I've used algebra in my work for more thann 45 years and I've NEVER heard anyone explain this so clearly before. Thank you for 'teaching an old dog new tricks'.
I've never seen sqrt(x²) = |x| before, but that makes so much sense! I was super confused when I got the wrong answer on your poll, thanks for explaining it.
My math teacher told us that this is actually the definition of the absolute value function, which means that you can work the other way as well: to cancel an absolute value, you can just square both sides since this creates the 2 possible answers by introducing a x^2 term (as per fundamental theorem of algebra) This is definitely not always more useful than the piecewise definition of the absolute value function but it is handy to know 🙂
@@hydroarxis there any downside to using sqrt(x²) instead of the piecewise function? I always find it handy being able to write it out in one term/line, especially when it comes to derivatives (arguably the piecewise function is more straightforward to calculate but I still prefer the single term
Here's the problem: x^2 = 9 -> sqrt(x^2) = sqrt(9) -> |x| = 3 -> x = 3, -3 - First know the trick to get plus or minus (+/-). You want to know the absolute value before you can take plus or minus the square root of any number. 🤓 Another thing: sqrt(x+1) = -2, when x = 3, sqrt(3+1) = -2 -> sqrt(4) ≠ -2, that's why you can never get a negative answer from an equation like this, it does not exist. This is why you should go back and check your answer. If it was sqrt(x+1) = 2, then we can say x = 3, so sqrt(4) = 2 would be your answer. Remember, x^2 = 9 is not x = 3, it's x = 3 OR x = -3. It's not just 3, but also -3. So as a reminder, x^2 = 9 is not the same as x = sqrt(9). Keep that in mind. Good luck! 😉
I think part of the confusion arises because we talk about "THE square root (or cube root, or 4th root, or...) of a number, meaning the principal square root (of which there is only one); but we also talk about square rootS (or nth roots) of a number to mean all the numbers that, when raised to the nth power, yield that number (as in "the nth roots of unity").
It mostly depends on what kind of math you are dealing with and how the squareroot is defined in some branches of math. In another video you kind of contradict yourself by converting a squareroot into a "power to a half". x^½ is like asking "what number do i need to multiply by itself to get to x" that includes both a negative and a positive answer. In this video you define the squareroot as only positive. The videos kinda contradict eachother just wanted to point that out
The problem with the fundamental theorem of algebra is that it implicitly assumes the degree n of a polynomial is a positive integer. As soon as we are forced to consider x^(1/2), all bets are off, as we can't have half of a solution. A similar problem rears its head if we that think the equation x = 1 + 1/x must have only one solution (or minus one solutions!). Of course, it has two. It's a pity that many mathematicians think that defining a function as having a single "output" is a mathematical law. It's not; it's merely a convenience that makes life easier in many fields of maths. Unfortunately, it makes many functions non-invertible other than over a very restricted range, which is less than ideal for problem-solving. Similarly, choosing the positive square root as the principal square root is a convention, not a requirement. You could imagine a mathematics where the negative square root is taken as the principal square root. Would it break everything? Most certainly not. The problem with the video you refer to is that the process of taking a square root is not the same as evaluating the square root function. The former is the inverse of the square function and leads to a set of two possible values (which are the negative of each other), while the latter, being a function, is constrained to a single, positive value, by convention. None of this even touches on the problems of single-valued functions when dealing with complex numbers. Do you really want to be working with a "function" that evaluates the cube root of -1 as -1 if we're considering real numbers, but evaluates the cube root of -1 as (1 + i√3)/2 if we're thinking about complex numbers?
Wow I can’t believe I finally found the one teacher who makes me love maths although I study it in Arabic I still translate my lesson to understand it from you
You really tried your best. Possibly the best explanation I've ever seen. Unfortunately, some students will never understand this. Students need to do as you said and realize that if the degree of the equation is 2 or 1, then you will get back 2 or 1 solutions respectively.
