funny 🤣🤣🤣

Fuuuuck

Makes sense. It's complex and perpetually negates your progress. Our governmental system only makes sense in our imagination.

🤣 they go straight to imaginary numbers

_Seems fair_ 🍷🗿 -Some Govt.

Yes many people in comment section do not know this

You are wrong💀💀💀💀if you don't know something then at least fact check yourself first before commenting online bruh embarrassing

​​​​​​​@venkovic No, OP is correct, multiplicative property of roots holds only for real numbers (in general, cube roots or higher are included), and you can't take a square root of a negative number in reals, so in reals √-2 is already wrong. "But we talk about complex numbers". Ok, if you insist, then saying that it holds for complex numbers "as long as at least one number is non-negative" is also wrong, because for any complex number z (and that includes 2) something like z≥0 (non-negative that is) doesn't make sense, since complex numbers are an unordered field. And also unless you specifically say that √ in your √-2 stands for the principal root you can't even say that √-2=i√2 (it should have ± in the front), it will be wrong. But for a principal root multiplicative property also doesn't hold just like it does not hold for an algebraic root in complex numbers. So all of this is completely wrong, this is not how you deal with complex or real numbers, for the love of God stop trying to mix the two. Basically, you here try to create your own function from real to complex numbers, use for it a symbol (√) that is already stands for like up to three to four functions and pass it as standard mathematics. This is not how any of it works

​@@kushaljoshi3862 Bruh, he's 100% on point, stop learning maths from some twats on YT, grab a good textbook on complex numbers and read it

No. It applies when ANY of a and b is >=0, because negative number under square root is fully legal notation (and means imaginary result).

lol

(North + wall) × leaf × wind - [summer + winter] /30people! Well, I guess you are right

Lmao, should have been the proper term for quaternions

Works best in hungarian 😂

All numbers are imaginary.

What is a complex plain. Is it X or Y?

​​@@sandromaspindzelashvili5767 A plane (C) consisting of the set of all complex numbers z of the form z = x + iy, where is the real part ( your every day real numbers ), y is the "imaginary" part, and i is the imaginary unit ( along the y-axis ) with the property i^2 = - 1, or i = sqrtc( -1), where sqrtc is the COMPLEX square root function. You can think of it as an expansion of your normal real numbers line into a plane to both sides of it - i.e. numbers that lie outside it ( or on it ). These complex numbers and complex functions have a lot of interesting and very useful properties - for instance in calculating otherwise difficult integrals and sums of various series, solving differential equations, AC power distribution, spectral analysis of analog and digital signals ( Fourier series & Fourier transforms, Laplace Transforms, analog and digital filters, control systems ), acoustics, physics ( oscillations ) and quantum mechanics etc. They also combine the properties of numbers with the properties of vectors (additions, subtraction, "length" ( magnitude ) - multiplication & division both involving a scaling of the magnitude and a rotation of the complex number ("vector") in the complex plane. All our normal numbers and operations on them are really just special cases of these complex numbers and these extended operations on them - it's like some sort of higher dimensional shadow world, so to speak 😉 - and they do actually make mathematical sense - satisfying certain conditions - , even though they may seem very weird at a first glance 😬 ). There are several really cool videos about these topics on KZfaq, so I suggest that you watch those, as they explain these fascinating concepts far better that I am able to do here - also graphically of course.

hey Schrodinger thank you for the electron wave equation 😊

​@@sohanchowdhury1312also thanks to wolf for his bite😢

And this explanation is still complex to us elementary learners but I love the detail and time you put into this

i am doing coputer engineering too first semester math first week of the first semester

Exactly

Dude how did you get into the college without knowing this?

@@naikubaid you can forget this sort of stuff pretty fast.

​@@BRAINSPLATTER16you have never studied a single day of your life, haven't you?

Wooooooooooooo That's a hell of an effort you've put in there.....i appreciate it.👍

You do not have to, you could also just embrace the square root as multivalued as god intended :3

I came to the comments to make the same complaints, then saw you’d already done a great job!

Lets be real.... i=ln(-1)/pi ;)

So applying the "privileged individual" status of -1 that I learned from the video, I can prove that sqrt(16)=-4. sqrt(16)=sqrt(-(-16))=i*sqrt(-16)=i*i*sqrt(16)=i^2*4=-4

Excuse it's for mathematicians not for normal people

Imaginary numbers are a tool used a lot in physics and esp in engineering to solve real world problems which would otherwise require rigorous calculations, high school sets u up with the basics, college is where u apply them.

