1st: Multiply both sides by 4. (Inverse operation of division) That gives you x² = -16 Because 4 times -4 is -16. 2nd: Square root on both sides (inverse operation of squaring) That gives you x = square root of -16... Not to be confused with x = square root of +16... Anyways, here you can just say "No Solution" or use the complex number system (explained wonderfully in this video) and arrive at the answer of plus(+) or minus(-) 4 times "i" (imaginary number), which is equal to the unsolvable square root of -1. The End.
An alternative answer to +/- 4i could also be: The point or vector (0,4) * +/- pi/2 = the point or vector (0,4i) within the complex plane. In other words, take the point (0, 4) and rotate it off of the real number axis (horizontal) within the complex plane and rotate it by +/- 90 degrees. It will then project onto the imaginary axis (vertical) at (0, 4i) or (0, -4i) respectively. Within the complex plane, the coordinate pairs are as follows: (Real, Imaginary) which is analogous to that of the standard 2D Euclidean Cartesian Coordinate Pair (X, Y) within the 2D XY Plane. An easy way to understand the complex plane and how it relates to rotations is simply that each real value has a complimentary imaginary counterpart. In other words, for every real value of x, there is an imaginary value x * i such that x and x * i are orthogonal or perpendicular to each other. It is a 90-degree rotation. A positive imaginary is a positive rotation, and a negative imaginary is a negative rotation within the complex plane. The only exception to this is when the value is 0. Then it just simply maps to the origin (0,0) in which there is No Real part and there is no Imaginary Part. The composition of a Real and Imaginary component as seen within the value 3 + 5i is a single value, a single term. Do not confuse this with basic arithmetic as having two terms. This is a single value, a single entity. When we see expressions or terms such as this, the 3 is the Real part, and the 5i is the Imaginary component. We use these components as a coordinate pair within the complex plane. So, the corresponding point or vector would be (3, 5i). Here we can map this point into the Complex plane. We trace over the Real horizontal axis by 3, then up or down vertically parallel to the Imaginary vertical axis by 5 units in the i direction. The point lies in the plane. It has a weight of or trajectory (magnitude) of 3 in the Real direction, and a weight of or trajectory (magnitude) of 5 in the Imaginary direction. This point (3,5i) by its term 3 + 5i is the vector that is drawn from the origin (0,0) to the point that lies in the plane at (3,5i). It sounds very complex, but in truth it is quite simple, quite intuitive, and easy to visualize and understand. However, it does require a basic understanding of vectors and their properties when applying transformations, onto them such as translations, rotations, and scaling. Knowing some Trigonometry can also aid in this but isn't required. Now, you might be asking, why would anyone want to or need to know this? Are you interested in how 3D Rotations are done? Are you interested in how Audio Processing or Image Processing works behind the scenes? Are you interested in how Electrical and Magnetic fields work, or how electricity works through a circuit? Are you interested in Cryptography? Numerical Analysis? And these are just the tip of the iceberg. This complex number system and the complex plane is the basis or entry point into understanding higher levels of mathematics such as the Fourier Series, Quaternions, and Complex Analysis. One has to first understand the Fourier Series in order to understand the Fourier Transforms. One has to first understand the Hamiltonians to understand the Quaternions or even the Octonions. And all of these do require some knowledge and understanding of the Complex Numbers which is based on the properties of the square root of (-1). Once one is able to understand that, and if you happen to get into specific fields of computer science that focuses on various algorithms, you may eventually come across the Fast Fourier Transforms and their inverses. This does involve the basic underlying principles and properties of Trigonometry, Linear Algebra, the Complex Number System, and even bits of Calculus. So, do not let others discourage you into thinking math isn't cool, or that it's useless. Math is very powerful, useful and effective. It is a universal language and a very powerful tool. You can do a lot with it if you take the time to understand it. You enjoy listening to music on your mp device? You enjoy watching Y.T. videos? You enjoy streaming movies from Netflix? You enjoy playing video games? You enjoy traveling in a vehicle, so you don't have to walk? Without people understanding and knowing math, none of that would be possible. So do be afraid or embarrassed to pick up a math book.
Ridiculously lengthy & convoluted explanation. Immediately write x = sqrt (-16) = +/-4xsqrt(-1). Sqrt(-1) is called i by a mathematical convention as it is not a real number ie. not part of the number line. Similarly zero and negative numbers took hundreds of years to become recognised as real numbers.
I'd immediately write x²=-16 and, with my level of maths, conclude that I clearly remember my teachers saying that this is impossible. Weird maths are weird.
@@Kualinar Not sure what you mean. What fraction? x = 4i and x = -4i are the two solutions to the equation in this video. Both those values for x lead to x² = -16, so x²/4 = -4
@@wernerviehhauser94 do they? Granted, my classroom is only tertiarily a math classroom, but translating my own experience with students, I grant that it may indeed take ten minutes IF they've never worked with complex numbers. If they know powers and the square-root function, then inducing with them that sqrt(-1) may be well defined and pose a use would perhaps take the length of this discussion. Not, I think, even a half-hour.