I've always thought it's a shame that the plus/minus symbol has been conflated with the square root symbol in mathematics. You are definitely educating and I think it's wonderful
@Prime Newtons: I was quite fortunate enough to had some great Maths teachers, while I was in school. After seeing your videos here, I got the same vibe as I was sitting in their class and learning all over again. Great explanations sir!!! ❤❤
Newton, your patience is incredible. Almost everybody makes this mistake, even math teachers. The math teachers who continue to make it after you've demonstrated why it's wrong are a little odd in my opinion. 😅
People might think sqrt(x+1)=-2 has some complex solution because of i, but notice that i=sqrt(-1) not sqrt(i)=-1, so there is still no solution even in the complex world.
Amm not, if sqrt() - is complex multivalued function it has solution x = 3. Because sqrt(z) = |z|*(cos((ang(z)+2pi*k)/2)+ i*sin((ang(z)+2pi*k)/2)) that has infinite repetitive values or/and 2 different values.
The reason this explanation is so good, is that it exposes the fact that maths is based on some human choices, in terms of definitions. Most maths profs would be reluctant to explain this.
Let say that sqrt(9) = X as you stated….then if we say that (sqrt(9))^2 = X^2 ….. we will have 9 = X^2 then we apply your solution for this problem and as result ….. we have 3 and -3 again…where is the error….lol
This has the same issue as if we did the following, using a practical example to show that it is not just for the sake of mathematical convention: Let's say a person's bank account has 10k in it, so x = 10k. That is an objective unchanging statement of truth in this situation. Let's just have some fun with that equation and square both sides, so now x^2 = (10k)^2. Now if we solved this equation, we'd end up with x = + or - 10k, but we know that the person's bank balance is 10k and they are not 10k in debt to the bank! By squaring both sides of the equation, we've created a solution that isn't a correct one. This is something you have to be careful of doing in Maths, and is what you've done in your example.
You have dealt with the my most confusing stage of maths that I had from my 7th standard..I went to my teachers but they said the same things ....They followed the norms like all other....You were different..... Thank you sir.....❤
the only thing i can think of is by assigning (x+1)^1/2 = -2 = y , then we square all the equations [ (x+1)^1/2 ]² = (-2)² = (y)² thus (x+1) = 4 = y² rearrange y² = 4 = x + 1 proceed by square root back all equations (y²)^1/2 = (4)^1/2 = (x+1)^1/2 as y² has two roots, the square root from its value should contain a positive and negative, thus y = +@- (4)^1/2 = +2 and -2 and also = +(x+1)^1/2 and -(x+1)^1/2 similarly -2 = -(x+1)^1/2 thus (x+1)^1/2 = 2 , and solving this equation x will equal to 3 and (3+1)^1/2 will give 2 same to previously statement in other words the equation will need to be assigned to a variable and go through square and square root process to solve it by algebra as a square from the left side of the equation is the same as the square root from the right side of the equation and also the square of a negative number will always equal to positive
Tell me one thing we are solving algebraic exponents with rational exponents and we come withh expression (x^2)^1/2..then from childhood we. Are writing as simply x...whose domain and range. We thought were -infinity to + infinity...means i was wrong..in solving algeebraic expression we assume base x as positive...i dont understand when we will apply mod and not when soving algebra of rational exponents???
I respectfully disagree. There is no constructive elementary proof of the "fundamental theorem of algebra", which is not algebraic but just an arbitrary post-modern claim. "Real numbers" and "real complex numbers" don't really exist and most certainly don't form a field, as non-computable and non-demonstrable numbers don't exist and cannot perform arithmetic operations. Mathematical truth and the foundations of mathematics are not speculative games of post-truth post-modernism aka "Formalism". We cannot draw sound theorems from false premisses. Begging the question by circular reasoning is dishonest sofistry, not honest and rigorous pure mathematics. Constructibility means ability to demonstrate a mathematical truth step by step. Elementary proof theory requires ability to give a proof by demonstration, which can be critically observed by mathematical cognition present in other partners of mathematical discourse. We can present a relatively simple demonstration of two-sided Stern-Brocot type construction in which the continued fractions of square roots on the Left and Right sides of the construction are different forms, and Boolean NOT-operations relative to each other.