Relatable 😂

me too, at least I thought so... I had a stroke and had to learn everything again. I needed my fingers to add up. But this I remember.....

whats the secret to getting rec letters?

Thank you~

if the video is correct, then 4=-4 is also true: 4 = √16 = √[(-2) x (-8)] = √(-2) x √(-8) = -4 The real result is both 4 and -4.

​@@gonzalobarragan8076 This is incorrect because the property √(ab) = √(a) × √(b) can only be applied when both a and b are non-negative. √[(-2) × (-8)] ≠ √(-2) × √(-8)

​@@gonzalobarragan8076 the property '√a•√b = √(a•b)' you used is only applicable for if a and b ≥ 0. Since in the given question, a and b are negative which means a and b < 0. Therefore the property cannot be applied in this case You should have paid attention in these small things also.

I haven't learnt really anything about iota, but if it equals -1, shouldn't i^2 therefore equal +1? -1 x -1 = 1. What am I missing?

True, I totally forgot that sqrt(-8) = sqrt(4). Very good

​@@fashnekwdym √(-8) = √4?

no? theres no unknown integer in the equation

I agree, square root of 16 is +/-4, so it's not correct

@@enchantedhamburger8934 what’s 4*4, ok good, now what’s -4*-4, u see it now don’t you

@@masteroogway2853 ye i know what you are reffering to, i may have just missunderstood this, but i never learnt that a square root is equal to +- of said number

But what if it's another square root.

it doesnt make sense

a shortcut reminder i would say

@@dekirou320 it’s factorizing, but with the negative symbol only

don't memorize this, it's wrong

When -1is taken outside the square root it becomes positive because -1 * -1 =+1

​@@detac1405bro wtf lmao? Have you ever studied complex numbers? i is sqrt(-1) not -1

Glad to share the knowledge

Very well explained!

You're most welcome. Thanks for the nice comment.

😂😂😂😂😂 love this

If you put 4 apple debts in a square and take away 1 line of apple debts, the whole thing will somehow convert into imaginary apples

Thanks

√16 is only +4. Not ±4.

​@@blankspace178 no it's not, sqrt(x) is a function from R+ to R+, hence the result is only positive.

@@blankspace178 I study Maths at uni, maybe this will clarify it for you: x^2=4 => x=±2 But √4=2

@Blank Space no of course it doesn't change anything.. but it's just a definition to avoid confusion. You can do your own maths with your own definitions, but I'm gonna stick with the ones I learned :)

@Blank Space "Although the question mentioned at the beginning has two solutions with different signs for even-numbered root exponents and positive radicands, the notation with the root sign always stands for the positive solution." This is from wikipedia, but I do understand your thought process, and it really doesn't matter, run with whatever you like. This will be my last reply, have a good one.

One of the rules in mathematics says that you can multiple root of a negative number. You have to first change it into a imaginary number by taking out the negative.

@@mrhtutoring doesn’t the square root of 16 have 2 answers? 4 and -4? So the answer here is -1 * those numbers meaning the answer remains 4 or -4?

@@overdose8329 Square root of 16 is only +16. When you have an equation x²=16, x=±4.

@@overdose8329 the square root function is defined in such a way that it only gives positive outputs. A common misconception is that it gives two answers, but that‘s simply not the case. After all, there is a reason we maticilously write out +-sqrt in quadratics and Not just +sqrt.

Isn't that because if you have for example √9=3 as a result from √x²=y, then x is either +3 or -3 since sqaring either results in 9, and y is thus ±x as a result of two values for x. It doesn't mean that y is negative. √x² = |x| according to what I was taught and I just checked that wolfram alpha affirms that

​@Retired Bore go ahead and simplify the quadratic equation then, I think you would find it contains an unnecessary ±

@Retired Bore as a software engineer, I don't even know what you mean by "mathematics for programming". The square root function is 'defined' to yield the positive solution

@Retired Bore there are multiple square root implementations for computers and so there's no "the square root function". And no, I'm not mixing them up. Feel free to look up the definition of square root, or functions in general and see they don't have multiple answers.

​@Retired Bore here you go "Every nonnegative real number x has a unique nonnegative square root, called the principal square root, which is denoted by √x" I pulled it from wikipedia "square root" so you can go and argue with the sources there instead of here.

because in definition, the condition says that number inside root must be positive.