@@HighKingTurgon well, let me think.... explained to them in 9th grade, with examples, about 45 min. Repeated in 10th and 11th, and still I need to repeat it in 12th grade.... yes quite a number of them do. Not because they are dumb. But because they don't pay attention since they don't care.
Well I can just look at this and tell there is an imaginary number involved. x^2 = minus 16 therefore x = SQRT(-16) = +/- 4 SQRT(-1) = +/- 4i In the complex number set this is 0 +/- 4i
It isn’t about the level of Maths . It’s about the field that you are solving the equation in . The equation has no reel roots but 2 complex roots In other words, it has no solution in Real but it hasp two different solutions in C. P
Hi, John, I want to know if there is an ebook on basics of Calculus, which I learnt in college but was a nightmare. I want to refresh my understanding. Please reply.
x² = -16 ► x² = 16i² ► |x| = 4i ► x = ±4i Square root of +16 is precisely +4. and not ±4. And square root of x² is, in the same note, |x|. Thus x² = +16 ► |x| = +4 ► x = ±4.
the square root of 16 is +/- 4 because it does not specify if it is the positive root or the negative root it is not the of the same note as |x|. Your proof needs work x squared = 16 but can be either -4 or positive 4 therefore cannot equal |x| because only one answer is given. |x| is only used when disregarding neg roots.
super simple first we must isolate the x by multiplying both sides by 4. That will give us x² = -16. This is not normally solvable without introducing imaginary numbers. What is an imaginary number? it is simply the rule i = √-1. So this means i² must equal -1. We incorporate the rule used above. x = √-16. We rewrite this to √16 * √-1. √16 = 4 and as I said above the √-1 is i so 4i is one of the solutions. Now since we had an x² that means we have two potential solutions a positive and a negative. So the other solution is -4i. Now we must prove our work: 4² * i² = 16 * -1 or -16 now we take -16 divide it by 4 and we do in fact get -4 now to prove the negative 4i solution (-4)² * i² again equals 16 * -1 or -16 This fully proves the solution is in fact plus or minus 4i
complex num : sqrt(-1) = i (X^2)/(4) = -4 , (4)*(X^2)/(4) = -4 * 4 , X^2 = -16 , X = sqrt(-16) , X = sqrt(-1 * 4^2) , X = i * 4 = 4i ANS : 4i ----> i complex number not a variable
How much ground do you want to cover? This guy would've put me to sleep. I had a chemistry teacher like this guy- I had to read at home. Its excruciatingly slow. Mmmmmmkay.
I used the LCM method first then I cancel out. Second approach I used was cross multiplication then I divide the base by four on both side I ended up with -4.
The big question is whether x is an element of C or not - of it is the solution is 4i or -4i otherwise there is no solution ... does not depend on math courses taken...
It is very simple. Answer is a mix of real part and imaginary part. Real part is 4. Imaginary is i. i×i as per the definition is i squared equals 1 in Maths. So the answer is 4+i.
I had the sound off and saw you mark ±4i as incorrect. But then you went on and solved for ±4i and explained the imaginary numbers. So I went back, turned the sound on and saw that you were "quoting" someone as saying they didn't understand that. I was prepping an argument about what is wrong with "±4i" anyway. Glad I didn't have to do that. GREAT VIDEO! Thanks!
The square root ot -1 is i. The square root of ab is the square root of a times the square root of b. The square root of -16=the square root of 16 times the square root of -1=4i.
@@isaactate1667 "The square root ot -1 is i." Right when you talk about complex numbers. Wrong (or at least incomplete) when you talk about quaternions.
You need to make the point that the domain is what determines which solution is the right answer. It isn’t just because I am in a lower level of algebra.
Came to my answer at the thumbnail, but it might be wrong. I'm getting the answer as the empty set. For (x^2)/4=-4 to be true, the result of x^2, x * x, would have to be -16 -- but anything multiplied by itself yields a positive result.
In Complex Numbers, there _is_ a number that gives a negative answer when it's squared. It's called "i" and it's the square root of -1. That sounds weird but it's mathematically valid.
x^2 = -1x16 x = √(-1x16) = √-1 *√16 = 4i, only then one can get x^2 = -16 Consider the question, "the area of a square is 16 sq units. Find the side of the square". While mathematically we can get +/-4, obviously -4 units does not make sense
Unless the problems specifies ± or - square root, the answer expected will be the principal (or positive) square root. The exception is when it's the square root of an unknown, ie. square root of "X". Then both + and - need to be considered.
My first answer is "- E -" because thats what my calculator said when I tried to hit "Square root of negative 16." So it's probably some imaginary number like Eleventy six, or Four Eye (4i), or Fifty Twelve. In fact, I'll just go with "Fifty Twelve" because it sounds cooler.
Actually, because you took the square root, there is a +/- before that. This is a + solution and a - solution to that where 4i is the magnitude and the + or - is the directional information.
