Why is the Square Root of Negative One equal to i?

[@itskshitij]

Can't believe I'm still learning math from Mr. Nieh 3 years after high school

[@joshualefort6849]

So cool

[@Reach3DPrinters]

Im learning math 20 years after high school! lol

[@snarevox]

yeah im 44 and this is the first im hearing about these number angles. its probably not even that "new math" everybody likes to blame stuff on.

[@rlf4160]

3? Try 50. Going with the phase angle (pun intended) would have eliminated several hours/months/years of consternation.

[@sridipta]

This is a beautiful video! This is the kind of stuff that needs to be part of the school educational material. Great job!

[@Drikett1]

Yes and no. Yes cause it explains the fundamentals in a greater form. No because how practical is this in people's lives? Speaking for those that are not mathematicians who learn this later anyways

[@bowlineobama]

@@Drikett1 It is practical because we are using the fundamentals everyday and a lot of people don't even know it. It is always good to know the truth about these things to better understand and appreciate.

[@bowlineobama]

I totally agree with you on that. It has taken me to this point in my life to discover this truth about numbers.

[@sridipta]

​​@@Drikett1I appreciate your point. But to me "statement without sound fact/basis" is useless and uneducative. Throughout out years in schools, we have been taught so many stuffs without being explained the whys. That's why i think we dont find the use of those "info" in real life. Its better to be taught less but taught right. Then everything. Then we would find practical relevance so many of these things in real life. I loved the explanation in this video. Such great job!

[@MattTippinishere]

Best 16 minutes of the week

[@KK-pw2dt]

Why are there no clear explanations like this ever seen. Excellent and I will watch it again

[@Life_42]

This has to be one of the most profounding video I've ever watch! Thank you sir!

[@FlatTopRob]

I had been look for an intuitive proof of understanding and finally I have my answer. Thank you sir! Definitely subscribing to your channel

[@PhilHibbs]

The thumbnail, and the “technically” at 10:58, are technically wrong. √-1 is i, not -i. The “square root” operation has only one output. Now if you’re solving the equation x²=-1 then there are two answers, but that’s different.

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@IDKisReal2401]

Yeah, when counting x^2=1 you don’t go x=sqrt(1), x=+-1, but rather x^2=1, x=+-sqrt(1), x=+-1

[@Scubadog_]

This is my first time hearing about imaginary numbers and it made a lot of things make a lot more sense, thank you.

[@zahrotin1750]

I watched this out of curiosity and Im glad I did, I have learnt so much from this. Probably going to watch this again in the future!! Thanks

[@danieljohnson270]

I can't begive to tell you how helpful this video was! I'm teaching my oldest and the math is quickly outpacing what I remember, but you've made this concept very understandable. Thank you.

[@eduardobarreto6095]

wow this is absolutelly the best video of math i've ever seen on youtube, thank you

[@ojake801]

As soon as you said it, I got it. Nice storytelling. Thank you 🙏

[@jesper856]

Wow your video blew my mind...thank you for explaining something that no other math teacher has ever done

[@jacemandt]

Nope. We need to be able to treat f(x)=√x as a function. So √9 can't have two answers, because that's not how functions work. The symbol "√9" means "What positive number can you square to get 9?" I promise that you will find exactly 0 algebra books in which the graph of f(x)=√x includes both the points (9,3) and (9,-3), which is what would have to happen if √9=±3. It doesn't. √9=3. Only (9,3) is on that graph. The domain of that function is non-negative real numbers. √(-1) isn't well-defined, because -1 isn't in the domain of the function. Yes, of course there are two different numbers whose squares are both 9, and there are two different numbers whose squares are both -1. But saying that √9 has two answers is an abuse of the √ notation.

[@MuffinsAPlenty]

I don't think the video is _as_ bad as you make it out to be here. Yes, typically, we use the radical to represent the principal square root. It's best to stick to that convention, but it is simply a convention, not some deep truth. Also, there's a perfectly reasonable way to define a principal square root function on the complex plane, so √(-1) could be perfectly well-defined, depending on how you define things.

[@zachansen8293]

@@MuffinsAPlenty Even the actual thumbnail of the video is wrong it's so wrong.

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@Ta_Mi_M_]

I am watching this video from Bangladesh.. You are the best teacher ever......

[@taylork2874]

Such a great video, you explain everything so well!! Thank you for this🙏🙏

[@paddydad]

I love this video not knowing the number line explanation for multiplying with negative numbers has been bugging me for a while! I didn't know that I would learn this while randomly clicking on a video about imaginary numbers.