That's true in basic algebra. However, algebra 2 and on, students are taught imaginary numbers. √-1=i And √-9=3i. 댓글 감사합니다.

Kamsabnida

That symbol means you only hold the positive not the negative. So -4 is the only answer

@@isjosh8064 No, both + 4 & - 4 are actually "solutions" to this expression - you need to check all the 4 ( = 2 x 2 ) possible combinations of the two (primary) branches of the complex square root function.

@@isjosh8064 Which symbol? 😉

@@Bjowolf2 When you say “solutions" I imagine you’re thinking of an equation like: x^2 = 16 where the solutions are +/- but those are what x can be to satisfy the equation. But /x = -5 has no solutions even though -5 squared is 25. The definition of domain of /x only allows a positive input and returns a positive input.

@@isjosh8064 Yes, for ordinary (real) square root this is true, but for COMPLEX square roots, which he is clearly working with here ( with -4 & -9 under the square root signs ), different rules come into play, since they have two possible (primary) branches. ( These socalled multi valued functions are permitted in complex analysis ). So sqrtc(-4) = +/- i x sqrt(4) = +/- 2i And likewise for sqrtc(-9).

Great 👍

i = sqrt(-1) sqrt(-2) = sqrt(-1)×sqrt(2) Not sqrt(-2) = +/-sqrt(-1)×sqrt(2)

@@moonchock4390 but y=sqrt(x^2) has two solutions. y=+x and y=-x. For sqrt(-1), this yields +i AND -i.

​@@YouHaveToBeTheChange, nope. The square root is a single valued function, defined as the principle branch of the inverse of the square. You've clearly learned that you need to remember the +- when solving a second degree polynomial equation, which is correct and important but doesn't apply to square roots.

​@@YouHaveToBeTheChange are you serious lmao

@@TheGlassgubben Sorry to burst your bubble, but the square root of -1 has indeed two values: +/- i.

What??? The whole point is that you can NOT use the square root factoring rule unless the factors are positive real numbers. His method of evaluating square roots of negative numbers is perfectly valid.

​@@Tulanir1 I did exactly as the video. For him ✓-2=i✓2 and that uses the same rule I'm using. If you can't truly use that rule unless it's a positive integer, then he can't even answer the problem.

​@@kevinmartincossiolozano8245 surely when you go from sqrt(i²*-16) to i*sqrt(-16) youre skipping an intermediate stage where you get sqrt(i²)*sqrt(-16) which means youve factored with a negative number

@@PotassiumLover33 That's exactly the same step done in the video. Because ✓-2=✓2✓-1, which means, they have factored with a negative number too!

look into imaginary numbers

Thank you!

so true

are you dumb? radical of a negative is a complex number

In the world of real numbers, sqrt(x)*sqrt(y) = sqrt(x*y) it's only valid if x and y are positive real numbers or at least x or y are 0. So, sqrt(-2)*sqrt(-8) = sqrt(-2*-8) is not valid, and so, having no solution at all. In the complex world, well... you do what was done in the video.

Doesn't dealing with the square root of negative numbers by definition mean we're dealing with complex numbers?

@@spectreone In the world of real numbers sqrt(16) is both 4 and -4. Positive real numbers have 2 square roots that are also real numbers.

no , square root can only give positive values

​@@dftsxy5 but -4×-4 is 16. So -4 is on of the answers. The other answer is obviously 4

@@soroushhaidary7934 square root is a function which means you have only one output (positive 4 in this case) you would be right if it was an equation like x²= 16 , only then x=4 v x=-4

@@soroushhaidary7934 You can also see how there are contradictions when you say sqrt(16) = +-4 Let’s assume sqrt(16) = +-4. Then because 4 = sqrt(16) and sqrt(16) = -4, our conclusion is that 4 = -4, which is false.

no

Incorrect, √(-2) × √(-8) ≠ √(-2 × -8) because the expression √(ab) = √(a) × (√(b) only applies when both a and b are non-negative. Since both a and b are negative in this situation, we have to use complex numbers to solve the expression. √(-a) × √(-b) = i√(a) × i√(b) = i^2 × √(a × b) Since i^2 = -1, and √ only returns the positive root, the answer to √(-a) × √(-b) only yields negative solutions. The end result is only -4, +4 is not a solution.

Yes but the √ symbol means positive root. Eg quadratic formula always include +-√ not just √

By the graph of the root it always greater than zero

√16 is only +4. Not ±4.