It's just a matter of definition and context. When someone says *the* square root it usually refers to the principle root. The positive and negative numbers are used in the context of complex numbers
Though that's not what the Newton guy argues in the video, D'Alembert's theorem does have proofs, algebraic or otherwise. Real numbers are a field, Complex numbers are a field, that a basic thing. Your premise is kinda weird since his video only tries to assess the confusion of some of his viewers with the principle root and absolute value.
@@TheRedRave From wiki about D'Alembert's theorem: "Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept." The roots of the confusion go deep into foundations of mathematics, and foundational issues cannot be solved authoritarian declarations. The claim that real numbers are a field is nothing but an authoritarian declaration, and even as such blatantly counterfactual. No matter who and how many times claims that Empreror's new clothes exist, well, they don't.
You have made a brilliant video. You make us black people proud. Yes, you could have added one more thing that is if you want to get the -2 result for 4 using the root function then people should use -root(4)=-2. That is why if some one needs both back then they can use +-root(x) function but yes |x| solves all of that and that is why we use abs(x) as mathematician. I am really happy, you explained really well. Racism should not have a place in the world. Allah created us all from two humans and that is why all of us are same and racism should not exist. May Allah bless you.
Sorry to be late to the discussion. But..... sqrt(9) = 3 This is the PRINCIPAL square root. This implies that there are other roots which are NOT principal? If there is only one solution why have a special label? I think you have done a brilliant job dealing with this topic. I don't disagree with your approach at all which as usual is rigorous. However, I am not convinced that the mathematical ambiguity has been laid to rest other than for academic mathematicians who appeal to particular definitions when convenient and ignore them otherwise.
Square root of a number - relation (can have more then one answer) Square root function - function by definition means at MOST one answer per input ('Primary branch')
Cant you define the square root function on the range of 0 to -infinity, then you would receive a negative for every input value so sqrt(x+1) = -2 would exist, but =2 wouldn't
well true it fails the vertical line test, but it passes the horizontal line test where y=x^2 fails the horizontal line test. That arm is there in the negative. the sqrt(9) is +/- 3 so x is +/-3
question, I haven't finished the video yet(doing this so I won't forget), what if the value or variable inside the square root was negative, should I get the negative because in a square root, I will always get two numbers, the number multiplied by itself, so for example √9 I will get 3*3 so what if I √-9 will I get 3*-3 and take the -3 as the answer?
A square root by definition gives a number which when multiplied by itself gives the number that had the square root taken in the first place. In your example, 3 and -3 are different numbers, so neither are the square root of -9. sqrt(-9) is actually 3i. If you had x^2 = -9, then you would have the two solutions of 3i and -3i. The definition of the principal root extends to complex numbers!
Trust the hat. The hat will not mislead you. Also, if square roots cover both plus and minus answers, then why does the quadratic formula include a plus or minus? (Of course, it also allows for imaginary results, so maybe that's a bad example.)
@@PrimeNewtonsSorry, life has just been busy. Nothing bad, I've just been unable to hang out on RU-vid. I have a favorite hat, but if I'm being honest, I'm not really here because of the hats.
It is precisely because the radical sign denotes only the principal (i.e. positive) square root that we need the "plus or minus" in the quadratic formula to get both solutions.
The square root function IS A FUNCTION. A function is defined such that for any argument x there is ONE MAPPING TO THE CODOMAIN f(x). If √a=±b, then f(x)=√x cannot be a function!
