And shout out to all the people still saying ± 6 because "the square root of 4 is ± 2". Really simple, x² = 4 then x = ± √4 = ± 2; but √4 = 2 and 2 alone. The ± symbol is always placed before the √ symbol when inverting a square.
@@WombatMan64that’s true if we’re inverting an x^2 term, but from what I understand the radical symbol denotes the principle root which is the positive branch, so -6 should be the only answer in this case. You could probably rewrite the thing with an x^2 term to get two solutions if you changed a bunch of things.
If we are taking the square root of negative numbers, then aren't there only complex solutions? The only square root of -4 is 2i and the only square root of -9 is 3i, correct?
Imaginary numbers aren't mathematically real, but in all fairness, the sets have a terrible naming system. Gauss himself suggested that imaginary numbers be called "lateral numbers" instead. Obviously, in a modern mathematical stance, imaginary numbers are not in the set of real numbers. But from a more debatable, linguistic standpoint, "imaginary numbers are real."
A real number may be defined as a + i*0, where a is a real part and i*0 is an imaginary part. That way there is a connection between real and imaginary numbers. Naming conventions are a bit misleading and don’t mean “real” as in “exist”. Would you call -5 a number that exists? Outside of math negative numbers can signify loss or deficit, but you might as well use positive numbers to measure the size of it.
Imaginary numbers show up in very real ways in electrical engineering. By real, I mean "physical things you can actually show on a meter or display". For example, the impedence of an antenna at various frequencies.
@3:06 The rule that sqrt(x) times sqrt(y) = sqrt(xy) - why do NO grade school or high school math teachers EVER bother to mention that this only applies to positive numbers???
It shouldn't be true. Start backwards: sqrt(36)=sqrt(6*6) or sqrt(-6 * -6), so why not sqrt(9*4) or sqrt(-9 * -4)? If this isn't true, then all square roots would be positive, but as we all know the correct answer to any square root is pos or neg
This is a question of definitions, and I do not feel that the definition given here for the square root of a number has been sufficiently justified. Yes, you CAN define the square root as a function that gives back a number with a positive/zero imaginary component, but... why? What reason do we have to discard the other root, if different? Why do we need to find only one answer to the problem given? Isn't it better to find ALL answers to the problem? In the case of Brilliant's question, the choices given make clear that only one answer is desired, and IN THAT CASE, I can get behind -6 being "the correct answer." But the poll question implies, through the presence of the plus-or-minus 6 option, that having multiple answers is valid, and as such, multiple answers should be accepted (if multiple answers exist, which in this case they do).
I like how you explain all the wrong answers and how people might have gotten to them -- I think it clarifies a lot of important misunderstandings, and even though I got the correct answer, I still learnt a lot (especially from the greater clarity in definitions, which is the most important thing in maths imo). thank you.
Maybe you're better at math than you've been giving yourself credit for. I bet if you keep watching this channel, or look for other opportunities to learn math, that you'll get pretty good.
I'm not very good at math either, and I figured that as sqrt(4) is 2 or -2, sqrt(-4) would also be 2 or -2, so I completely ignored the option to multiply the -2 with the 3 or the 2 with the -3 and ended up with 6... Like I said, I'm not very good at math :P
I think it's disingenuous to go: "we should definitely be considering complex numbers" in one breath and "let's ignore the other half and just consider the principal square root" in the next. Also, who decided Wolfram Alpha was the final authority on how to do math. It's a calculator. An advanced calculator, but a calculator nonetheless. And decisions were made when programming it. Without an understanding of what went into making that decision, I'm not quite ready to say they didn't make a mistake. Edit: All these justifications about established convention and the definition of the square root operator are great and all, but the fact remains: he gave none of them. He gave a reasoning about it not being a function (by the strict definition of only producing one output), but that felt like him deciding on his own that it had to conform to that standard. After all, if you were solving for a value, you wouldn't care that it doesn't match the definition of a function, why would you care here? He said nothing about established conventions.
Wikipedia also defines the √¯ symbol as the principal square root and if you want both halves you'd want to write ±√x̄, I'd guess it's a consensus in the entire math community, so yeah
@@yukimoeI suppose that makes sense in the same way we write "6" rather than "+6" when writing positives, still think that it feels somewhat arbitrary tho
I do agree, somewhat - the root symbol typically means the principal root. Which is what the wolfram alpha function uses too. But I'd suggest if you take the secondary root for one, you should for the other as well, and that's still -6. I do think +/6 would be a valid choice though, with an explanation. But like undefined, you're making an assumption that most people wouldn't make, so you'd want to be upfront about that when you answered. Like, no real solution is a fine answer too. As long as you say there is no real solution, not just undefined.