Regular short doses of math problems keeps my synopsis alive and well. Thank You,,, keep them coming.. How about for kids and their cell phones a system that parents can turn on that requires a short math problem every 60 min or social media lock up==except for emergency use..
I enjoy your videos. When I was young, we were taught +- results for square roots. In some of your videos, I see you talking about only the positive root being the answer. In today's video, its plus/minus. What's the full rule for determining the answer to something like sqrt(16)
I'm not surprised that you're confused, because he is using the notation in a sloppy, careless and incorrect manner. The √ symbol is DEFINED to mean "principal square root". So for example, √25 equals 5 and does not, EVER, equal -5. This gives a clear and precise way to use the √ symbol: The principal square root of a is √a The other square root of a is -√a And if you want to refer to both square roots you write ±√a √25 is 5 and only 5 -√25 is -5 and only -5 ±√25 is ±5 In the video we are dealing with the equation x² = -16. He's correct when he says that we know we're looking for two solutions because we have a quadratic. However, he is INCORRECT at the 18 minute 15 mark in the video when he says that "for our purposes √16 is ±4, because we're solving a quadratic equation". That is, to put it politely, complete bullshit. √16 is not, ever, ±4. The equation is x² = -16, therefore by the definition of the √ symbol, x = ±√(-16). That is how you know that you need to consider both square roots - because the √ symbol has ± in front of it. There is a standard formula for solving any quadratic equation. If you Google "quadratic equation formula" you will find it straight away. And even if you look at the formula and you don't understand it, you can immediately see that it has a √ sign in it, and in front of that √ sign is ±.
All quaternion solutions are missing. The concept of the domain is missing in the explanation. And there's a missed opportunity to introduce fields and a little Galois theory.
There is a glaring error in your notation towards the end of the video (around 18 minutes 15 seconds) and it is guaranteed to confuse people. You CANNOT say "For our purpose √16 equals ±4 in this equation". That is a direct contradiction of the meaning of the √ symbol. You do lots of videos where you explain to people that the √ symbol specifically means "principal square root" and therefore refers to just a single value. Since you so clearly understand that fact, why on earth are you confusing your students by violsting that fact in your own notation here???!!! Given x² = -16, it is complete nonsense to say "x = √16 × √(-1) but in this case I'm going to ignore the fact that √ means principal square root and allow myself to treat √16 as ±4". That's is mathematically illiterate. If x² = -16 then x = ±√(-16). THAT is how you signify that you are considering both square roots. You do not randomly decide to violate the definition of the √ symbol. You simply put ± in front of the √ symbol. Surely you know this. Surely you are familiar with the formula for solving a quadratic equation, which has ± in front of the √ symbol for precisely this reason!
@@thomaskole1391 Nope. The meaning of the √ symbol is clear and precise. The guy in the video knows this. He makes videos that explain this. He is just failing to follow his own teaching here. You seriously learned that the same symbol can sometimes mean one thing and sometimes mean something else??? That's shockingly bad teaching!!! Within minutes of this video being posted, someone was already in the comments asking how we are supposed to know when √ means just the principle square root (like this guy explains in other videos) and when √ is allowed to mean both square roots (like this guy uses it in this video). Other people have posted exactly the same question under other videos where this guy makes the same careless mistake. It is completely understandable why somebody who watches this guy's videos would ask that. But that question literally could not arise if this guy used the notation correctly. His shoddy, careless use of notation is confusing people in a completely predictable and completely avoidable way. We're solving a quadratic equation here. Why do you think the standard formula for solving a quadratic has ± in front of the √ sign? It's not for the shits and giggles. It's because it needs to be there.
@@MattStMarie-bm5sqMany people don't fully understand the precise meaning of the √ symbol, but that's generally an omission in teaching rather than being actively taught the wrong thing - they just never got to the point where that was made clear, or it got confused with the fact that numbers have two square roots. Actively teaching that √x means "both square roots of x" is shockingly bad and I'm sure that's rare. This symbol has a defined meaning in mathematical notation. It's defined that way for a reason. It gives us a clear and precise way to refer to one of the square roots, a clear and precise way to refer to the other square root, and a clear and precise way to refer to both of them. The principle square root of x is √x The other square root is -√x And if you want both then you write ±√x The example a lot of people will know is the formula for solving a quadratic equation. That requires both square roots, hence the ± in front of the √. I don't think I have *ever* seen that written without the ±. What would the alternative be? How would we refer specifically to one of the square roots (or specifically the other one) if √x meant both of them?
As step 2 of the equation I get x^2 = -16. And you can not get a real answer by taking the square root of a negative number. Mye guess before watching this video is that the answer is Ø (NULL). (Or possibly (not likely) +/-4))
Because of the negative you have to have the imaginary number of i And then you just square root 16 which is 4 so the answer is 4i .. as far as I can remember 30 years ago
The answer is NOT a real number. Answer : i*4 and i*-4 Even scientific calculators can't digest complex numbers like ✓-1. My calculator will display a small E in the bottom left corner and a zero as a result.
