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 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").
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.
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 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
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.....❤
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. 😅
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
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!
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
to me the main confusion came from the fact that the definition of the square root was always presented as: "the number which squared will give back the original number". given this definition, the result of a square root could be a negative number. if this definition collides with other fundamental rules in mathematics, then it should be taught in a different way, with a different definition. reminds me a bit of why 1 was excluded from being a prime number, because it conflicted with another fundamental rule.
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 👍.
Good video: It's important to understand the difference between the _mapping_ of y=sqrt(x), which includes both square roots, but is not a function, and the function y=sqrt(x) which is required to have a single solution in order to be a function.
Yes, I think it must be more stressed that sqrt function is a function and a function by definition never returns 2 different outputs for the same input.
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.
I’d love a wider discussion on negative numbers. When are they relevant. Two examples - if look at temperatures in Celsius then negative values are obviously essential. Then you go to geometry and the Pythagorean problems - a 3,4,5 triangle is never going to be 3,4,-5!
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! 😉
A philosophical question: how do we know i is not the same as -i. I’d like to think Rafael Bombelli arbitrarily chose one of the pair to be the positive/primary “i” and we have all just followed along since.
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')
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|
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
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.
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?
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.
How does the choice for how you decide to orient your 2 axis affect the square root function? Like if we invert the y-axis so negative numbers are now on top… Per the fundamental theorem of Galois any algebraic equation with rational coefficients that is satisfied by a positive root will also be satisfied by the negative root. But could we not let the negative root be (x+1)^1/2? And therefore the positive root would be -(x+1)^1/2? It also seems to me that a positive or negative sign is relative to a defined direction. Like a joystick might have its axis inverted such that you have to pull back or down to make the object go up.
It's amusing that you made that +-sqrt(x) mistake in a video that you posted 4 years ago, "What are natural numbers" and that you are now teaching us about it. Well, of course! "Never stop learning", you say! By the way, was it fun to start using block lettering? I still see you draw some letters/words cursive and it reminds me how hard it must be, I feel like there must be some psychological warfare going on in your head as you teach us stuff! Hehe...
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
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!
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 ?
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 💗💌🌹
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
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 TH-cam. 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.
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???
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!
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.
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'.
Wow, thank you!
you're defining mathematics for the people who think they know math. awesome keep going.
This single/handedly explained everything my teachers didn't know how to explain and I didn't understand! Thank you so much, God bless you! ❤️🔥
Amen
It was one of the greatest fundamental challenges which any math student could ever deal with , you've made it clear , God bless you.
I love your style. You are such a great teacher!
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").
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.
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.
Is it standard to assume the range of the square root function to be 0 to infinity, and that's why you always get a positive value?
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
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.....❤
You’re such a great teacher. I watched your videos virtually everyday and never got tired of watching them. Great job.
Those are some impressive blackboard skills! Your beautiful caligraphy is enough to earn a sub. Thank you, TH-cam recommendations.
So simple, but so tricky also! Thank you very much for your sharing!
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
Seriously, I'm not a math student, buh you are making me love math than my course
May the peace, blessings, and mercy of God be upon you. I have benefited a lot from you.
You're very good . Keep going !❤😊❤
@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. 😅
This is a charming youtube channel, incredible enthusiasm.
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
Beautiful class. Very complete. Bravo teacher!
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!
It's always nice to refresh yourself on the logic. Much appreciated!
Such a great teacher! Happy to be here!
It's like I'm watching Omar Sy (as Lupin) explain maths. Brilliant AND entertaining!!
Keep teaching!
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
Same
to me the main confusion came from the fact that the definition of the square root was always presented as: "the number which squared will give back the original number".
given this definition, the result of a square root could be a negative number.
if this definition collides with other fundamental rules in mathematics, then it should be taught in a different way, with a different definition.
reminds me a bit of why 1 was excluded from being a prime number, because it conflicted with another fundamental rule.
Oh man what a smart guy,never stop luv U
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 👍.
Excellent explanation. Unfortunately in many schools it isn’t explained us this way. Congratulations for your great help 👍
You are a superb math teacher.
Wow, thank you!
You are brilliant.
A million thanks 🙏
Wonderful, absolutely strong logic, mesmerizing❤❤❤
I can just say awsome 🎉🎉🎉🎉
Thank you
Keep going .....
03:43 perfect explanation. Thank you. 🙏☘️
I think at 6:00 the graph of √y=x would the part graph of y=x^2 which is in first quadrant.
Correct. He got confused.