7:47 No. Not everybody. Complex numbers are not ordered, so predicates () are not defined for them, you can't universally choose (i) to be the principal sqrt(-1). In some schools it's taught that if z = r * (cos(t) + i * sin(t)), where -pi < t
"Complex numbers are not ordered, so predicates () are not defined for them, you can't universally choose (i) to be the principal sqrt(-1)" - As it happens this is located on the ordinate of the complex plane, therefore on a straight line with real coordinates, so can't we ?
It would be, if it weren't for the fact that the rest of the video is about finding why people might believe the other answers are right and showing why they are wrong.
+6 and -6 both are correct according to the usual definition of the radical sign meaning. For a real positive number its square root is the positive one, but for a complex number the square root of z is defined as the solution of the equation x^2 = z. In a similar way, the cubic root of -1 is the solution of x^3 = -1, i.e. the set ( e^(i*pi/3) , -1, e^(-i*pi/3) )
@angeljimenez3362 I don't think so. The square root symbol is meant to refer to the principal value in all cases. For a negative number, it is the one with positive imaginary part.
for real number, you can choose the principal square root because IR have an order, so you say sqrt(p) is the greater or equal to 0 solution of x^2=p but for complexe number, there is no order, so no good way to choose a principal square root, so there is 2 square root in the complexe set last argument : we all know that 1 = e^i2kpi where k is an integer by the law of exponent, sqrt(1) = e^ikpi with k is an integer so it’s both 1 and -1
Finally someone in the comments who gets the difference between a mathematical object (the complex field) and some representation of that object using non-unique (re+im) components
> but for complexe number, there is no order, so no good way to choose a principal square root That's just false. Principal square root is perfectly defined for all C. And √ symbol defines specifically principal square root. There is no room for different answers in this problem.
@@lerarosalene Not so simple. I think your confusion arises from the way complex numbers are introduced in highschool and engineering classes as "pair of two real numbers". But those pairs are NOT complex numbers, those are just some representation of the complex number field that you can do calculations on. Now of course one can define a principal square root as an operation on your specific representation, which may be handy for some engineering problems or whatever. But that definition is non-mathematical. The reason is that you can map Re and In in many different ways onto the complex field and get the same mathematics back (but your number pairs would look very differently). In fact, you cannot even tell the difference between i and -i by any equation involving field operations only. And don't say that i is the squareroot of -1, because that would be cyclic reasoning.
@@valentinziegler1649 stop smoking whatever you are smoking. √ symbol is defined to be principal square root and principal square root is also precisely defined. This whole problem is about notation and people like you not understanding it.
But they are distinguishable. For instance, the principle value of the argument of i is pi/2, while for -i it is -pi/2. Or, -i is a root of the polynomial z+i, while i is not. Etc.
@@johnreid5321 The principal value of i being pi/2 simply means that exp(i.pi/2) = i. But since exp(-i.pi/2) = -i, that fails to distinguish i from -i: that is, if you replace all occurrences of i by -i you get the same thing. Similarly -i's being a zero of the polynomial z+i: replace i by -i throughout, and get the equally true statement that -(-i) is a zero of z+(-i). Again i and -i can be switched and true statements come out true again, and once more nothing serves to distinguish i from -i. (But you have to switch ALL occurrences of i of course)
In the original question where both 6 and -6 are options, and +-6 and undefined are not, my answer is -6, as I feel it's the most correct. In your version of the question, where all choices are potentially correct I would say +-6. If +-6 is one of the options, that implies to me that such answers are allowed under the writer's definition of square root.
Using the root symbol (√) is defined as taking the positive root. Taking to a fractional even power (like ½) gives the positive and negative root. 4^(½) = ±2 but √4 = 2
@@pageboysamThat's not true either. 4^(1/2) is the same as √4. It us only the principal square root. No combination of elementary arithmetic symbols produces more than one value without explicitly using ±.
Yeah this is the key here. I too was emotionally attached to my answer of the initial question, too much to realise that the new options for answers changes things, so whilst before the video I chose -6, by the time I was 75% of the way through this video, my choice changed to +/-6
• When dealing with square roots of negative numbers in the complex number system, each square root operation returns a single principal value. • The principal square roots of -4 and -9 are 2i and 3i, respectively. • Multiplying these roots results in 6i^2, which simplifies to -6. Therefore, in the complex number system, the answer to -4 x -9 is -6, not ±6. There should be no ambiguity here.
There is no such limit to negative numbers under the root. We all have happily used this with variables, not even wasting a single thought about signs.
±6 is objectively correct, because there's no such thing as a principal square root in complex numbers. There is n sollutions to n-th root of any complex number and there's absolutely no reason why we should pick one instead of the other
If there is no such thing as a principal square root in complex numbers, why is "i" defined as only +sqrt(-1) and not positive and negative? If it were positive or negative it wouldn't be unique and would vastly change results. Complex numbers are two-parters which makes them a dot on a field and not a position on a string. If you didn't have exact coordinates on a field, you'd end up in the wrong place. That's also why they are vastly different to the remaining number definitions. Imagine it like a geo coordinate. If you try to navigate to europe, e.g. longitude 7, latitude 47, you better hope they are defined as positive. If you miss it, you might end up in the Ivory Coast or somewhere in the middle of the atlantic ocean.
square root is defined to only have one output so everyone who says +- is wrong. that only happens with the absolute value operation and a lot of people like to pretend theres an absolute value around square roots when there shouldn't be.