"scientific" isn't a very meaningful designation for a calculator. Calculators can be branded "scientific" when they don't really do very much more than the basic + - × / operations. But there are plenty of calculators that can work with complex numbers and have no problem at all with √(-1) = i.
I think, on purpose 😎 the domain is missing in the question. Multiply by 4 then x^2=-16. Uncontroversial step. Now, can a squared number be negative? Nope, not in the Real numbers. But, if we enter the complex world, yes. Sqrt(-16)= + or - i , times sqrt(16) = +/- 4i. Tablet teacher, what educational level is your target audience? 5th grade, no answer at all. 9th grade, no Real answer. 12th grade, I hope, the answer will be complex. I’ll leave alone the other infinitely many solutions. Though I think I was taught Euler’s formula around 12th grade… after watching, I did not expect this vid to go into cross multiplication. That’s way below complex math 🫠. Unexpected mix of arithmetic, algebra 1, 2. Going forward, could you please also add some college level content as teasers. At the outer end, maybe try and explain why the complex numbers C are a closed field, while R is actually not, and do it at a 12th grade level. That would be interesting 🤔
Actually no solutions is correct, because x² is always positive. The complex you get as the solutions are invented. No real solutions. I'm able to solve this, but I'm only interested in REAL solutions, NOT the invented or imaginary ones.
Arabic Numbers were invented. Degrees, of angles, was invented. i = √-1 was invented. Sin( ) & Cos( ) were invented. They were all invented. It is called mathematics.
@@givenfirstnamefamilyfirstn3935 That's true, but we're looking for the two square roots of -16. A square root of -16 is a number which, when multiplied by ITSELF, gives -16. 4i multiplied by itself gives -16, and -4i multiplied by itself gives -16.
1st: Multiply both sides by 4. (Inverse operation of division)
That gives you x² = -16
Because 4 times -4 is -16.
2nd: Square root on both sides (inverse operation of squaring)
That gives you x = square root of -16...
Not to be confused with x = square root of +16...
Anyways, here you can just say "No Solution" or use the complex number system (explained wonderfully in this video) and arrive at the answer of plus(+) or minus(-) 4 times "i" (imaginary number), which is equal to the unsolvable square root of -1.
The End.
The solution is 2 ×i ×i. That is 2 I squared
Isquareis is -1
It is 4i.
(x^2)/4=-4 ; X^2=-16 ; X= √-16 ; x= √-1 •√16 ; i•4
4i and (-4i)
An alternative answer to +/- 4i could also be:
The point or vector (0,4) * +/- pi/2 = the point or vector (0,4i) within the complex plane.
In other words, take the point (0, 4) and rotate it off of the real number axis (horizontal) within the complex plane and rotate it by +/- 90 degrees. It will then project onto the imaginary axis (vertical) at (0, 4i) or (0, -4i) respectively.
Within the complex plane, the coordinate pairs are as follows: (Real, Imaginary) which is analogous to that of the standard 2D Euclidean Cartesian Coordinate Pair (X, Y) within the 2D XY Plane.
An easy way to understand the complex plane and how it relates to rotations is simply that each real value has a complimentary imaginary counterpart. In other words, for every real value of x, there is an imaginary value x * i such that x and x * i are orthogonal or perpendicular to each other. It is a 90-degree rotation. A positive imaginary is a positive rotation, and a negative imaginary is a negative rotation within the complex plane. The only exception to this is when the value is 0. Then it just simply maps to the origin (0,0) in which there is No Real part and there is no Imaginary Part.
The composition of a Real and Imaginary component as seen within the value 3 + 5i is a single value, a single term. Do not confuse this with basic arithmetic as having two terms. This is a single value, a single entity. When we see expressions or terms such as this, the 3 is the Real part, and the 5i is the Imaginary component.
We use these components as a coordinate pair within the complex plane. So, the corresponding point or vector would be (3, 5i). Here we can map this point into the Complex plane. We trace over the Real horizontal axis by 3, then up or down vertically parallel to the Imaginary vertical axis by 5 units in the i direction. The point lies in the plane. It has a weight of or trajectory (magnitude) of 3 in the Real direction, and a weight of or trajectory (magnitude) of 5 in the Imaginary direction. This point (3,5i) by its term 3 + 5i is the vector that is drawn from the origin (0,0) to the point that lies in the plane at (3,5i).
It sounds very complex, but in truth it is quite simple, quite intuitive, and easy to visualize and understand. However, it does require a basic understanding of vectors and their properties when applying transformations, onto them such as translations, rotations, and scaling. Knowing some Trigonometry can also aid in this but isn't required.
Now, you might be asking, why would anyone want to or need to know this?
Are you interested in how 3D Rotations are done? Are you interested in how Audio Processing or Image Processing works behind the scenes? Are you interested in how Electrical and Magnetic fields work, or how electricity works through a circuit? Are you interested in Cryptography? Numerical Analysis? And these are just the tip of the iceberg.
