As fun as it is for me to revisit high school & college math with these videos, the thing that leaves me most impressed and entertained is watching him seamlessly switch dry erase colors on the go like it's nothing.
I'm curious to see how people incorrectly solved for x=3 without realizing something was off when the equation simplified to 18=0. Assuming they weren't aware of i, it would make far more sense to conclude there is no solution.
I prefer using your second method because it makes certain I get both solutions. As a tutor, forgetting the plus/minus seems too common. Also students get confused about why we need both values when solving an equation but only use the principal root in other square root problems; if I just factor, it’s a non-issue.
The problem there is middle school maths teachers who tell students to only use the principal root, to make things easy for them (easy for the middle school maths teacher that is). TabletClass Math put a video out doing that, trying to prove that an equation was impossible to solve when in truth it had a complex solution, that YT recommended to me last week. I was not a happy bunny.
@@StephenBoothUK I'm no expert on the history of math education but I'm guessing the use of the principal root isn't a recent idea for middle school, and I don't understand how that makes things easier for the teacher. Also, if you don't have the convention of using the principal root then what is the right answer for sqrt(121) - sqrt(9) - sqrt(4)? Or do you claim there are multiple right answers?
@@StephenBoothUK For the real numbers, the square root of x is defined as the principal root to x²=a. TabletClass Math provides instructional videos mainly to help people who struggle with basic math. For those, introducing complex numbers could be detrimental to their understanding. And, to be fair, when operating within the set of real numbers there are no solutions, when the discriminant is less than 0.
Rather than ±, it should be x₁ = √-9 and x₂=-√-9, the ± thing is just an abreviation that can be misleading unless they have done it with x₁ and x₂ a lot of times before. If you are teaching middle school students you would be doing a disservice by teaching them ±, they are not ready for being lazy in those details, they should understand first and abbreviate later, not the other way around.
As someone who lives in the real world, with real world numbers, I don't care what imaginary things people come up with, numbers or otherwise. They're simply convenient notions certain people use to justify nonsense. 😂 It's a non issue because you don't have negative 3 apples, people don't give birth to negative 2 children, and I've never been into a shop and been charged charged negative £5 for a frozen pizza. Have you? Some of this may be reflect my actual views, but I quite enjoy being negative. 😉
It's mostly because of the convention of mathematicians refusing to believe that you could take the square root of a negative, and calling it imaginary to mock them, when we later discovered that they had uses in physics
He doesn’t mean “real world” as in real-life, but rather the real number domain. It becomes more clear when he describes the complex number domain as the “complex world”
The use of complex numbers in physics is not because they have a connection to the "real" world, but because the math methods used require it. We never user the imaginary solutions like Tachyons, since even if they existed, they can not interact with out measurable universe.
Imaginary doesn't mean pretend it means it follows a different set of rules than the so called real numbers you're used to. The imaginary unit i is defined to be the square root of -1 and from that definition comes a whole lot of extremely useful properties for science and engineering. These numbers may behave differently from the reals but that doesn't make them pretend.
i feel like i²=-1 is just some mnemonic to help define i in such a way as to make or mathematically calculable. cause i sure looks like it's just a stand in just to make sure people don't lose track of roots made from negative numbers. the point is, it feels like the whole branch of math was just made up just to deal with rooting negative numbers and we just happened to figure out how to use it to our advantage after the fact
@@fomxgorl It's a general thing. You can say that integers “don't really exist” they're “made up” to deal with general subtraction. Rationals “don't really exist” they're “made up” to deal with general division. Reals “don't really exist” they're “made up” to deal with limits. Same thing here. And as with integers and rationals and reals, complex numbers turn out to be interesting in their own right; for example, 1 turns out to have k kth roots, and complex analysis (the underlying theory of doing calculus on complex numbers) has all sorts of useful properties that real analysis lacks. The entire field of mathematics is about tracking down the properties of definitions, and following them where they naturally “want” to go regardless of our preconceptions. It turns out that logic and physical reality are very closely related, so it's almost inevitable that mathematics turns out to be useful; for that not to be the case, reality would have to not make sense. I guess mathematics isn't for everyone, but just as physics isn't really “made up” because you can go measure in the real world, mathematics isn't “made up” because you can go check the proof. Meanwhile, what things “mean” is up to you; meaning is not a property of the world, but of your interaction with the world.