When you're using the quadratic equation you're essentially completing the square. And in completing the square you need to take a square root of a variable squared. A variable squared has a value that could be traced back to two original values for that variable. We use the plus minus because we are *dismantling* something that could be either option.
@@Philip-qq7ql That isn't a flawed argument though, since the square root function is specifically defined that way, the reason being you want to be able to refer to only the positive or only the negative numbers that when squared equal another number as opposed to always referring to both. That said, we could have created a different notation to specify when a square root was positive or negative and kept the base square root as implying both, but it just didn't turn out that way.
@PopeVancis expecting a grade 1-3(7-9yr old) to compute arithmetic above 50......have adults forgotten this is about learning ...esp mathematics semantics....
Around 7:20, while finding the inverse of y = x² as y = √x, it actually should come out as y = ±√x, and since we're splitting y = ±√x into two separate functions, I wish you had also mentioned y = -√x is a full-fledged function on its own; because then √9 would indeed be equal to -3 where y = -√x; but since we don't work with negatives, the principal value is what's taken into account
@@MrSummitville And no one said otherwise. Do you know how to find the inverse of a function (y = √x is the inverse of y = x²) -- and that's what he did when showing the graphical visualization at 6:30 (but skipped the math behind it) which could be what's confusing you
No. √9 will never equal -3. That's not what the √ symbol means. There are two numbers that square to 9: 3 and -3. The expression "√9” only refers to one of them, never both.
@@nboothWho are arguing against? Nobody here tried to claim that √x could be a negative number. Perhaps you misunderstood the OP. He was saying the inverse of y = x² is y = ±√x. The "√x" is still always a positive number, but the inverse function itself needs a "±" to account for all solutions. This is the same reason that the construction "±√..." appears in the quadratic formula.
Whomever it was that chose the words “real”, “imaginary”, “rational”, “irrational”, etc., to describe those types of numbers, did a HUGE disservice to mathematicians for the rest of history. How can we convince non-mathematicians of the fundamental correctness of our results, when we tell them that we arrived at our conclusions using “imaginary” and “irrational” numbers?? Or that when you multiply one “imaginary” number by another “imaginary” number, you get a result which is somehow not imaginary??
I think "imaginary" was a term coined by those trying to discredit their use to solve quadratics and cubics I don't think these terms could be any better, they are inconvenient for non-mathematicians sure but they all have a use In their wordings
Every new type of number gets a pejorative name: "negative", "fractional (Latin for 'broken')", "irrational", "imaginary", because the previous generation of mathematicians are uncomfortable with it.
Those successively less useful "numbers" get more and more dismissive labels because they are further and further removed from what numbers are and how numbers behave. It's not pejorative, it's descriptive.
Mathematician here. Complex numbers are based on the imaginary unit i, which is defined by i^2 = -1. If you, on the other hand, define i = sqrt(-1), you get into trouble: i^2 = sqrt(-1) * sqrt(-1) = sqrt(-1 * -1) = sqrt(1) = 1. That's incorrect! So suddently there are rules about sqrt() and other functions that you cannot apply. Rather than trying to memorize what you can and cannot do, it's better to never write sqrt(-k) with some positive integer k in the first place. Complex numbers don't allow you to write sqrt(-5), they simply give you i, the number when squared gives -1. That's it, and it turns out that's enough, too, to do all the complex magic.
Since complex numbers DO allow -5 to have square roots, it seems odd to conclude that they don't allow you to write √(-5). I know that you can write it as i√5, but your argument seems to say that being able to write √(-5) is redundant, not that it's prohibited.
@@gavindeane3670 I see where you're coming from. The complex square root function does indeed exist, but is very tricky! For one, it's multi-valued (exept for z=0), so sqrt(-5) isn't just "a number", but can be one of two (sqrt(5)i and -sqrt(5)i in this case). Likewise, sqrt(-1) is i and -i. It's not useful to do any calculation with that. Would you agree?
@@nschloeObviously numbers have two square roots. But the principal root is defined for complex numbers (it wouldn't need to be called "principal" if it was only defined for real numbers - we could just call it the positive one) so I don't see how √(-5) is problematic notation. I can see how there would be a problem if the notation √(-5) was ambiguous as to whether it meant 5i or -5i, but that's no different to the problem we'd have if √4 was ambiguous as to whether it meant -2 or 2. We invented all these squiggles and shapes that we call "mathematical notation" so we can choose what it means.