This complex number system and the complex plane is the basis or entry point into understanding higher levels of mathematics such as the Fourier Series, Quaternions, and Complex Analysis. One has to first understand the Fourier Series in order to understand the Fourier Transforms. One has to first understand the Hamiltonians to understand the Quaternions or even the Octonions. And all of these do require some knowledge and understanding of the Complex Numbers which is based on the properties of the square root of (-1).
Once one is able to understand that, and if you happen to get into specific fields of computer science that focuses on various algorithms, you may eventually come across the Fast Fourier Transforms and their inverses. This does involve the basic underlying principles and properties of Trigonometry, Linear Algebra, the Complex Number System, and even bits of Calculus. So, do not let others discourage you into thinking math isn't cool, or that it's useless. Math is very powerful, useful and effective. It is a universal language and a very powerful tool. You can do a lot with it if you take the time to understand it.
You enjoy listening to music on your mp device? You enjoy watching Y.T. videos? You enjoy streaming movies from Netflix? You enjoy playing video games? You enjoy traveling in a vehicle, so you don't have to walk? Without people understanding and knowing math, none of that would be possible. So do be afraid or embarrassed to pick up a math book.
4i^2 = -16 and so does -4i^2
Ridiculously lengthy & convoluted explanation. Immediately write x = sqrt (-16) = +/-4xsqrt(-1). Sqrt(-1) is called i by a mathematical convention as it is not a real number ie. not part of the number line. Similarly zero and negative numbers took hundreds of years to become recognised as real numbers.
Agreed, it's really not complicated.
except for you, there is nothing really ridiculous here...
@@iamhe999
It's ridiculously long.
It literally just requires 2 or 3 calculations.
Multiply by 4
Then take the sqrt
+-sqrt(-16) is +-4i
@@iamhe999 no, he's right.. Its a snooze fest. Mmmmmkay?
I'd immediately write x²=-16 and, with my level of maths, conclude that I clearly remember my teachers saying that this is impossible. Weird maths are weird.
What is the name of the chalkboard software you are using?
Depends on whether complex numbers are permitted. If no, the answer is the null set. If yes, x= I (square root of -1.
More accurately involves square root of -1.
If complex solutions are permitted (which they must be, because the question says nothing to prohibit them) then x = ±4i
@@Kualinar Not sure what you mean. What fraction?
x = 4i and x = -4i are the two solutions to the equation in this video.
Both those values for x lead to x² = -16, so x²/4 = -4
There is no reason to assume that complex numbers are not permitted, and the square root of -16 is 4i.
Not complete since -4i is also a valid solution
X is 4i or -4i. Does that need ten minutes' explanation of one problem?
You'd be surprised. A lot of today's students need half an hour of explanation for x^2 = 4 => x=2 or x=-2 .......
@@wernerviehhauser94 do they? Granted, my classroom is only tertiarily a math classroom, but translating my own experience with students, I grant that it may indeed take ten minutes IF they've never worked with complex numbers. If they know powers and the square-root function, then inducing with them that sqrt(-1) may be well defined and pose a use would perhaps take the length of this discussion. Not, I think, even a half-hour.
@@HighKingTurgon well, let me think.... explained to them in 9th grade, with examples, about 45 min. Repeated in 10th and 11th, and still I need to repeat it in 12th grade.... yes quite a number of them do. Not because they are dumb. But because they don't pay attention since they don't care.
I wanted to aske the same question there is no way to negtive result.
@@MrSergioKon as I'm not quite sure what you were asking, let me ask something - do you know what complex numbers are?
Well I can just look at this and tell there is an imaginary number involved.
x^2 = minus 16 therefore x = SQRT(-16) = +/- 4 SQRT(-1) = +/- 4i
In the complex number set this is 0 +/- 4i
Very well explained
It isn’t about the level of Maths . It’s about the field that you are solving the equation in . The equation has no reel roots but 2 complex roots
In other words, it has no solution in Real but it hasp two different solutions in C. P
Hi, John, I want to know if there is an ebook on basics of Calculus, which I learnt in college but was a nightmare. I want to refresh my understanding. Please reply.
that is why you should keep your old text books unless seriously outdated.
@@MattStMarie-bm5sq you are right. I donated them after completing high school.
Well explained. :)
x² = -16 ► x² = 16i² ► |x| = 4i ► x = ±4i
Square root of +16 is precisely +4. and not ±4. And square root of x² is, in the same note, |x|. Thus x² = +16 ► |x| = +4 ► x = ±4.
the square root of 16 is +/- 4 because it does not specify if it is the positive root or the negative root it is not the of the same note as |x|. Your proof needs work x squared = 16 but can be either -4 or positive 4 therefore cannot equal |x| because only one answer is given. |x| is only used when disregarding neg roots.
-4 x -4 =+16 same as +4 x +4
super simple first we must isolate the x by multiplying both sides by 4. That will give us x² = -16. This is not normally solvable without introducing imaginary numbers. What is an imaginary number? it is simply the rule i = √-1. So this means i² must equal -1. We incorporate the rule used above. x = √-16. We rewrite this to √16 * √-1. √16 = 4 and as I said above the √-1 is i so 4i is one of the solutions. Now since we had an x² that means we have two potential solutions a positive and a negative. So the other solution is -4i.