You dont find a use for them unless you go in a heavily technical field. People like to pretend these fields do highschool level math and then are surprised when theres more to maths after functions
@portobeIIa no doubt. They show up in wave equations a lot where i describes the phase. That's where I use it the most. I've never actually tried to find an imaginary root since highschool
Sure you can… you take the square as a result of a function and you take negative as a value of that number (as in 3 is +3 from 0 & -3 is minus 3 from 0)
math stopped representing reality long ago.... the result is modern society, an entire world that believes they live on a spinning ball spinning through an infinite vacuum despite being at rest in a pressurized system. It must be true because 'the math works"
Many areas of physics and enginerring such as signal analysis, circuits, quantum physics and etc. It’s main uses are when working with sinusoids due to euler’s formula, as well as fourier and laplace transforms.
They are used for turning differential equations into algebra, when solving electrical circuits involving more complicated than just resistances and simple sources. Essentially, the real and imaginary components are used for representing sine amplitude and cosine amplitude, of current and voltage waveforms. They are also used for generalizing the performance of capacitors, resistors, and inductors, with the concept of impedance, that allows for extending Ohm's law to apply to energy storage elements (capacitors and inductors) in addition to energy dissipation elements (resistors). While a go-to example is electrical, they also apply for mechanical vibration analysis with the same principles.
Another noteworthy application of complex numbers, is where they were discovered. There is a cubic formula, just like the quadratic formula, that just uses arithmetic, powers, and roots as the master key to solve any cubic equation. It works directly, when finding either the only real solution, or when finding the distinct solution in the case of two real solutions. However, when there are three distinct real solutions, it produces square roots of negative numbers at intermediate steps. There definitely are real solutions to such examples that you can verify by trial and error, or by graphing. But to get to them with the algebraic cubic formula, it requires a detour to the complex numbers. When it is all resolved, you'll have 6 total cube roots to find, and among them, there will be 3 complex conjugate pairs that add up to the 3 real solutions, as confirmation that they are still consistent with the algebraic cubic formula (Cardano's formula). There is a trigonometric cubic formula that avoids this detour, and finds the 3 real solutions more directly (Francois Viete's cubic formula).
-3 X is the benefactor. 3 is the value. Negotiate is the affliction. When X is charged to something, it will affect a change of minus 3 to the value. Yeaaaaaah. Now you see? You CAN negative a square because the negative is a subtract value and positive is an add value the number is the amount.
I'm used to seeing you tackle more difficult stuff, this is a nice change of pace. Is what I would have said if I didn't notice that this is a different channel with a basics behind the main name. 😅
You absolutely blew my mind with the factoring solution. I've ALWAYS been forgetting to do +/- when I take the sqrt of both sides in my college math courses, but showing it here as a result of factoring is going to help me remember for sure! Also showing that even adding a quadratic can be interpreted as the difference of squares, just minus a negative number was a paradigm shift for me.
That was exactly how we were introduced to the value i back in Grade Ten math. But you're mistaken that NATURAL NUMBERS are the squares of complex numbers. Natural numbers are restricted to the positive integers, and the square root of a positive integer has NO imaginary component. Squaring a complex number may yield a negative integer. For that matter, it may yield a negative real number such as -π. The correct statement, therefore, is that REAL NUMBERS are the squares of complex numbers.
@@starfishsystems imaginary component is always there it is just expressed as 0i - real numbers are a subset of complex numbers and the real numbers are always there as 0 + 5i etc
0:24 formally that is not quite correct. sqrt(x^2) equals sqrt(-9) but does not equal -sqrt(-9). But of course +-sqrt(-9) are the solutions of the original equation.
Before we get going, first blush is i3. 3^2 =9 so to get -9 (so we can add 9 to it and get 0) a negative times a negative is a positive so no normal number will do we need something to do so so we need 3 times sqrt( -1) or i to modify the 3 so that when squared it become -9.
Not true, a negative number is only turned into a positive if the exponent is also raising the negative which would be written as for example (-3)², when written as -3² it's just raising the number instead of the negative. Even with the first example, negatives cancel each other out so odd powers such as 3 and 5 would still equal a negative.