@@gavindeane3670 no, non-negative numbers only have one square root. Quadratic equations however have 2 solutions. And that is what I think is the greatest misconception. Even Presh here goed from calculating square roots to solving quadratic equations. Jumping from calculating √-4 to solving the equation x² = -4 does not justify the use of i.
@@gavindeane3670no, it doesn't allow that. Complex numbers are used to solve equations like x² = - 5 which results in x = +/- i√5. So 2 answers/solutions/roots.
It is undefined for the system of real number but if you are taking complex numbers, -6 would be correct answer √-4 * √-9 = 2i * 3i = 6i² = -6 So the correct answer according to system of complex numbers is -6
06:39 The information "sqrt(x) is not a function!" is wrong. sqrt(x) IS A FUNCTION. The graph show at 06:39 is NOT a graph of sqrt(x). It is a graph of+-sqrt(x). And this +-sqrt(x) is NOT a function.
He isn't saying that the accepted definition of sqrt(x) is not a function, nor is he claiming the original graph (pre-pruning) to be a graph of the acceptedly-defined sqrt(x). He is saying that if instead sqrt(x) was alternatively defined to included both possible roots, and not just the principal root, then the alternatively-defined sqrt(x) would not be a function.
@@nwoDekaTsyawlA By convention, the √ symbol defines a single-valued function. √a refers only to the principal branch solution of the equation x^2=a. By this convention, √(-4)√(-9) has a single value of -6. This would not apply to other ways of writing. (-4)^0.5 * (-9)^0.5 = ±6
The fact that the square root symbol refers to only the principal square root is similar to the fact that 1 is not a prime number. It might not fit with your world view, but mathematicians have adopted is as a standard because it makes mathematics easier, and you should too.
There is a big difference between : - the function square root, which by definition of all functions has one and only one value where it is defined, eg sqrt(9)=3, and sqrt(-9) is not valid and has no precise meaning. - the solutions of the equation x^2=-9, which has two solutions 3i and -3i. Well defined maths have no logical flaw, you just need to apply right definitions to each concept.
@@verkuilb Glad to be better in Math than in English 🙂 Only one mistake is still very good, as English is not my mother language 🙂 Sometime you should think about assumptions before being ironic 😞
@@StephTBM4yeah he is one of those people who will see that his/her son scoring 95 percent in exams and still complain that spelling of blah blah is wrong in the report card instead on focusing on the result😂😂😂😂
That is not the issue. The issue is much more complicated. If you take your domain as complex numbers for the relation f(z) = sqrt(z) it is multivalued. To make it a nice function you restrict the domain and that restriction tells that -1 has an unique square root and it is i. If you need more answers you are talking about roots of unity and that is totally different thing than just taking a square root.
@@SkegAudio Because it isn't a function if you don't do that. It's double valued relation and generally n:th root is n-valued relation and as function you take the first branch. If you want to study the double valued relation it's done by forgetting the 2 and just think how n-valued root handles. And for that you only need to know how the roots of unity work. Remember that any complex number is ae^bi, where a and b are any two real numbers. Then we can discard the a as it is just a multiplicative factor and focus on the e^bi and then n-roots of that are the roots of unity offsetted by e^(bi/n).
@@topilinkala1594 Look while OP is indicating that complex numbers have n nth roots, the √ symbol itself has a specific conventional meaning in complex analysis - it denotes the principal square root. When we write √(-4), we're specifically referring to 2i, not ±2i. It's similar to how √4 means +2, not ±2, even though both ±2 are solutions to x²=4. If we wanted to indicate both possible roots, we'd need to explicitly write ±√(-4)⋅±√(-9). The √ notation alone has a well-defined meaning that gives us one specific result: (2i)(3i) = -6. You're absolutely right that both roots exist mathematically! But the question is about what the specific notation √(-4)√(-9) evaluates to, given standard mathematical conventions.
The actual thing here is, that the task solving sqrt(-4) within complex numbers can easily be rewritten to sqrt((i^2)*4), which then translates to | i | * sqrt(4)
I mean, it depends on the agreement on what branch you can take for sqrt, you could even take it as "multivalued function" However, given no context we should take the principal branch, probably the test it was taken assumed principal branches too...
I am on the -6 camp never understood why 7:00 - 7:22, finally someone explained to me why only positive roots are used. My HS teacher only told me "just its the rule" but didnt expound more. Thanks!
Nope, at pre-calculus you learn that √x*√x=x and hopefully that √x*√y=√xy.
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The problem is one of definition, how do you choose which root is the principal root of a number? Easy to do for the positive reals, but why choose i as root(-1) and not -i? And what about root(i) or root(-i) or any other complex number, which of the two "roots" count as the principal one? Choosing arbitrarily is fine, of course, but there would be a continuity problem in the resulting function (from C to C). This is typically why I try to avoid using the square root symbol with negative or complex numbers, to avoid ambiguity.