Good video: It's important to understand the difference between the _mapping_ of y=sqrt(x), which includes both square roots, but is not a function, and the function y=sqrt(x) which is required to have a single solution in order to be a function.
Yes, I think it must be more stressed that sqrt function is a function and a function by definition never returns 2 different outputs for the same input.
I never see a good math teacher like you
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.
I’d love a wider discussion on negative numbers. When are they relevant. Two examples - if look at temperatures in Celsius then negative values are obviously essential. Then you go to geometry and the Pythagorean problems - a 3,4,5 triangle is never going to be 3,4,-5!
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! 😉
Very thorough. Niiicce!
It is very good that you discuss this topic !
Hey . forget about that people .
These are awsome
Thanks🎉
You did a great job at explaining this!
A philosophical question: how do we know i is not the same as -i.
I’d like to think Rafael Bombelli arbitrarily chose one of the pair to be the positive/primary “i” and we have all just followed along since.
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')
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|
Keep going sir❤❤❤
Nice explanation
Thanks Sir 🙏
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 PRIME NEWTONS ❤.
FROM INDIA 🇮🇳
you're helping me love math again
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.
That video deserves a Nobel Prize. Not for Mathematics, but for Peace. ✌️ 😂
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?
Great work
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.
I know all this, I just like watching you explain simple maths
How does the choice for how you decide to orient your 2 axis affect the square root function?
Like if we invert the y-axis so negative numbers are now on top…
Per the fundamental theorem of Galois any algebraic equation with rational coefficients that is satisfied by a positive root will also be satisfied by the negative root.
But could we not let the negative root be (x+1)^1/2? And therefore the positive root would be
-(x+1)^1/2?
It also seems to me that a positive or negative sign is relative to a defined direction. Like a joystick might have its axis inverted such that you have to pull back or down to make the object go up.
Now I understand that I don’t know mathematics
It's amusing that you made that +-sqrt(x) mistake in a video that you posted 4 years ago, "What are natural numbers" and that you are now teaching us about it. Well, of course! "Never stop learning", you say!
By the way, was it fun to start using block lettering? I still see you draw some letters/words cursive and it reminds me how hard it must be, I feel like there must be some psychological warfare going on in your head as you teach us stuff! Hehe...
Circles next?
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
And it would be great if you can name a book to read more on this sir
Bro sounds frustrated lmao
Many thanks for this video! 🎉😊
Why don't just
+√ -√ or ±√
??
Another way, you can also use difference of two squares
x²-3²=0
(x-3)(x+3)=0
x=±3
How is you black board so clean my one turned white
Thank you😭
On solving 9 = x^2 why not do (x+3)(x-3)=0 to get both solutions?
Could you solve this problem for complex solutions? That would be a nice way of thinking outside the box for most people.
How does this hold when you write 4^0.5?
Thanks!
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
The basic explanation should be as you has said:
1st order equation has one solution
2nd order equation has 2 solutions, and so on....
I think the elegant way to solve 9 = x² is:
9 = x²
x² - 9 = 0
(x + 3) (x - 3) = 0
x + 3 = 0 or x - 3 = 0
x = -3 or x = 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!
Why not just make it a multivalued function? It seems kind of confusing to have the inverse operation of squaring not have the same priorities.
Also who said that each side of an algebraic equation had to be a function?
I was kinda annoyed on your behalf for that short... can't believe people were acting like YOU were the stupid one 😂
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 am sure there is something outside mainstream high school or college math where that is correct.
3:50 point
Number one case is a radical equation. Radical equations are radical
Can i travel back in time and have you as my math teacher? Please? ❤
I don’t have Apple Pay; how can I send my thanks.
this only applies when x is real yes? since z = 4i^4 -1 = 3 would yield sqrt(z+1)=-2.
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 💗💌🌹
Math is precise and square root of -9 is not 3, but its "3i". "i" stands for imaginary number. i = √-1. You can search it on google for more info
@@1918w-j4r umm thats kind of not the point of the comment
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
The entire confusion stems from the fact that
√(x²) = |x|
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.)
Where have you been? I'll get your favorite hat for the next video.
@@PrimeNewtonsSorry, life has just been busy. Nothing bad, I've just been unable to hang out on TH-cam.
I have a favorite hat, but if I'm being honest, I'm not really here because of the hats.
Let's just make it a bonus.
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.
@@Steve_StowersI think that was the point OP was making, albeit with an odd use of a rhetorical question (if I'm correct)
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 feel like my life is a lie
this is perfect
best content
Principal root. End of video.
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!
No one explained this problem before.
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.