[@endwayphysicist]

The square root is a function, when I ask for the n-th root of any complex number z, you are not solving x^n = z. Instead, we apply the *function* f(z) = z^1/n which corresponds to a unique value, the principal root, therefore sqrt(-1) = i. On the other hand, the solution of x^2 = -1 is ±i

[@marcusgloder8755]

Exactly! 😎👍

[@dono42]

I was about to say the same thing. I was impressed by the video, but was a little surprised to find such an basic mistake.

[@carlosmanuel3876]

After all these years you gave me the right explanation thank you I finally understand why don't they teach this in school

[@zachansen8293]

This is *NOT* right and the poster doesn't know what square root means.

[@jimkaszyk847]

Thank you for this concept of Phase Shift! We use imaginary numbers to explain the new energy we discovered. It is negative entropic and therefore not from here or the law of entropy would not work. It has the properties to act at a distance, pass through matter like magnetic energy and create structure.

[@dougmarkham]

Just wondering if anyone knows which Mathematician or natural scientist first used degrees in arithmetic? Be interesting to read more about that...

[@marcusgloder8755]

Hello everyone, the square root of -1 is not i or -i, but i. Only i. The following applies: √(x²) = |x| This means: The result of a square root is always positive or zero, more precisely: an absolute number. This also applies to complex numbers. In particular, the result of a (principal) square root is not identical to the solutions (“roots”) of a quadratic equation. Best regards Marcus 😎

[@zachansen8293]

Thank you, this video was pissing me off

[@evgtro8727]

How positivity or the concept of absolute value is applied to complex numbers? Could you explain, please?

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@bowlineobama]

Good point. Context is important in math.

[@peterkings2012]

Square Root of 9 is not -3. By Pythagoras Square roots represents the. Longest distance on a right angle...distance as in absolute values cannot be negative.

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@largewallofbeans9812]

@@ottobass9193the square root (or really any root/exponent) always returns one number: the principal root. There is a difference between “square root of x” and “numbers that, when squared, equal x”.

[@ottobass9193]

@@largewallofbeans9812 that's right only for real numbers In set of complex numbers the rule are different from the real numbers Like when in natural numbers you can't go lower than 0 Geometry of complex numbers allows for rotations more solutions of the square root, which simply has no parallel in real numbers

[@largewallofbeans9812]

@@ottobass9193 literally no one says this. If anything, the complex plane only solidifies why square roots should only yield one value.

[@ottobass9193]

@@largewallofbeans9812 you think only with real numbers Complex numbers are more complex than real numbers They extended some functions and works in completely different ways Every math doctor can tell this Is not only about complex plane Just search "Fundamental Theorem of Algebra": In complex numbers any polynomial of degree n has n roots. If your idea is correct which is the positive solution for sqrt(-1+i)? The positive solution is clear for real numbers, is not so easy for complex

[@kernelsoe]

Please do explain more maths stuff and science alike! Thanks a lot for the awesome educational videos!!!

[@noeldc]

Mind blown! Subscribed!

[@n019n]

i saw "i" in a video and i was wondering what it meant. now i understand my entire recommendations section.

[@senthilpuliadi6599]

Sir pls make a video on the application ....I know this doesn’t have 1000 views ,but pls make .you make everything crystal clear pls

[@tuhocr]

thanks for useful video. Which software that used for creating this lecture please?

[@Sciencenme]

Thanks! Written using Microsoft one note. Screen captured using OBS. Edited using premiere.

[@zack_120]

Genius, the polar rule is creative! Keep posting more videos👍 Just curious, why 180°/2 (12:33)? without, doesn't 180° negate 1 so that √-1 = -1 ?

[@Sciencenme]

180° =-1, 90°=i , i is not equal to -1

[@whitesun264]

If maths is the framework / skeleton behind the visible world, does this progression on from a linear 0,1,2,3,4 etc hint at the existence of another dimension that we can't see?

[@nicdheen2365]

Beautiful explanation sir

[@moodz9217]

I don't usually comment at all but I had to thank you for this absolute perfect explanation!!

[@Sciencenme]

Glad it was helpful!

[@Reach3DPrinters]

VERY COOL! it makes a lot of sense, in an imaginary way. But legit, very nicely explained, thank you.

[@Life_42]

Just saw the list of videos on your channel; subscribed!