All you need is the definition of the radical symbol. From Wikipedia - The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. Additionally, I don’t think the tree pruning for functions explanation is relevant. There’s nothing in the original question that requires a function. We’re simply asked to evaluate an expression. -6 😎
I’m a bit closer to agree with the Principal Root argument. However, as a counter example in the quadratic formula +/- is part of it even when the radicand is negative. There is no enough information in the problem to exclude the positive answer. What am I missing?
It’s just the definition of the radical symbol. The two square roots of a negative number are both imaginary numbers, and the square root symbol refers to the principal square root, the one with a positive imaginary part. So when evaluating the expression, we use the positive imaginary part for both radicals and the answer is -6. The answer is very straightforward. The quadratic formula is a different situation. In this case we’re solving an equation (not simply evaluating an expression) and it will have two answers (unless the radicand is 0). Here we want to include both roots, positive and negative, no matter if the radicand is real or imaginary. Since the radical symbol only refers to the positive square root, we must add the plus minus sign to specify both roots and find both solutions.
@@johnhoslett6732 Interesting. So you are saying "Simplify the following term" and "x equals the following term, solve for x" are two different things. But who says that I am not allowed to pick the second option to solve this problem?
@@manfredlemke4671they are different things. If you take a square root of something (call it a) you get two possibilities: ±√a. If you're given a square root √a, that *always* refers to just one of them.
many people might see 2 negetives in multiplication and get the answer as +6 but it actually goes like this: since i=sqrt(-1) we get, -> [{i*sqrt(4)}{i*sqrt(9)}] -> (i^2)*(2)*(3) and since {sqrt(-1)}^2=-1 the answer is -6
The problem is that you considered i as sqrt(-1), it ain't. i is the number such as i² is -1, while the answer for sqrt(-1) is i and -i , both branches should always be considered, unless the question amde has a context and only one branch makes sense. is nice that you talked about functions and all, but this isn't a question about functions, is a question about an equation. I don't know how you gringos learn equations, but when i did in primary school, if we had to answer a problem of the kind: "what are the roots of 3x² + 4 - 2" we would solve 3x² + 4 - 2 = 0 and we knew we could put that on the in the square equation formula (A.K.A. Bhaskara) and we could find up to two roots, unless they coincided or one or both couldn't be calculated in real numbers (it was primary school after all). So, it is + or - 6. when we looks at sqrt(-4)*sqrt(-9) we immediately know that we could have four answers because each square root can have two answers and we know that two of hose answers would coincide in the same number, and the other two as well therefore we can have up to two unique answers. then when we expand that to sqrt(-1)*sqrt(+4)*sqrt(-1)*sqrt(+9) we know this expression, written as it is can give us eight answers and we know we will have, at most, two unique answers because we know that (+or-)i*(+or-)2*(+or-)i*(+or-)9 have eight anwers.
@@noname_atall there is no equasion here! "√-9*√-4" is a simple arithmatic expression. There are no equasions or functions involved. Even in the realm of complex numbers, simple arithmatic expressions have SINGLE values.
But the "√" is _not_ defined as the solution to a quadratic equation. It's defined as the principle square root. This symbol always denotes a single positive number. Recall the quadratic formula that you referenced. Note that it includes the construction "±√...". If "√" could be either a positive or negative number, why would the quadratic formula have to include the ± here? If it was already implicit in the √ symbol, it wouldn't be a necessary to include it in the formula.
I think Presh's observation that the (±2i)(±3i) camp would be the hardest to convince is exceptionally astute, as I understand but continue to disagree with his belief that the principal square root bears on the evaluation of this expression. Given the PEMDAS stipulation that exponentiation supersedes multiplication, the ± has to be produced twice and therefore winds up in the final answer, as well. Denying half the complete answer in order to solve an irrelevant manufactured issue (the not-a-function issue) is logically fallacious. The expression has two valid values, not one. // Do I sound terribly opinionated? Well, I probably am -- except for the *terribly*. Cheers. :)
@@jim55price the expression has one value. ALL arithmetic expressions involving addition, subtraction, multiplication, division and powers (including the √ symbol) ALWAYS produce a single value. Even in the context of complex arithmetic no simple expressions ever has more than a single value. There is no equasion here. There are no functions. √-4*√-9 is a simple arithmatic expression with a SINGLE value of -6.
@@nbooth Its not just arithmetic operators, _all functions_ should return a single value, since by definition a function is a one-to-one (or many-to-one, but never one-to-many) mapping between two sets
@@nbooth Your comment is preposterous, given that far simpler expressions, e.g. ±6, ±x, ±i, have two values. I've no idea what you're going on about with your claims of "ALL" and "ALWAYS". All it takes to illustrate your error is a single "±".