Now we must prove our work:
4² * i² = 16 * -1 or -16
now we take -16 divide it by 4 and we do in fact get -4
now to prove the negative 4i solution
(-4)² * i² again equals 16 * -1 or -16
This fully proves the solution is in fact plus or minus 4i
complex num : sqrt(-1) = i
(X^2)/(4) = -4 , (4)*(X^2)/(4) = -4 * 4 , X^2 = -16 , X = sqrt(-16) , X = sqrt(-1 * 4^2) , X = i * 4 = 4i
ANS : 4i ----> i complex number not a variable
If you go to a music channel that explicitly claims to teach music and complain about their being explanations about music... need I say more?
How much ground do you want to cover? This guy would've put me to sleep. I had a chemistry teacher like this guy- I had to read at home. Its excruciatingly slow. Mmmmmmkay.
I used the LCM method first then I cancel out. Second approach I used was cross multiplication then I divide the base by four on both side I ended up with -4.
The big question is whether x is an element of C or not - of it is the solution is 4i or -4i otherwise there is no solution ... does not depend on math courses taken...
0:14 The picture doesn't seem to me to match the description. are you dividing by -4(description) or by 4(chalkboard)
Irrelevant. Same formula...
X^2 = -16
answer contains "i"
i = sqrt(-1)
It is very simple. Answer is a mix of real part and imaginary part. Real part is 4. Imaginary is i. i×i as per the definition is i squared equals 1 in Maths. So the answer is 4+i.
I squared is -1
@@pas6295 I suggest you watch it again starting at 17.30.
4xi not 4+i
People who are not interested please stop saying these comments
The title is wrong. It says “divided by -4” which doesn’t appear in the equation on the screen.
I had the sound off and saw you mark ±4i as incorrect. But then you went on and solved for ±4i and explained the imaginary numbers. So I went back, turned the sound on and saw that you were "quoting" someone as saying they didn't understand that. I was prepping an argument about what is wrong with "±4i" anyway. Glad I didn't have to do that. GREAT VIDEO! Thanks!
Can you tell me what graphics app you're using for your 'white board' please? . I use onenote but it's very linited.
*x^2=(4i)^2 x=±4i*
the video is so long . the démonstration is simple than that
x² /4 = - 4
x² = -1 • 4²
x² = i² • 4²
x = 4i ; x = - 4i
X^2/4 = -4 It's probably easier to just multiply both sides by 4 here
X^2 = -16
X = square root of -16
I'm going to say no solution
The square root ot -1 is i. The square root of ab is the square root of a times the square root of b. The square root of -16=the square root of 16 times the square root of -1=4i.
@@isaactate1667 "The square root ot -1 is i."
Right when you talk about complex numbers. Wrong (or at least incomplete) when you talk about quaternions.
@@isaactate1667 your missing a root it's 4i and -4i
You need to make the point that the domain is what determines which solution is the right answer. It isn’t just because I am in a lower level of algebra.
Came to my answer at the thumbnail, but it might be wrong. I'm getting the answer as the empty set.
For (x^2)/4=-4 to be true, the result of x^2, x * x, would have to be -16 -- but anything multiplied by itself yields a positive result.
Any REAL number multiplied by itself is positive.
In Complex Numbers, there _is_ a number that gives a negative answer when it's squared. It's called "i" and it's the square root of -1. That sounds weird but it's mathematically valid.
x^2 = -1x16
x = √(-1x16) = √-1 *√16 = 4i, only then one can get x^2 = -16
Consider the question, "the area of a square is 16 sq units. Find the side of the square". While mathematically we can get +/-4, obviously -4 units does not make sense
Unless the problems specifies ± or - square root, the answer expected will be the principal (or positive) square root. The exception is when it's the square root of an unknown, ie. square root of "X". Then both + and - need to be considered.
My first answer is "- E -" because thats what my calculator said when I tried to hit "Square root of negative 16." So it's probably some imaginary number like Eleventy six, or Four Eye (4i), or Fifty Twelve. In fact, I'll just go with "Fifty Twelve" because it sounds cooler.
u are a genius ..u BS 20 min and make a video ..
As I recall from Algebra ll, x = 4i
Actually, because you took the square root, there is a +/- before that. This is a + solution and a - solution to that where 4i is the magnitude and the + or - is the directional information.
Squaring a real number does not result in a negative number. Gotta go complex. 4i or -4i
Regular short doses of math problems keeps my synopsis alive and well. Thank You,,, keep them coming.. How about for kids and their cell phones a system that parents can turn on that requires a short math problem every 60 min or social media lock up==except for emergency use..