[@bowlineobama]

Something is bothering me. I see that i * i = -1. But, could it also be +1 as well? I really like your comment on it. This is how I came up with +1 as an alternate answer. See if you agree. Let i = Sqrt of -1. Then Sqrt (-1 ) * Sqrt ( -1) = Sqrt (mag 1 phase 180 * mag 1 phase 180). Then it is equal to Sqrt of (mag 1 * phase 360), which is equal to mag 1 phase 360/2 = mag 1 phase 180. This gives one answer, -1. I understand that. Since two -1s are under the Sqrt, it should have two possible answers. Correct? The other answer should be +1 as well. Correct? Because, adding one more phase cycle to 360, which makes 720 and then dividing it by 2 gives back to 360, which is +1. I would like your reply back please. Thanks.

[@cyberagua]

@bowlineobama > Let i = Sqrt of -1 That's not correct: √-̅1̅ = ±𝒊 = {+𝒊; -𝒊}, so √-̅1̅ × √-̅1̅ = (±𝒊) × (±𝒊) = ±1 = {+1; -1}, exactly as you pointed it out.

[@cyberagua]

@@bowlineobama That's correct that √-̅1̅ × √-̅1̅ = (±𝒊) × (±𝒊) = ±1 because √-̅1̅ = ±𝒊, not just 𝒊 but 𝒊 × 𝒊 = -1 only just like (-𝒊) × (-𝒊) = -1

[@abrazilianandanamericansol580]

My professor asked to prove why the square root of negative a is negative a. I really liked the video! Will come back as I homeschool by child!

[@abulfadlahmadi]

Thank you sir, for increasing my knowledge

[@chuckeycheese3215]

Outstanding Video

[@karamashi520]

excellent explanation thank you so much

[@frankman2]

this is beautiful and all, but I have a question: didn't -1 x -1 = +1 come before imaginary numbers? If so, how did they justify it then?

[@SergioC7799]

Professor, but square root of 9 is equal to l3l = 3. How square root of 9 can be equal to -3? For me this is like if 3 = -3. Thanks for the excellent video.

[@Sciencenme]

There are two roots (or correct answers) to the square root of 9, that's why sqrt(9)= +/- 3. Think about the quadratic curve, as long as it opens past though the x-axis, you will always have two points that touch the x-axis (or two real answers). If you rephrase your question to y=x^2 - 9, and try to solve where y=0, you will have two solutions to x. I'm not a professor, but thanks for the acknowledgement :)

[@SergioC7799]

@@Sciencenme I would solve it this way: x^2 - 9 = 0 => x^2 = 9 => sqrt (x^2) = sqrt 9 => lxl = 3 => x = 3 or x = -3. sqrt 9 continues 3.

[@benja2998]

​@@SergioC7799 i solve it this way 9 ÷ 3

[@chickencat101]

​@@SciencenmeThe radical sign indicates the non-negative value only, as it's a function -- the principal square root. You would need "plus or minus" on both sides of the equation to have a true statement.

[@ottobass9193]

​@@chickencat101Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@z.m.4331]

Amazing explanation :D this video is a gem (as a tip, you really should set up a Kofi / Patreon / KZbin "thanks ($)" button!)

[@Sciencenme]

Thanks for the tip! I do have a patreon, but it seems neglected: www.patreon.com/scienceNme

[@andreasg.9997]

So would i^2 equal negative 1 then?

[@रोहित1]

Such an underrated video i feel so bad 😩 thank you so much🥺😭❤❤❤❤

[@mathepunk]

By some definitions of the square root the square root of 9 would be ±3. However, roots of positive real numbers have been defined as positive. For example solving the radical equation (x+4)^(1/2) = x-8 by squaring both sides gives rise to a quadratic equation with two solutions x1=12 and x2 = 5. When inserting the solution x2 = 5 into the original radical equation the right hand side equals -3 and the left hand side equals square root of 9. If you say that square root of 9 equals ±3, x2 = 5 would also be a solution. This, however, is not the case. Square root of 9 is defined as +3. You can solve the original radical equation for example with wolframalpha and you will find that there is no solution x2 = 5 the reason being that square root of nine does not equal -3. The square root of -1 usually is taken as +i, but there is no clear cut definition as is the case for square root of 9. In this case you are definitely leaving the realm of real calculus and things turn murky. For exampla the square root of 3 - 4i can be taken as ±(2 - i).