@jim55price yes the ± symbol allows you to write two values with a single expression. Congratulations. {-1, 1} is also an expression with more than a single value and [0, 1) is uncountably many. Those aren't arithmetic operations however. They're ways of writing sets, which is what the ± symbol does as well. Raising a number to a power is an arithmatic operation that produces a SINGLE value. That's why we need the ± symbol in the first place.
@@jim55price you didn't read what I wrote. I said all expressions involving the following operations (+, -, /, *, ^) yield a single value and you're claiming some other operation is a counterexample to that.
because them being the root of a negative number changes how the maths works it is more that the product of 2 roots is more a guide and a quick shortcut that can be used under some circumstances rather than a good rule
Pretty much just to avoid this specific problem having the answer 6. If you're using the version of square root with both branches present then the rule is fine, if you use the one with only one branch, sometimes the rule gives you the other branch. Since 6 would be the positive and thus correct branch for sqrt(36) but not for sqrt(-4)*sqrt(-9).
@@Stereomoo It would also end up working if there was only one negative number, TBF. Like radical(-9)*radical(4) = 3i*2 = 6i And radical(-36) = 6i Same answer. It's only deceptive when there are multiple negatives, because you'll end up on the non-principal branch. Actually, going to more than 2 numbers... Like, 4 negatives and 5 negatives both work. i^4 = 1, i^5 = i, both on the principal path if you multiplied everything together. 0,1 work; 2,3 don't; 4,5 work; 6,7 don't; etc.
@@durandle9226 in my limited knowledge of pure math, i consider the whole explanation unnecessarily complicated, because sqrt(-4)*sqrt(-9) could be simplified to sqrt ((-4)*(-9)) which simplifies to sqrt (36) = 6
this is why i don’t like how roots of even numbers are ONLY positive. it just throws various rules out the window. it’s like mathematicians were like “how do we do something that seems reasonable, but actually ends up being annoying?” like you could solve it like: sqrt(-4)sqrt(-9)=x (-4)(-9)=x^2 36=x^2 and then wait a minute. you don’t actually take the square root of 36. what you do is: +-sqrt(36)=x and THEN you have to check both values of x to see if they’re valid
As one among the 11% who voted +or- 6, my reasoning is different. Specifically, I note that real numbers have no preference for + i vs - i; they treat + i as equal and opposite to - i. A function not defined with any imaginary numbers cannot possibly distinguish a complex number from its conjugate. Thus, sqrt(-1) = +or- i, as there can’t be a preference for either + i or - i. With that, even if we use principal branch square roots, the answer can still be positive or negative. However, if we do have a preference either way for + i or - i, the answer either way would indeed be strictly -6.
I don’t fully buy the argument against undefined (although my answer is -6). First of all, x usually denotes a real variable (compare to z). Second, not all problems imply that you should extend the domain, for example if I say “can you factor x^2 + 1” then “no” seems like the answer you would usually expect unless you had already established that you are working over C.
But even then the correct answer would be to specify "if real: undefined" which would be correct. You are doing a disservice to yourself by assuming properties without writing them down. Then when validating your work nobody can know why you got the answers wrong (because not showing which assumptions you made).
I'm perfectly happy with ±6 as an answer. I have no reason to assume the root here is a principal root, as the question has nothing to do with functions, so I don't have to handwave away half the answers bc they're inconvenient.
Its odd to me that the segment that went off on people who only used Real numbers is the followed by an assumption that we should take the positive values for both roots. sqrt(4) =±2 is not a function, true, but sqrt(4) = -2 is. Its is completely possible to get sqrt(4)i * sqrt(9)i -2i * 3i -6i^2 6 There are 4 potential routes for the equation. 2 give you -6 and 2 give you 6. Therefore the general solution is ±6. The answer is only -6 if you add a qualifier of using principle roots.
The question is not to find solutions for a quadratic function, where considering the negative root part is required, but it is about calculating the root, which is defined to be positive only.
Mathematician have agreed that whenever your are talking about "the square root" your are talking about "the principle square root." And especially in this case, as when that sign (radical sign/radix) is used, it specifically denotes the principle square root.
@@arejaybee there's no equasion here. We just have arithmetic operations on complex numbers. The expression has a single value: -6. (Yes the √ does mean principal square root).
@@neuralwarp sqrt(-4) != ± 2 as that is not a valid function. From wikipedia: In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. you are trying to solve a quadratic equation for example: y^2 = x => y = ±√(x) but note how the ± is only on one side not both sides, and that it is also outside of the √(x) function.
04:17 Fair point. By "default" people will assume we are in a domain of real numbers as this is something we are all into it on a daily basis. But do a survey but rewrite the question like this: sqrt(-4+0i)*sqrt(-9+0i) (or just write that the domain is complex numbers) And I bet the percentage of people picking -6 will increase drastically.