@@Sekiberius24 Are you replying to travissmith or to me?!? Either way, I don't understand what your comment has to do with what we wrote. What _exactly_ is "not true" in what he or I wrote? Please quote the specific part.
Good work; your algebraic manipulations are absolutely correct .. finally somebody who knows what they're talking about with this type of mathematics .. there's only one thing here though, that I'd like to point out .. I wouldn't phrase it as "not being in the real world" bc we would not be able to explain many things mathematically in branches of physics that actually are real world phenomena without the i operator at our disposal ... in electrical concepts, it would create difficulty, although we use j instead of i, but everything else about the operator itself remains consistent. Mathematicians use i for the square root of -1, which creates 4 useful identities when considering i to be raised to the second, third, and fourth powers. Have a great day!
The usual proposal is "lateral numbers", since they form an axis orthogonal to the Real number line. I prefer to call them "spinny numbers", or if I'm being more serious "elliptic numbers", since they perfectly model 2D rotation and most uses for them are really just types of rotation in disguise. "Elliptic" also specifically comes from elliptic geometry due to the existence of "hyperbolic numbers" (more often called "split-complex numbers") which are a close cousin and model hyperbolic geometry as well as complex numbers model elliptic or spherical geometry.
My idea: wall numbers, as opposed to floor numbers (the reals). It references the axes of the complex plane and is a bit of a pun, as in "apartment complex"
huh, I never thought of using i the way you used it in the second method. I always saw it as just a notation formality of sorts for negative square roots. That's honestly pretty cool!
I love coming back to these kind of math problems as an adult! It just feels like my brain is able to interpret them so much better than I could as a teenager.
@@coldheaven8007 x^2+9=0→x^2=-9=+or- 3i As written the answer has to be an imaginary number. But they had a good fix if they were asking how to modify the expression.
Squaring a positiv number or a negative number will result in a positive number always. On the other hand the square of i will return -1. Multiplying a postive number with a -1 will result in the negative of said number. So yes (+3i)^2 and (-3i)^2 does end up to be -9 in both cases.
Screaming at inanimate objects and thumbnails is a sign of madness. Before long you'll be using imaginary numbers too! Let's hope your psychiatrist only charges negative numbers. 😂
You're so good at explaining things intuitively. I knew the answer was ±3i, but I didn't know how to arrive at that answer from the ground up. Now I know two ways!
That takes me way back to school. I completely forgot about imaginary numbers since I don’t use it at my job and I’m feeling nostalgic just remembering what I did in specialist maths and in Engineering maths decades ago
Since we couldn’t solve square root of -1 with real numbers, we added the complex numbers with i being defined as i = (-1)^1/2, so therefore i^2 = -1 per definition
Secondary school teachers in the UK seem to have a vendetta against negative numbers, saying things like they are the "opposite" of positive numbers. The problem you posed in another video about "-3^2 = -9" now has taken hold whereas 5 years ago the answer would have been given as +9. There are going to be lots of problems for kids doing engineering and physics.
I mean, negative numbers are defined as the additive inverse of positive numbers, no? In that sense negative numbers are intuitively the opposite of positive numbers, though I do agree it’s important to build on that and define negative numbers using more precise terms after students have an intuitive grasp of what they are.
@@rainbowhorsecake4376 If I am testing a piece of electronic equipment and the voltmeter probe reads "-6.32V" that is a value. It is not the "opposite" of anything! Negative numbers are as legitimate as positive ones, nothing subordinate or secondary about them.
@@nahoj.2569 A reading of -6.32 is not the same or the opposite of +6.32. If you are working on electrical equipment, the two numbers are very different!
Thank you for a great video. You gave an easily understood explanation and also reintroduced me to an old friend, Math. It has been too many years since I last spent some time with that friend.😊
When I taught Algebra, I avoided using the descriptor "moved to the other side of the equal sign". I hammered on the students that solving equations can be easy, if you remember that you MUST do the same thing to both side of the equal sign, to keep both sides equal to each other. I shuddered, when I heard you say to "move the 9 to the other side". NO! You subtract 9 from both sides!