Someone who does not listen to feedback and is too busy justifying himself to change. A hopeless case.
save that for app developers
X^2 ÷ 4 = -4
X^2 = -16
X = 4i and (-4i)
My first thought is 4i... sorry, I forgot -4i (it's almost 40 years since I graduated)
Why do you take more tan 10 minutes to arrive at x^2 = -16 ? After that, we can quickly deduct that the answers are 4i and -4i.
he is explaining the process
X**2=-4x4, x**2=-16, x=square root of- 16, an imaginary number
Χ^2=-16=>
Χ=(sqr-16)
No solution
Very simple and short
I enjoy your videos. When I was young, we were taught +- results for square roots. In some of your videos, I see you talking about only the positive root being the answer. In today's video, its plus/minus. What's the full rule for determining the answer to something like sqrt(16)
I'm not surprised that you're confused, because he is using the notation in a sloppy, careless and incorrect manner.
The √ symbol is DEFINED to mean "principal square root". So for example, √25 equals 5 and does not, EVER, equal -5.
This gives a clear and precise way to use the √ symbol:
The principal square root of a is √a
The other square root of a is -√a
And if you want to refer to both square roots you write ±√a
√25 is 5 and only 5
-√25 is -5 and only -5
±√25 is ±5
In the video we are dealing with the equation x² = -16. He's correct when he says that we know we're looking for two solutions because we have a quadratic. However, he is INCORRECT at the 18 minute 15 mark in the video when he says that "for our purposes √16 is ±4, because we're solving a quadratic equation". That is, to put it politely, complete bullshit.
√16 is not, ever, ±4.
The equation is x² = -16, therefore by the definition of the √ symbol, x = ±√(-16).
That is how you know that you need to consider both square roots - because the √ symbol has ± in front of it.
There is a standard formula for solving any quadratic equation. If you Google "quadratic equation formula" you will find it straight away. And even if you look at the formula and you don't understand it, you can immediately see that it has a √ sign in it, and in front of that √ sign is ±.
shows the complexity of playing with numbers.
All quaternion solutions are missing. The concept of the domain is missing in the explanation. And there's a missed opportunity to introduce fields and a little Galois theory.
basic math review this is not nessecerily for the introduction of higher mathematics.
There is a glaring error in your notation towards the end of the video (around 18 minutes 15 seconds) and it is guaranteed to confuse people.
You CANNOT say "For our purpose √16 equals ±4 in this equation". That is a direct contradiction of the meaning of the √ symbol.
You do lots of videos where you explain to people that the √ symbol specifically means "principal square root" and therefore refers to just a single value.
Since you so clearly understand that fact, why on earth are you confusing your students by violsting that fact in your own notation here???!!!
Given x² = -16, it is complete nonsense to say "x = √16 × √(-1) but in this case I'm going to ignore the fact that √ means principal square root and allow myself to treat √16 as ±4".
That's is mathematically illiterate.
If x² = -16 then x = ±√(-16).
THAT is how you signify that you are considering both square roots. You do not randomly decide to violate the definition of the √ symbol. You simply put ± in front of the √ symbol.
Surely you know this. Surely you are familiar with the formula for solving a quadratic equation, which has ± in front of the √ symbol for precisely this reason!
I took advanced algebra in college and learned it the same way it's shown in the video.
I believe you're incorrect and/or confused.
@@thomaskole1391 Nope. The meaning of the √ symbol is clear and precise.
The guy in the video knows this. He makes videos that explain this. He is just failing to follow his own teaching here.
You seriously learned that the same symbol can sometimes mean one thing and sometimes mean something else??? That's shockingly bad teaching!!!
Within minutes of this video being posted, someone was already in the comments asking how we are supposed to know when √ means just the principle square root (like this guy explains in other videos) and when √ is allowed to mean both square roots (like this guy uses it in this video). Other people have posted exactly the same question under other videos where this guy makes the same careless mistake.
It is completely understandable why somebody who watches this guy's videos would ask that. But that question literally could not arise if this guy used the notation correctly. His shoddy, careless use of notation is confusing people in a completely predictable and completely avoidable way.
We're solving a quadratic equation here. Why do you think the standard formula for solving a quadratic has ± in front of the √ sign? It's not for the shits and giggles. It's because it needs to be there.
@@gavindeane3670 most people were taught that it can be used for both roots I agree with thomaskole here.
@@MattStMarie-bm5sqMany people don't fully understand the precise meaning of the √ symbol, but that's generally an omission in teaching rather than being actively taught the wrong thing - they just never got to the point where that was made clear, or it got confused with the fact that numbers have two square roots. Actively teaching that √x means "both square roots of x" is shockingly bad and I'm sure that's rare. This symbol has a defined meaning in mathematical notation.
It's defined that way for a reason. It gives us a clear and precise way to refer to one of the square roots, a clear and precise way to refer to the other square root, and a clear and precise way to refer to both of them.
The principle square root of x is √x
The other square root is -√x
And if you want both then you write ±√x
The example a lot of people will know is the formula for solving a quadratic equation. That requires both square roots, hence the ± in front of the √. I don't think I have *ever* seen that written without the ±.
What would the alternative be? How would we refer specifically to one of the square roots (or specifically the other one) if √x meant both of them?
Look at the statement. You are saying divided by -4 but instead it's +4?