[@duykhanhpham5803]

thank you so much, i love this explanation

[@prantodatta904]

So much helpful and interesting

[@JoepertN]

this was so helpful tysm

[@ashishkumarnath3563]

Beautiful 👏👏

[@GSU_Panther_Nation]

Adding and subtracting angles should be included as part of mathematical education that prepares students for trigonometry in precalculus

[@faithwangui7200]

Thank you sir. Do you have videos on logarithms and derivatives? I loved your clear explainations.

[@Sciencenme]

Not specifically on logarithms, but applications of it, yes: in dB system: kzbin.info/www/bejne/majToGSMZrh-e8k in analyzing data correlations: kzbin.info/www/bejne/iWjcnHiVapaLr8k I did create a calculus video 18 years ago, but the pace is pretty slow. I'll upload it if you do want to see it.

[@snarevox]

@@Sciencenme hi, i have a question... when you were explaining multiplication with the adding of angles, you said something like even an angle of 1080° will end up being positive since it equals 3 full rotations or a phase shift of zero at the end. my question is, wouldnt all real numbers have an angle of either 0° or 180°?? and if thats the case, when would you ever come across an angle greater than 180°? like the 1080°...the only time i can think of is multiplying 2 or more numbers together with at least two negative numbers involved? am i way off or ? i guess what im asking is like what even is 2 with an angle of 720°? i know its the same as just 2 with an angle of 0° ...so if 0° and 180° are the only angles a real number can truly have, are these larger angles only there to be divided by 180 to find out how many rotations its going to make to figure out if its going to be positive or negative??

[@snarevox]

@@Sciencenme also, i was playing with my calculator and found out that -2 squared is -4 not 4 like i thought -2 times -2 should be.. then i also found that (-2)squared is the 4 i was looking for. do you have a simple answer for why this is? sorry to keep bugging you.

[@Sciencenme]

@@snarevox the point is, if you limit your output range to 0°~360°, you might miss out on other valid answers. Two waves are in-phase if they are shifted 0° or 360° from each other; however there are infinite possible solutions including 720°, 1080°, but also -360°. In certain applications, other possible answers should be factored in, otherwise erratic things may happen. It's kind of like testing software code, if you're into that.

[@Sciencenme]

@@snarevox i cannot account for all calculators, but scientific calculators have two minus buttons - one is subtract, and the other is 'negative'. If the IC for the calculator has been programmed correctly the 'negative' should be associated with the value typed in e.g. "(-2)" as opposed to "-(2)"

[@bigtiger1964]

sqrt(9) = 3; by definition the square root is only the positive solution of x^2=9. On the other hand the equation x^2=9 has got two solutions: sqrt(9) and -sqrt(9), often written as ±sqrt(9). If sqrt(9) took also the negative value, the ± sign would be redundant.

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@jeevesponzi5257]

Thank you soooooo much!,,

[@CJDAM]

How did I never learn about phase shift wtf

[@AyushKumar-ue2oz]

Awesome , i will be starting a new jourmey with u

[@weldonwebb889]

Thank you for your presentation. You are amongst others have simplified this concept such that i was on the precipice of grasping. George Gamow uses this in his book. To be blunt, This should be taught to the intellectual appropriate or simplified. Thare is a called Wff'n'Proof which teaches logic at the age of five. Is there way to teach this through Gov't(public) or private schools?

[@ness8661]

thank you ♥️

[@Pedrooo19]

WOW this is really blown my mind

[@nuwindudamsara7147]

You are a great teacher sir💛🎉Thanks for your effort❤❤❤

[@Sciencenme]

It's my pleasure :)

[@googolian]

All numbers, including negative numbers, travel only in one direction on the number line ie from left to right

[@the_linguist_ll]

That only works for the real numbers, higher algebras lose ordering

[@isaacprosper6295]

this is awesome i subscribed immediatelt

[@PatrickDuffy-u3s]

In electrical engineering, "i" was already taken, so it's "j" for us.

[@JimmyMatis-h9y]

"i" always smoked a "j" before doing my complex analysis homework.. 🤭

[@weldonwebb889]

George Gamow "1,2,3...Infinity uses complex graph to find treasure

[@RudeReaper]

Here from watching "the platform 2"

[@TheSmokeySky]

BOOOM there goes my mind

[@ZMDD2017]

WHAT THIS IS SO COOL???