Hate to break it, but not giving any alternative where you can is a bad way of teaching. Yes, from the "correct" standpoint, there is no such thing as "moving" numbers, but some people can simplify calculations in their head just by doing that exact process
Imaginary numbers are part of the real world . Just because "i" cannot be projected on a 3D graph it sometimes is forgotten that the "i" plane is laterally displaced. It still exists as any electrical/engineering student will confirm.
Real numbers are used in the real life, on cash registers, scales, spreadsheets and calculators. But of course, most of those uses of numbers are just completely made up: the measuring system, the monetary values and etc. Imaginary numbers are used, fittingly, in basically anything that you can't see directly or that isn't obvious at first sight. You need complex numbers for basically any useful application of math that interacts with the natural, physical world at a significant level of detail. It allows us to use formulas to describe real world events that simply wouldn't be possible without a hidden, imaginary value to _round_ things up. It really adds a whole new _dimension_ to what math can do.
I'm personally an electronic engineer. Complex numbers (imaginary numbers) are very commonly used to denote the phase shift in AC circuit analysis. It's actually so commonly used that we use a 'j' instead of 'i' in rectangular form, as the lower case 'i' is already used for electric current.
That's easy because I'm good with imaginary numbers. I often imagine I'm a billionaire. Now in the real world it's not true, but in my imagination....... Lol
@@Ggdivhjkjldepends on country, I've noticed the more eastward you are the harder it is but the more you know, asia being the hardest, east-south europe being plenty easier and west europe/north america being the easiest or around east-south europe's level depending on school Overall I wish we had a universal education system
@@Ivan_Santos-Perez It's the result of a degrading educational system that heavily focuses on memorization instead of applying your knowledge, we study 1st language/history/chemistry more than we study math and that seems to be a common thing in most of the world, now don't get me wrong at least for stuff like chemistry and history there is ways to use your knowledge so it's not a lost cause, but something like 1st language taking priority over math is just sad as of now, unless you are just someone with such a mind the most they should be teaching is what/why/how languages, grammar, vocabulary, and some more, not the entire history of the language including but not limited to literature, figures of importance, etc. It's embarassing, really, especially since I am stuck in this generation
@fitmotheyap yeah, I know why it's happening, and it's up to you to educate yourself on what school misses out on, but with ai muddying up the internet, you guys are kinda screwed
@@MirlitronOneWell, if it's OK to use imaginary numbers to explain problems, it's OK to use other imaginary things to explain the world around us. For example, gravity is simply the result of billions of really tiny Smurfs, throwing velcro ropes onto everything physically and pulling it back down. Earthquakes are caused by large rock monsters farting, and the universe was made by "God", who then buggered off to do something more useful. See... imaginary things can explain ANYTHING.
@@another3997 do... do you honestly think that just because they're called imaginary numbers that they don't exist? Do you also eat urinal cakes because they have cake in the name? Surely you realize how silly of a statement this is to make. If you don't like the name "imaginary" call them quadrature numbers, or orthoganal numbers. If you don't understand them, that's fine, don't come out here pretending that everyone else is wrong because of your own personal incredulity. Be better than that, cheers.
ok so I knew about imaginary numbers but the factoring bit you showed on the right hand side of the board hat probably wasn't firecrackers or whatnot you just heard, it was my mind being blown away. Thanks so much I watch this stuff to keep my brain sharp and I found this really interesting.
I was almost one of those people who solved it wrong but I got stuck at remebering how any negative number squared becomes positive. Although I was familiar with the concept of imaginary numbers, I never would have solved using them until he reminded me that the square-root of a negative number is imaginary.
They are not saying -3, because they are looking at it too quickly. The human brain works that way, it's not because they don't know the actual answer.
But.. The actual answer ISN'T that x=-3 I would hade liked to put an exclamation mark after the above to stress what I wrote, but it would immediately follow a mathematical expression, and that would have messed it up completely :-)
I've solved your equation like 1^x=2 , without seeing your video. But i got only principal value i.e , without getting any "n" on principal value, could u please tell me, is it necessary to give general solutions?