Not correct. The square root of 16 is 4. -4 is not the square root of 16. -4 is one of the solutions for the quadratic equation x^2=16. Square root of 16 is a number and a number cannot assume multiple values while the equation will have two solutions

It's specifically √x not x^2=y , the answer in second one is x=±√y Where √y is the principal solution x^2=y , This is done because √x is defined to be one-one function, for each x you put you can get only one answer, the negative ans is discarded, this principle soln is put in x=±√y.

When we ask for all the square roots of 16, it's ±4. But when we write it with a square root symbol such as √16, it's only the principal square root, which is +4. Hence, the reason when we simplify expressions such as √4+√9, we simplify it to +5, not ±5 or ±1.

No, √x is defined as the principal value of x^(1/2), which has 2 roots. Now to take √i, we need to take the principal value of i^(1/2). In exponential form, i is e^i(pi/2 + 2pi*n), so i^(1/2) = e^i(pi/4 + pi*n). For √i, we set n=0 to take the principal value and get e^i*pi/4 or (1+i)/√2.

It’s not lol. That’s only when you’ve introduced a square root yourself.

More like education system failure, √n is always positive lol.

@@omnipresentcatgod245 he probably meant a square power

​@@Dra3oonBoth are correct but only because we included complex numbers in the beginning. If you only have sqrt(16) then is 4, but if you have Sqrt(-16) × sqrt(-1), then its both -4 and 4.

@@goldbeni thank you

but √(x²) = |x|

4÷√-2 = -2√-2 = -√-8 incorrect -4÷√-2 = 2√-2 = √-8 correct So, the answer is -4.

sqrt(16) = +4 Not +- The solution of x² = 16 is x = +-sqrt(16)

​@@ComposedBySam your comment make not sense.

@@IoDavide1 square root by definition refers to the positive root of x²=16. square root as an operator itself doesn't give you both the roots. Because if that was the case then we couldn't raise both sides of an equation to fractional powers. Suppose sqrt(1) =+-1 1=1, square rooting both sides (raising powers in both side to 1/2) We would get 1=-1 as a solution. Hence by convention 16 to the power 1/2 (ie. Sqrt(16)) gives only 4 (the absolute value)

@@ComposedBySam this second comment make less sense then the first. The square root has not determined sign, so you have always two results: + and - You seems still confused with primary school definitions

@@IoDavide1 then do a favor. Try harder to understand what I wrote. And if you cannot google it. the solution of x² = 16 and square root of 16 are different things

Normally, it depends on how you _define_ the √□ operation. And as long as you explain the full chain of reasoning and it starts with a reasonable definition, I'd say the reasoning is valid. Of course, if you have a test-like question where you have to produce the same text as what's written on the sheet labeled "correct answers"... Then it's a game of either "memorize this particular textbook and not another textbook" or "guess the answer". The answer here can be any of "4", "-4", "±4", "expression is ill-defined".

Square root is a function, meaning it can only return one result, the positive one (for real numbers) You need to take both values when you have x²=4 x=±2 but it's because x²=4 x²-4=0 (x+2)(x-2)=0 x=±2

@@gabrieleymat6332 A "√□" is a symbol. Nothing internally inconsistent would happen if you were to define it as "all numbers that produce □ when multiplied by themselves". For example, in a residue field modulo 7, 3²=4²=2. In this case, trying to find a "principal" value of √2 would be very much futile. And even if you insist to take a positive value for real numbers (why? because a particular textbook said so? how you would arrive to this conclusion if you were inventing all math from scratch?), you still have the same problem with complex numbers: i and -i don't differ in any reasonable way. In some cases, it may be important to take the value in "the same direction" for both roots: you can write √-1x√-1 = ixi *or* √-1x√-1 = (-i)x(-i) but not √-1x√-1 = ix(-i). Like when solving cubic equations, you get an expression ³√a+³√b and then you have to sum the "right" pairs of cubic roots of a and b.

Nobody denies the square roots of 16 are 4 and -4, but √ does NOT yield the square roots - it yields the POSITIVE square root. If you want both, you need to write this x² = 16 -- > x = +/ - √16 because √16 = 4 There is no discussion on that - it's simply using the defintion of the function/symbol √ If you use a different definition, you're solving a different problem. If you solve a different problem, you get a different answer.

And you pretend to be a "mathematic fanatic", what a joke !!! And the Earth is flat, that's it ??? Lol

@@extrams0 forgive me for discussing it :P bad definition is bad

Simple. No money? No purchase. Avoid credit at all costs in your whole life!