Ok sqrt(x) = -2 has no solution. But for a long time people were saying sqrt(-1) = x had no solution. And then we gave it a name and discovered a new dimension for number. Could we call j so that sqrt(j) = -1 Have people try it ? does it make nice stuff like i or is j just a complex number I am not good enough to calculate ?
I think the absolute value step isn't terribly necessary to include (although it is completely explanatory of all parts), so that's why it's possibly not taught everywhere. Meaning, you can say that when you square root both sides, you can write: 9=x^2, (±)sqrt(9)=x, (and this is fine because like another user said, the quadratic formula has x = two solutions) Which it then follows that: (±)3=x. You insert the plus or minus BEFORE applying the square root function in order to satisfy the fundamental theorem of algebra that your original function of degree 2 will have 2 solutions. I think this is actually factually equivalent to the notion that sqrt(x^2)= |x|
Respected sir☺️, I have a new perspective about the equation [x=√9]. Please correct if I am wrong, I'll be genuinely so happy to get my mistakes corrected... Sir you see, mathematical equations is a way to express a PRACTICAL SITUATION... It is not incorrect to say that it is a "Language". I wholeheartedly feel that this is indeed a language... Coz we are "Expressing", and that is what a language Essentiality does! So, we see that in many languages, a word may have a different meaning in different situations. Here's an example: Statement: "I saw a man on a hill with a telescope." This statement can have two different meanings: 1. The person saw a man who was on a hill, and that man was using a telescope. 2. The person saw a man while they themselves were on a hill and was using a telescope to observe him. So, same happens with Mathematics, [TWO DIFFERENT PRACTICAL SITUATION MAY HAVE THE SAME MATHEMATICAL EQUATIONS]. And this condition is also with the equation [x=√9]. This equation DOES have ONLY one situation, but whether it is '-3' or '+3', that depends on the Practical Situation which is being represented. So, I think it is not essential that "Square root of a number is the principal root number", it is just a norm, because, this types of equations fits with the most of situations normally(norm-ally)... So, yes the fundamental theorem of algebra is a true principal, but this norm is to be taken just as a 'Norm'! At the end, I want to say that:- I do love mathematics (and Biology also), but I opted For biology(and not for maths) in my high school because I hated the boring style of majority of maths teacher around me. Otherwise I could take both of them together if I wanted. But you are unique. Your passion for mathematics touches my heart 💗💌🌹
I think I understand this, its essentially a case of determining whether youre looking for an input value or an output value, and if its an output, there cant be more than one value for any given input (otherwise ±n is fine) Edit: never mind, I do now - I did not see that 3=|x| coming, that was some finesse
Thanks for this very good explanation of the difference of the square root of number, being always positive, as against x^2 is equal to a number, hence the square root of that number is either positive or negative. Great 👍.
I agree with what you said in the video. Thanks for clarifying this for other people. I wish more teachers would explain it this way, instead of just saying "you need the ± square root" because thats what is most likely causing the confusion
this is really cool to understand, but its a shame that anyone in school has to just ignore it cause our exams will still expect us to assume that sqrt(x) has two solutions.
Firstly, let me start by saying that I enjoy watching your videos. I am an old electronic engineer with a master degree, who passed all these algebra and calculus exams. However, I have a feeling that is because I am disciplined and I know how to memorize, apply and follow the rules. One of the feelings, that I cannot get rid of, is about negative numbers and the algebra around them. When we count things we get positive (natural) numbers (once something is gone we cannot count it). Than we were taught that The Zero is a useful 'reference point' on any scale as is temperature, pressure, distance and such. Temperatures do go below zero (zero being chosen conveniently) as the time before a rocket launch may be "negative" to describe what was happening before and after the launch. Yet, if we multiply -2 degrees Celsius by 2 we get -4 degrees and we do get +4 degrees when we multiply -2 by -2 (multiplicator of -2 signaling negative/opposite direction of multiplication). The two multiplications of the value of -2 degrees Celsius caused the multiplicand to grow 2 degrees in negative direction (down) and 6 degrees in positive direction (up)...(one may expect that -2 x -2 equals zero or +2). The feeling grows worse when I see that same paradox being applied across all the physics and electronics... one day being stuck in quantum mechanics, multidimensional reality/worlds etc. Please, comment the subject and enlighten my retired soul.