I would say, that that's up to you. Nobody will hold you responsible for, how you solve an equation you find on youtube. That being said, I would find it more satisfying to find the full set of solutions, if I could, and to formulate a general solution would definitely be nice. In a formal setting, it would depend on the wording of the problem.
wait isn’t it 3i. Cause it’s the square root of negative 9, which is basically the square root of 9 x -1. Turn that square root of -1 to i, then square root that 3. You get 3i
@@HenshinFanatic They don't teach negative roots and imaginary numbers in 9th grade US math. They do go up to pre-calculus as an elective, but mandatory math classes top out at basic geometry here. So maybe this is "9th grade math" where you're from, but it's not for me.
Honestly I'm just glad it turned out you needed imaginary numbers to solve this. I was looking at this and thinking wait, this makes no sense, you can't simply solve that.
I will say I am unable to solve this, mainly because it's been so long I don't even remember if i was exposed to 'i = -1', and thus unable to parse through the square roots needed to solve this one. (I went through a very truncated calculus course after getting speed-blitzed from being unable to algebra correctly to 'competent at trig.' Many things were skipped or briefly gone over but otherwise not focused on) But I at LEAST knew that the answer was not '3'. because I recognize cancelling out +9, meant X^2 = -9. And you don't get a negative nine out of 3*3. People need to be taught how to sanity check, even if they don't know how to finish a problem. And the biggest sanity check in math is: Anything you do forwards, can be done backwards to where you started.
In engineering, we don't like calling it imaginary, since it is quite useful in very real differential equations and simplification via Laplace and Fourier transformation. Rather, we call it complex. You'll also see a "j" used in place of the "i" to discourage calling it imaginary.
What is a practical use for an equation like this? I enjoy math, but I’ve never pursued anything much beyond basic algebra. As fun as these can be, I like to know what, if any, practical purpose it has. Or is it just fun math stuff?
Anything with oscillation, imaginary numbers tend to be used as they're easier to use in a lot of cases. As peter here has mentioned, electronics is the most common example, if you have an AC circuit you can model the oscillating voltage as a complex number with Eulers identity, and the impedance (like AC resistance) with a complex number, doing this simplifies your problem significantly. it is in fact possible to do this without imaginary numbers, but that requires solving differential equations, which is possible but can become progressively tedious especially with larger and more complex circuits/systems. Like everything in math, they're a tool, they just make our lives easier.
The only thing I’ve encountered in mathematics so far that i cannot think or find much practical use for is ‘volumes of a revolution’. i however, is quite important.
"...not in the real world." I'm sorry, what? Imaginary numbers are actual numbers that have that name because they aren't on the "Real number line". Imaginary numbers are used to represent very real world measurements. Orthogonal motion or phase representation in wave phenomena (EM, sound, water, etc.) would be a big one.
I was initially disheartened but it let to me being sickened with an abject fear to the extreme extent that it had me repair to the vomitorium after I had read the title.
Solving x^4=1
ruclips.net/video/8qhGzsCyklQ/видео.html
As fun as it is for me to revisit high school & college math with these videos, the thing that leaves me most impressed and entertained is watching him seamlessly switch dry erase colors on the go like it's nothing.
You’re so right! I need to dedicate some time to mastering his technique.
Dry erase markers not highlighters
@@screenmonkey oops! good catch. fixed it.
Miss those teachers/professors.
@@nikdog419 I miss the delicate sound of felt scratching the glass. Shhhh....
I like that shirt.
It's "algebra" in Arabic for anyone wondering.
It is indeed algebra in Arabic, good catch!
Now write Vedic Math in Sanskrit!
🤣
btw if anyone's wondering its pronounced as "Al - Jabar"
Algebra was created by an Arab
What's the variable?
I'm curious to see how people incorrectly solved for x=3 without realizing something was off when the equation simplified to 18=0. Assuming they weren't aware of i, it would make far more sense to conclude there is no solution.
My guess is that they "half solved" it in their heads, and didn't check that their solution makes sense.
I feel like they meant -3
@@my3rdface389 -3² = 9
Maybe yeah.
9+9 = 0
Surely wouldn't have looked right
Honestly I got no clue what an imaginary number is
It is so sstisfying how he can seamlessly swap between the black and red marker as he writes
He should name his channel based on that technique!
Mathematical sleight of hand.
