2. If it were then x^2=i so x=sqrt(i) and sqrt(x^2)=sqrt(sqrt(i))
Merdock
Yes, so the equation given in my post would mean that √(i)=1, which obviously isn't true. Hence her response to my question, "It's a trap"
NineLongWall
The “equation” in your post is incorrect. It should read either: |sqrt(x^2)| = |x| Or: sqrt(x^2) = +/- x.
NineLongWall
You have to either apply absolute to both sides, or indicate that there are two possible values of x
Merdock
In my previous discussion with her, I had several problems with said equation, but she brushed them off. I changed her opinion on it today
10001110101periodictablewithasantapieceofmind
AmyBernadettePenny
RandomnessSlayer
sqrt[sqrt(i)^2] = |sqrt(i)| ; sqrt(i) = |sqrt(i)| ; (1+i)/sqrt(2) = |sqrt(i)| ; how do you get 1? what am I missing here?
RandomnessSlayer
Is the notation ||x|| as in the L2 norm? Or is it abs as I have interpreted it as?
RandomnessSlayer
Ah because the terminology is garbage and changes definition from real to complex. Ha