Guess I am one of the hard to convince troublemakers :) Thought it was interesting that you pulled out excellent reasons to not restrict answer to only real solutions, and then pulled out functions (which were not mentioned in problem) to justify restricting answer to only positive roots. Given the answers from Brilliant, only one of those was correct under any rational interpretation. Your answers would depend on context. In a class that had just covered roots, focused on them being + or -, and now was talking about imaginary numbers, the best answer would probably be +-6. In a section on functions, or in a book/class where the convention is explicit to only consider positive roots unless specifically flagged +/-, the answer is -6. Without understanding the context of where it is being asked, both are right and that would be a crappy/unfair question on a test. Especially to people who haven't sat in a math class in 30 years.
I think it's because the square root symbol is defined as a function, so he explained what a function is. Pruning the tree is part of the definition of a function. It isn't really the same as not restricting answers to the reals.
how do you get from 6i² to -6. I would just leave it at 6i² and therefore click undifined as the answer because the one with an i in it is not an option.
It's so funny! I immediately evaluated it to the correct answer because when I was in school, I remember my teacher saying something like "If you multiply 2 negative roots, just root both of them before finding the product and just preserve the negative." I TOTALLY forgot WHY that works, though, and now I understand why again!
The undefined solution is largely given to students learning square roots initially. In order to not overwhelm students, the teachers simply want students to say it is undefined. But, once students learn of complex numbers, negative square roots can be taught in their entirety and should be viewed as the actual correct answer.
I see myself as average in Maths. But when I saw the thumbnail my answer was thought 6i^2 (I forgot that i^2 equals to -1). So I skipped to the end of the video to see the answer. I got it wrong but was surprised to find out that my answer was part of the solution. So say what you want, but I'm proud of that.
I solved it as -6 but I have an argument for undefined which you didn’t mention. One can say it’s undefined as there was no domain of x mentioned so I can not be sure what was meant. The reason for this is that most of the time real numbers are used as domain and that is the reason for fair assumption that this is also the case.
Exactly. I stopped watching at the half-baked integer mathematics analogy. The domain is the real numbers unless otherwise specified. If this question came up in a textbook in a chapter about complex numbers I wouldn't expect them to state it again in each problem.. but in isolation it's intentionally ambiguous about what domain to use -- there is no right answer because you can't know which of the two right answers it wants.
@@seedmole I disagree, the domain is all numbers unless otherwise specified. Without that assumption the fundamental theorem of algebra breaks. Unless there is something that specifically prohibits them. The characteristic equations of second order differential equations wind up with complex roots even if you can use them to work your way to purely real valued functions.
That makes no sense. If you know about complex numbers then you know how to evaluate this expression. You do not need the author's permission to use that knowledge.
The thing is, ±6 comes with the assumption that we can take both square root values both times. That's (usually) a bigger assumption than "there's a square root that is explicitly and obviously that of a negative number, we can use i", which makes ±6 feel still correct, but not quite as correct as just -6. Especially when you can tell that the i's are gonna "cancel each other out" into -1, it feels much neater to remove that uncertainty even if it takes two weird steps, than if you were to take two more uncommon weird steps and still have this "uncertain" result in the end.
*If we can have 2 x value for one y value then why can't we take2 y values for one x value? Why it can't be simply a function * If √x√y=√xy only if (x,y>0) Then √-1√4= √-4 is false because -1
What's the "correct" value of sqrt(4)? Is it 2 or is it +-2? If the latter, the answer to this question is +-6. In a synthetic question, sqrt(4) = 2. In real life problems, it can be +2 or -2 or even both; depends on HOW you got there and why you need to evaluate it.
In all my courses, we had to ensure to include both the plus and minus or be counted wrong. In real world problems we cannot afford to ignore a possible set of solutions.
√4 = 2 The radical function is implied to be positive unless otherwise noted (as is literally every number as 5 is assumed to be +5 and not -5) Confusion arises from the solution of x^2 = 4. In order to solve on must take both the positive and negative square roots of 4(notated as ±√4). The solution of x^2 = 4 is x = ±2 This is problematic as people conflate the equations x^2 = 4 and √4 = x, when they are, in fact, different.
The problem is many people have never learned that +/- is supposed to be added in front of a sqrt if a square is inverted. For those people their tools will be "not enough data given" to know if it's -6 or +/- 6. Had it been multiple choice like or brilliant, almost all the =/-6 people would have likely selected -6.
In the survey run by Brilliant, less than 50% got it right, so "most people get it wrong" is technically correct. When the video maker re-ran the survey, 55% got it right. Presumably, this says something about the math knowledge of folks who watch this channel vs. folks who follow Brilliant.
@jmr5125, and John. Generally speaking while something over 50% is a majority, that does not mean that it is most as most requires nearly all. I'd be looking in the 80s or 90s for most. However, if we are talking comparatives, many, more, most and comparing groups containing different numbers then many (of a large group) might require only 20%, with a group containing 35% being more and a remaining group containing 45% of the total might indeed represent the most. Ain't words fun too.