I can't do a rigorous proof, but I think the idea that a negative times a negative equals a positive is just a consequence of the use of negative numbers, otherwise math becomes inconsistent. The demonstration I like best is 10 x 10 = 100 (11 - 1)(11 - 1) = 100 121 - 22 + (-1 x -1) = 100 99 + (-1 x -1) = 100 Therefore, in order for math to stay consistent, you need to accept that -1 x -1 = 1.
@@jeffstryker2419 Thank you so much for responding...you are a rare person to find this math-case worth a discussion. Yet, you confirmed an approach of a follower rather than a philosopher/creator... the same like me. However, the bigger question here is that math-processing that is based on quite a few convenient methods, like this negative-numbers one, may be an obstacle for the future scientists. It seems to me that math (as it is) allows discussion that one thing may be in a couple of different places at the same time. WHAT IF "we" would not get stuck there if we had not accepted those convenient assumptions, that make no sense even for the biggest scientists of modern time. Never the less, I am so grateful that you commented. I hope you will pass the "case" to your esteemed colleagues and someone comes up with a better explanation then ours. All the best!
technically, (x -> x^2) : (-inf,inf)->[0,inf) isn't an invertible function (it isn't one to one), thus a "square root" doesn't exist but that's problematic, so we solve that! we ARBITRARILY redefine the function of square by throwing away all of the negative values in the domain, thus we get: (x -> x^2) : [0,inf)->[0,inf) and now this function is invertible, so we can get the square root! The range of any inverse function is always the domain of the inverted function, and thus: sqrt : [0->inf)->[0,inf) as is very easy to see, the sqrt function only outputs nonnegative numbers. The important thing to notice is that this is ARBITRARY! I could have just as easily thrown out the positive x values instead of the negative values and gotten (x -> x^2): (-inf,0]->[0,inf) and that too is a perfectly legal invertable function that has an inverse who's range is (-inf,0] (ie, its output is always nonpositive). But convention states that we take a positively domained square function and thus we get a positively valued square root function! * You could just as well take that negative sqaure root and a lot of things will still work just fine as long as you're consistent. *** Heck, you could technically take any weird domain for the square function, as long as it is one to one on that domain, you would be able to define any cursed abomination of a square root function to your hearts content! For example, we can take the domain D := (2N + [0,1)) union (-2N - 1 + [0,1)): Where N is the group of all natural numbers including 0 ie, D is the group of all positive numbers whose whole part is even, together with all the negative numbers whose whole part is odd we will get (x -> x^2) : D->[0,inf) and it is one to one and it is invertible (I will leave that proof as an exercise to the reader) And now our square root function's range is D. So now if our square root function's output is odd than it will be negative, and if it is even it will be positive. That's the power of an arbitrary definition. And because it can get so cursed, that is why it is so important to set a common convention for it, so when we're doing other things where the positive square root suffices we won't have to think about it.
If √ not negative has no solition then tell that to the engineering communities. Then there is no need to do "right hand rule" of checking thumb location where the fingers curl around an axis. Mathematics like this can't can't define gradient cross vector J mathematics type engineering. If always positive to be a solution then the right hand rule engineers use is not definable with vectors.
Does the logic of “principal square roots” apply to when we’re working with complex numbers? Generally we’ve (or just me) learnt to solve √(a₁+b₁i) = a₂+b₂i to always end up with a ±(a₂+b₂i) looking answer
i just despise people who think square root of a number is both plus and minus conjugates of it see it like this when x^2 = lets say, 9 its asking for numbers whose squares are 9 when is x= root 9, it only has one solution i.e 3