Me watching with monochrome filter on: 😬
I prefer using your second method because it makes certain I get both solutions. As a tutor, forgetting the plus/minus seems too common. Also students get confused about why we need both values when solving an equation but only use the principal root in other square root problems; if I just factor, it’s a non-issue.
The problem there is middle school maths teachers who tell students to only use the principal root, to make things easy for them (easy for the middle school maths teacher that is). TabletClass Math put a video out doing that, trying to prove that an equation was impossible to solve when in truth it had a complex solution, that YT recommended to me last week. I was not a happy bunny.
@@StephenBoothUK I'm no expert on the history of math education but I'm guessing the use of the principal root isn't a recent idea for middle school, and I don't understand how that makes things easier for the teacher. Also, if you don't have the convention of using the principal root then what is the right answer for sqrt(121) - sqrt(9) - sqrt(4)? Or do you claim there are multiple right answers?
@@StephenBoothUK For the real numbers, the square root of x is defined as the principal root to x²=a. TabletClass Math provides instructional videos mainly to help people who struggle with basic math. For those, introducing complex numbers could be detrimental to their understanding. And, to be fair, when operating within the set of real numbers there are no solutions, when the discriminant is less than 0.
Rather than ±, it should be x₁ = √-9 and x₂=-√-9, the ± thing is just an abreviation that can be misleading unless they have done it with x₁ and x₂ a lot of times before.
If you are teaching middle school students you would be doing a disservice by teaching them ±, they are not ready for being lazy in those details, they should understand first and abbreviate later, not the other way around.
As someone who lives in the real world, with real world numbers, I don't care what imaginary things people come up with, numbers or otherwise. They're simply convenient notions certain people use to justify nonsense. 😂 It's a non issue because you don't have negative 3 apples, people don't give birth to negative 2 children, and I've never been into a shop and been charged charged negative £5 for a frozen pizza. Have you? Some of this may be reflect my actual views, but I quite enjoy being negative. 😉
i dislike the sentence "not in the real world" because there are imaginary numbers in physic models...
It's mostly because of the convention of mathematicians refusing to believe that you could take the square root of a negative, and calling it imaginary to mock them, when we later discovered that they had uses in physics
Because it isnt in the real numbers
Non in the real(number) world
He doesn’t mean “real world” as in real-life, but rather the real number domain. It becomes more clear when he describes the complex number domain as the “complex world”
The use of complex numbers in physics is not because they have a connection to the "real" world, but because the math methods used require it. We never user the imaginary solutions like Tachyons, since even if they existed, they can not interact with out measurable universe.
Imaginary doesn't mean pretend it means it follows a different set of rules than the so called real numbers you're used to. The imaginary unit i is defined to be the square root of -1 and from that definition comes a whole lot of extremely useful properties for science and engineering. These numbers may behave differently from the reals but that doesn't make them pretend.
i feel like i²=-1 is just some mnemonic to help define i in such a way as to make or mathematically calculable. cause i sure looks like it's just a stand in just to make sure people don't lose track of roots made from negative numbers. the point is, it feels like the whole branch of math was just made up just to deal with rooting negative numbers and we just happened to figure out how to use it to our advantage after the fact
@@fomxgorl It's a general thing. You can say that integers “don't really exist” they're “made up” to deal with general subtraction. Rationals “don't really exist” they're “made up” to deal with general division. Reals “don't really exist” they're “made up” to deal with limits. Same thing here. And as with integers and rationals and reals, complex numbers turn out to be interesting in their own right; for example, 1 turns out to have k kth roots, and complex analysis (the underlying theory of doing calculus on complex numbers) has all sorts of useful properties that real analysis lacks.
The entire field of mathematics is about tracking down the properties of definitions, and following them where they naturally “want” to go regardless of our preconceptions. It turns out that logic and physical reality are very closely related, so it's almost inevitable that mathematics turns out to be useful; for that not to be the case, reality would have to not make sense.
I guess mathematics isn't for everyone, but just as physics isn't really “made up” because you can go measure in the real world, mathematics isn't “made up” because you can go check the proof. Meanwhile, what things “mean” is up to you; meaning is not a property of the world, but of your interaction with the world.