Even if we extend the principal branch of the square root to complex numbers, it is usually not defined on a ray. That ray is typically negative real numbers. If we extend it continuously to negative numbers (and remove another ray instead), a choice must be made as to whether sqrt(-1) is i or -i. Fortunately, for either choice, sqrt(-4) * sqrt(-9) evaluates to -6.
They way I like to think about it is the sqrt(x) is really a shorthand for (x)^(1/2). That square root symbol implies powers and parentheses, so you have to do them before you perform any multiplication. Unfortunately, many people are taught in school that sqrt is "fundamental".
I answered -6 on the poll. No other answer even occurred to me because my background is in physical chemistry and physical chemists deal with wave functions which assign a physical meaning to the imaginary number.
I got the right answer of -6 myself, however I think undefined is for some people a very reasonable answer. Not everybody was taught complex numbers at school, so I would not assume that this is widespread and common knowledge. In 9th grade for example we were taught at school that square roots of negative numbers are undefined and in any test undefined for this question would have been marked as correct answer.
An important distinction he made is that although a number has more than one square root, the square root function has only one answer which is why we are careful to write +/- in front of the square root symbol in the quadratic formula for example.
I come from France where √-1 clearly isn't defined. I like the debate though. I undertsand that it's a convention in some countries that √-1 is i. However, if I had to define √z where z isn't positive, I would call it "A root of z" in stead of "THE root of z". Then √-4 = ±2i and √-9 = ±3i, then √-4√-9 = (±2i)×(±3i) = ±6. Finally, I would agree that it all depends an the convention "we choose" to accept : -> undef IF √-1 is undef -> -6 IF √-1 = i -> ±6 IF √-1 = ±i (with this definition, for any z = r·e^(α) in ℂ, with r>0 and α real, √z = (±√r)e^(α/2) -> this even means that √1 = ±1, so we lose a lot of unicity, but get back the "√a√b = √ab" rule )
Finally my BSEE came in handy as I was able to get the correct answer straight away. BTW, I applaud your decision to be Unsponsored. Thanks for sharing.
I understand the -6, but i learnt to never write √-1or any other negative number. A bit like 3y is defined by 3 × y, but y3 is not defined. So √x for x< 0 is defined, but √-3 is not - or at least to be avoided at all cost. One proof that i am right is the existence of "i" : if it was ok to write √-1 nobody would be using i.
So he should be 🙂. The reasoning for "undefined" on the basis that we aren't told complex numbers are allowed is, notwithstanding that it appeared quite commonly in the comments, completely brain-dead.
It's "undefined" , not because it's "undefined under real numbers", but because the square root function, as written in the video is not defined for negative numbers.
sqrt(-4) * sqrt(-9) = i * sqrt(4) * i * sqrt(9) = i*2 * i*3 = i^2* 6 = -6. BUT i * sqrt(4) * i * sqrt(9) = i*-2 * i*3 = i^2 * -6 = 6. Remember that the square root function produces both a negative and positive result!
All of this comes down to convention being used. Traditionally the square root is defined as the positive root of a positive number. And thus sqrt(-1) is not i, but undefined using that convention.
I agree with you in that it depends on what you're assuming. What assumptions you're most comfortable with is open for discussion. If you approach this by looking at branch cuts then you do get two answers but if you choose a particular branch cut then you only get one. The two valued solution can exist but it isn't useful so why not just choose a branch cut and get something useful to work with.
When I saw the thumbnail, I immediately got -6 by thinking about geometric averages rather than actually multiplying anything. I know that the geometric average of 4 and 9 is 6, so it stands to reason that the geometric average of -4 and -9 is -6. I thought it was going to be something sneaky, but I was disappointed.
Yes, many get the concept of the principal root wrong. Some thing whenever you solve a root you end up with two solutions. That is not true. That's only true when you have an unknown under the root. When you have a constant known value, there's only one value. So you get a +- whenever you cancel sqrt(x²) to +-x since we don't know what x actually is.
It’s kind of amusing to me, that if you ignore the arbitrary “this has to be a function“ argument, and what seems to me, equally arbitrary “you can only multiply radicands if they’re positive” (which I’ve never heard before), you can get the correct answer another way: sqrt(-4)*sqrt(-9) = sqrt(36) = +/-6
it is not correct to use an ambiguity like that in the first place solving the sub step of " i root 4" is not something to consider the 4 might be the product of 2 negative numbers we KNOW it is a product of positive numbers in that state because we started with the square root of a negative number
Limiting to the principal root is the definition, not the assumption. Also, "wolfram alpha said so" is a valid argument. WolframAlpha just follows the definitions we gave it. 1+1=2 is defined. We follow that definition. So do we follow the definition of a square root.