[@zachansen8293]

12:10 Probably should have looked up what the square root "symbol" means. sqrt(9) is *NOT* +/-3. It does NOT mean X^2=9 solve for x. That's why you say +/-sqrt(9) if you want +/-3 as the answer. Imagine making a video about math and not even having ever seen the quadratic equation. If this video's definition was correct, it wouldn't have +/-sqrt(...) it would just be sqrt(...)

[@ottobass9193]

Square root operation have one solution only in set of real numbers, not in set of complex numbers

[@kuroraven4128]

The fact that video finally has what i'm EXACTLY looking for is just 😔✨

[@zachansen8293]

This video is *WRONG* as it fundamentally doesn't know what square root means... so the entire explanation is wrong.

[@autoauto2000]

❤❤❤ grandios🎉🎉🎉

[@evgtro8727]

That also means that i can be both -sqrt(-1) and sqrt(-1).

[@Sciencenme]

i or 1/i, if you want to avoid that negative sign debacle as read in the comments.

[@evgtro8727]

@@Sciencenme Sorry, I did not understand you. My statement was about i, not 1/i. Could you explain? I have just made a simple algebraic conclusion from your video. If sqrt(-1) = i and sqrt(-1) = -i then this also means that i = sqrt(-1) and i = -sqrt(-1) at the same time. Surprisingly, this is not contradictory and is the key point of the definition of i.

[@Universe4-U]

Nice now i have new better way to imagine the imaginary numbers.

[@zlosliwa_menda]

Wait, there is a country where negative numbers are introduced in junior high?! That's freaking 2nd or 3rd grade material! Wtf?!

[@xa5468]

Mind blown rn

[@irwanpross1493]

Gila,. Mantab boss

[@wraith6776]

I thought you'd just say that math needed a symbol to equal the square root of -1. The same way pi was chosen to approximately equal 3.14. You really explained how the calculations worked out. Great video.

[@zachansen8293]

Pi was not "chosen" pi was observed.

[@gregedwards1087]

Mathematics is both crazy and logical, like most languages.

[@RalKrish]

Oh my God bro you are mind blowing learning with intuition is more important than learning in tuition

[@zachansen8293]

But the math is *WRONG* sqrt(9) is ONLY 3. It is never +/-3

[@dag410]

🤯🤯🤯 dude!

[@rohannbacolod7814]

Yes mind blown

[@undernetjack]

Thank you

[@benquinneyiii7941]

Real number line

[@joeylo2302]

2/3:smaller than one Him:2/3=1.5

[@raytsang250]

√-1 is i, by convention, instead of -i. There is no point to debate on this convention. Come on. Furthermore, a single mathematical expression always gives one single value only.

[@ottobass9193]

The real definition of i is i^2 = 1 and not the square root

[@ninjapirate123]

I thought square root of negative number doesn't exist

[@JimmyMatis-h9y]

Doesn't exist in the set of Real numbers* Welcome to the Complex plane! (Complex as in made of more than one part not as in complicated)

[@BuleriaChk]

there are no negative numbers; they only exist as differences between positive numbers. -c = a-b. b>a b-c = a a=a a-a=0 If there are no negative numbers, there are no square roots of negative numbers. i:= sqr(-1) i^2= sqr(-1)sqr(-1) = sqr[(-1)(-1)] = sqr(1^2) = 1 -1

[@theophonchana5025]

Square root of -1 = invalid input

[@supersonik76]

Cant believe I'm just now learning about phase shifts. This is unreal.

[@JonathanMarcy]

This still doesn't explain it to me. Negatives don't have square roots. Creating another dimension on a number line doesn't actually solve it. That number line still has a positive and negative relationship. I can understand numbers being offset from the main number sequence in degrees of rotation, that's just a way to represent multiple dimensions on one number line. But changing the principles of mathematics to essentially ignore a negative I don't. Essentially all your doing is creating a way around acknowledging a negative. Why does this solution have its own rules? For instance an explanation I've seen for the square root of negative 8 is: -1×√4×√2 -1×2×√2 -2×√2 But -2×√2 is not only not a number times itself, it also doesn't equal 8, let alone -8

[@sailexw6414]

A negative multiplied by a negative is more easily understood using language; "I want you to do the reverse of taking away 2, 3 times"

[@isaacprosper6295]

if maths was explained like this to me in school i would have achieved more than elon musk

[@zachansen8293]

If it were taught like this to you, then you would have been taught wrong. This video is based on a complete misunderstanding of what sqrt means.

[@theophonchana5025]

Square root of -1 = error