@@BlackHoleSpain Yikes, I can’t think of anyone, literally no one who’s successful that doesn’t have an amazing credit score. Paying cash is fine if you want to stay poor.

Everyone makes jokes like this, but in practice, there are second order differential equations that appear in the study of finance and economics. Their solutions sometimes require finding the roots of a polynomial which are often complex numbers.

@@cdula26 Credit score literally doesn't matter once you're making consistent reliable income. If you have cash, people have goods and services they want to offer you.

@@acex222 Even my real estate mentor who brings in a net income of $800,000 a month has a great credit score and doesn't pay cash for his 500k cars or 5 million dollar yacht. It's better to finance things and use all the cash you would have spent on buying more assets and investing the rest.

Thank you for all the great comments.

Yes. AI is going to have the whole math system held in perfect order. No more rounding off numbers.

Yeah, so the answer is technically ±4. Or maybe we just missed something

They are radicals which have only positive solution That's waht i understand atleast

@@georgesas7090 no the square root is a function meaning every input has one output. You are thinking of solutions to x^2 = 16. Which is different than sqrt(16) as sqrt(16) and -sqrt(16) Are both solutions to that equation

​@@ChessThingsOfficial It's not, the square root funcion is defined assigning only positive values.

@@eldins1813 Oh okay thanks!

Thank you

Maybe he forgot his mathematics because he stayed in murica for so long. 🤣🤣🤣 √16 = + -4

​@@BlackSakura33 maybe you forger that square root of as positive no. is positive For eg. √4=2 and not -2 But if you put as x²=4. Then x has values as 2 and -2 And please stop hating and stereotyping other countries, coz we are also being hated and stereotyped at international level Spread peace

​@@kartavyasharma7266thats only true when x is a real number. If you inclide the complex plain, both negative and positive solutions are real solutions. And because we included the complex plain at the start, we cant ignore it. Look at the start of the problem, sqrt(-2)=i×sqrt(2) AND -i×sqrt(2) Same for the 8, and then we can see that theres actually 2 solutions, + and - 4

You only take the positive root when solving squares

My pleasure!

Square root of 16 is only 4 which is why this question is tricky

@@Adventurer-te8fl no, sqrt(16) is + or - 4 actually

@@let1742 what ??? Sqrt is always positive, sqrt(16) is equal to 4 only

@@let1742 Completely incorrect a square root cant be negative. Its always positive. So it is 4. Thats why imaginary numbers get used.

@@BlackCat-fx9kb you are right that sq root should be positive but here you see ✓16 = ✓(+4)^2 and it can also be ✓16= ✓(-4)^2 therefore they both are positive only. So sq root of 16 will give +4 or -4.

This is actual maths wdym

@@HatterTobias it's a joke I know

No it's just +4

It is a constant discussion, usually mathematicians say its only +4 and physicians say +-4

​@@thelord9099(-4)^2 = 16 so √16 = -4, and the same way for 4^2

The square roots of 16 are indeed those, but the symbol used, √, is defined to give only the principal root, the positive one, so he is actually correct, and I do not believe there is any discussion in the math community about this. Furthermore, saying the answer is +/-4 would also be incorrect because +4 is not the correct answer.

Can you give an example of "set of reals?" I need an example to answer your question

-4 is itself a real number. so it is in the set of reals.

​@@sytherplayz but the √-4 isn't in the set of real numbers

@@lazygod1854 √(-4) is not in the set of solutions.

@@sytherplayz yes but I think "solutions" isn't the right word in this case because we aren't solving a polynomial equation but nonetheless we both are saying the same thing that the √-4 doesnt belong in the sets of real number system.

Actually he quite messes with exponentiation properties. sqrt(-2)·sqrt(-8)=sqrt((-2)·(-8)) . I wouldn't trust my kids to this teacher

​@@ovidiucroitoru2290that property only works for square roots of positive numbers Also for the OP : no, √16 = x and x^2 = 16 is not the same thing, x for the 1st is 4, while x for the 2nd can either be -4 or 4. The guy in video talks about the 1st

@@ovidiucroitoru2290 you do realise sqrt of negative numbers are undefined in real numbers which is why complex numbers exist right

​@@ovidiucroitoru2290You would get the square root of -16 with that result which is undefined in real numbers. Hence why one needs to express it as a complex number like he does in the video