You dont find a use for them unless you go in a heavily technical field. People like to pretend these fields do highschool level math and then are surprised when theres more to maths after functions
@portobeIIa no doubt. They show up in wave equations a lot where i describes the phase. That's where I use it the most. I've never actually tried to find an imaginary root since highschool
There is no such thing as imaginary numbers or numbers that do not follow the rules we are used to.
How to solve x^3=8
ruclips.net/video/7ac4fp7M4t0/видео.html
Imaginary numbers? Yeah, right. Maybe that will be useful to count your imaginary friends.
2
@@JhonJairoBernal-i3j I should try to pay my bills with imaginary money.
X= sq. root^8
X=2
2^3=8
@@martytu20 it is cubic root, not square
My initial reaction: "but... you can't negative a square"
Imaginary numbers: Am I a joke to you?
@@wohlhabendermanagerI pretend they are, so I can laugh through my tears
Sure you can… you take the square as a result of a function and you take negative as a value of that number (as in 3 is +3 from 0 & -3 is minus 3 from 0)
math stopped representing reality long ago.... the result is modern society, an entire world that believes they live on a spinning ball spinning through an infinite vacuum despite being at rest in a pressurized system. It must be true because 'the math works"
Well you lack... Imagination
Get your "algebra" t-shirt 👉 amzn.to/3A8Ed4k
@@bprpmathbasics awesome. As an Arab I appreciate that it's written in Arabic
@@bprpmathbasics I was hoping for Algae Bras, too.
@@charlescox290 noooooooooooo
Because voltage is tension or pressure, it's expressed as tension or negative tension.
When else are "imaginary" numbers used in the real world?
Many areas of physics and enginerring such as signal analysis, circuits, quantum physics and etc.
It’s main uses are when working with sinusoids due to euler’s formula, as well as fourier and laplace transforms.
Anywhere that involves oscillation, or rotation, imaginary numbers are extraordinary useful.
They are used for turning differential equations into algebra, when solving electrical circuits involving more complicated than just resistances and simple sources.
Essentially, the real and imaginary components are used for representing sine amplitude and cosine amplitude, of current and voltage waveforms. They are also used for generalizing the performance of capacitors, resistors, and inductors, with the concept of impedance, that allows for extending Ohm's law to apply to energy storage elements (capacitors and inductors) in addition to energy dissipation elements (resistors).
While a go-to example is electrical, they also apply for mechanical vibration analysis with the same principles.
Another noteworthy application of complex numbers, is where they were discovered. There is a cubic formula, just like the quadratic formula, that just uses arithmetic, powers, and roots as the master key to solve any cubic equation. It works directly, when finding either the only real solution, or when finding the distinct solution in the case of two real solutions.
However, when there are three distinct real solutions, it produces square roots of negative numbers at intermediate steps. There definitely are real solutions to such examples that you can verify by trial and error, or by graphing. But to get to them with the algebraic cubic formula, it requires a detour to the complex numbers. When it is all resolved, you'll have 6 total cube roots to find, and among them, there will be 3 complex conjugate pairs that add up to the 3 real solutions, as confirmation that they are still consistent with the algebraic cubic formula (Cardano's formula). There is a trigonometric cubic formula that avoids this detour, and finds the 3 real solutions more directly (Francois Viete's cubic formula).
@@Ninja20704Imagine knowing this yet not spelling “engineering” correct?
That t-shirt, that poster in the back and writing with two marker i like it all
Thank you!
Good observation! I had to go back and look at the poster!
Thank you for the throwback to math lessons! It might not have been enjoyable back then, but I'm glad I still got the correct answer!
Solving x^3=64, the omega way! ruclips.net/video/1Lq5fsG48W4/видео.html
The omega way..
I thought I was smart & put - + lol
is it 4³=64?
@@pvzgamerlegisniana6492 Yes. It's basic stupid shhh you can do in your head. 4x4x4.
x^2=-9
√x^2=i√9
x=+-3i
-3
X is the benefactor.
3 is the value.
Negotiate is the affliction.
When X is charged to something, it will affect a change of minus 3 to the value.
Yeaaaaaah. Now you see?
You CAN negative a square because the negative is a subtract value and positive is an add value the number is the amount.
Damn, I do not miss maths.