As step 2 of the equation I get x^2 = -16. And you can not get a real answer by taking the square root of a negative number. Mye guess before watching this video is that the answer is Ø (NULL). (Or possibly (not likely) +/-4))
I totally forgot even algebra. After almost 50 years using only for very basic simple needs...
x^2/4 = - 4
F(x)x^2 = 2x
2x/4 = - 4
2x = - 4(4)
2x = - 16
x = - 16/ 2
x = - 8
- 8(2) / 4 = - 4
- 16/ 4 = - 4
- 4 = - 4
The equation can not be formed naturally as minus times minus is a plus
Bring back fun math memories
These are fun math memories. And now you reply, "thanks for staying up with the rest of the class." Or not.
Agree with most of the comments. Ridiculously lengthy explanation
X2 = root -16.. imaginary number is confabulation
Trivial. Complex solutions.
(x^2)/4=(-4)
x=√(-16)
"x squared divided by - 4 = 4 What is x =? How well do you know Algebra and Quadratic Equations?" What's wrong with this?
Irracionales, x=-ix4, x=ix4
In the case of "complex number"
Correct goes √(-4)4
Because of the negative you have to have the imaginary number of i And then you just square root 16 which is 4 so the answer is 4i .. as far as I can remember 30 years ago
and -4i
take both roots pos/neg
√-4×4
You took the long way, start by multiplying each side by 4 - much easier to get to the answer.
No, x is equal to 4 and -4i. My mistake on previous submission.
The answer is NOT a real number. Answer : i*4 and i*-4
Even scientific calculators can't digest complex numbers like ✓-1. My calculator will display a small E in the bottom left corner and a zero as a result.
"scientific" isn't a very meaningful designation for a calculator. Calculators can be branded "scientific" when they don't really do very much more than the basic + - × / operations.
But there are plenty of calculators that can work with complex numbers and have no problem at all with √(-1) = i.
X=4i. Complex numbers
X = 4i where i = sqrt (-1)
I think, on purpose 😎 the domain is missing in the question. Multiply by 4 then x^2=-16. Uncontroversial step. Now, can a squared number be negative? Nope, not in the Real numbers. But, if we enter the complex world, yes. Sqrt(-16)= + or - i , times sqrt(16) = +/- 4i. Tablet teacher, what educational level is your target audience? 5th grade, no answer at all. 9th grade, no Real answer. 12th grade, I hope, the answer will be complex. I’ll leave alone the other infinitely many solutions. Though I think I was taught Euler’s formula around 12th grade… after watching, I did not expect this vid to go into cross multiplication. That’s way below complex math 🫠. Unexpected mix of arithmetic, algebra 1, 2. Going forward, could you please also add some college level content as teasers. At the outer end, maybe try and explain why the complex numbers C are a closed field, while R is actually not, and do it at a 12th grade level. That would be interesting 🤔
X^2/4 = -4 ; x^2=-16 ; x= √-16 ; x= ±√16•√-1 ; x= ±4•¡
4i, where I = √-1
Awesome 👍👋💪🌎❤️
+ or - 4i
You are right. Even Terrance Tao got it wrong.
+/- i4
Oh God, you are a teacher?
X = i4, i being the imaginary number i.
X = ± 4i
√(-4)4=√-16=4i
The correct answers are " yes, I can."(Short answer ) or " yes, I can solve it."(long answer), after all, it is a yes or no question😅😅😅
Actually no solutions is correct, because x² is always positive. The complex you get as the solutions are invented.
No real solutions.
I'm able to solve this, but I'm only interested in REAL solutions, NOT the invented or imaginary ones.
Arabic Numbers were invented. Degrees, of angles, was invented. i = √-1 was invented. Sin( ) & Cos( ) were invented. They were all invented. It is called mathematics.
4i or -4i
No wonder people struggle with maths. This is a ridiculously laboured description of a simple problem.
The final step
X^2=-16
X=4i
X1= 4 ; x2= -4
Is +- 4i the same as 4i?
No. +4i and -4i are different numbers just like +4 and -4 are different numbers.
Though +4 * -4 is the same as -4 * +4 ?
@@givenfirstnamefamilyfirstn3935 That's true, but we're looking for the two square roots of -16. A square root of -16 is a number which, when multiplied by ITSELF, gives -16.
4i multiplied by itself gives -16, and -4i multiplied by itself gives -16.
I did this in my head, and it appears to me to be: +/- 4i
I solved this in my head in 4i tenths of a minute!
4i. Took me a second.
Thank you, Mr. Obvious.
4
-1
+4i & -4i
X = i 4. You can see it right away.
Line
±4i in about five seconds in my head. Way too sleepy. Yes, it's a quadratic, so there are two roots. In this case, both are complex.
If one solution is complex, then the other must be complex too since complex roots (solutions) come in pairs.
You are all wrong, the answer is 42 !! Or you have to prove me that squr of (-1) is NOT 42 !!
X= -4 or +4
X^2=-16
Forgot my basic multiplication! 4i
I always forget the plus or minus