I prefer the prequel, Math.
Just remember he’s wrong and you’re good 👍
I'm used to seeing you tackle more difficult stuff, this is a nice change of pace.
Is what I would have said if I didn't notice that this is a different channel with a basics behind the main name. 😅
You absolutely blew my mind with the factoring solution. I've ALWAYS been forgetting to do +/- when I take the sqrt of both sides in my college math courses, but showing it here as a result of factoring is going to help me remember for sure! Also showing that even adding a quadratic can be interpreted as the difference of squares, just minus a negative number was a paradigm shift for me.
I've never seen someone think of natural numbers as squares of complex numbers, that's amazing!
That was exactly how we were introduced to the value i back in Grade Ten math.
But you're mistaken that NATURAL NUMBERS are the squares of complex numbers. Natural numbers are restricted to the positive integers, and the square root of a positive integer has NO imaginary component.
Squaring a complex number may yield a negative integer. For that matter, it may yield a negative real number such as -π.
The correct statement, therefore, is that REAL NUMBERS are the squares of complex numbers.
Yeah, I learned it in grade 5 as the square root of a negative number, but I think you're thinking of "real numbers"
@@starfishsystems imaginary component is always there it is just expressed as 0i - real numbers are a subset of complex numbers and the real numbers are always there as 0 + 5i etc
Al-jaber 🗣️🔥
🗿
Its the guy who made Algebra its named after him
@@blintzy6969hell naw bruh
The inventor of Al Jabr aka algebra is Al Khawarizmi
I love how a wizard inventing his magic called his magic book Al-jaber.
@@toma.pudding that's what I'm saying
0:24 formally that is not quite correct. sqrt(x^2) equals sqrt(-9) but does not equal -sqrt(-9). But of course +-sqrt(-9) are the solutions of the original equation.
Love the t shirt :) algebra
Before we get going, first blush is i3. 3^2 =9 so to get -9 (so we can add 9 to it and get 0) a negative times a negative is a positive so no normal number will do we need something to do so so we need 3 times sqrt( -1) or i to modify the 3 so that when squared it become -9.
You forgot -i3.
Not true, a negative number is only turned into a positive if the exponent is also raising the negative which would be written as for example (-3)², when written as -3² it's just raising the number instead of the negative. Even with the first example, negatives cancel each other out so odd powers such as 3 and 5 would still equal a negative.
@@Sekiberius24 Are you replying to travissmith or to me?!? Either way, I don't understand what your comment has to do with what we wrote. What _exactly_ is "not true" in what he or I wrote? Please quote the specific part.
Good work; your algebraic manipulations are absolutely correct .. finally somebody who knows what they're talking about with this type of mathematics .. there's only one thing here though, that I'd like to point out ..
I wouldn't phrase it as "not being in the real world"
bc we would not be able to explain many things mathematically in branches of physics that actually are real world phenomena without the i operator at our disposal ...
in electrical concepts, it would create difficulty, although we use j instead of i, but everything else about the operator itself remains consistent. Mathematicians use i for the square root of -1, which creates 4 useful identities when considering i to be raised to the second, third, and fourth powers.
Have a great day!
he is talking about the real number world - not the actual real world
@ the square root of negative one is imaginary no matter how you look at it lol
Oh my goodness the way he seamlessly transitions from black to red is really satisfying.
Dude did the impossible and made math legitimately interesting for me to watch/listen to. Mad respect man, keep up the great work.
This mans marker skills are still so slick, plus thank you for explaining the +- part, never seen it written out
We need to rename imaginary numbers
The usual proposal is "lateral numbers", since they form an axis orthogonal to the Real number line. I prefer to call them "spinny numbers", or if I'm being more serious "elliptic numbers", since they perfectly model 2D rotation and most uses for them are really just types of rotation in disguise. "Elliptic" also specifically comes from elliptic geometry due to the existence of "hyperbolic numbers" (more often called "split-complex numbers") which are a close cousin and model hyperbolic geometry as well as complex numbers model elliptic or spherical geometry.
Imaginary numbers are sometimes referred to as pseudo scalars. Is that a better name?
@ProjectionProjects2.7182
"imaginary numbers" is the same as "pseudo scalars" which is the same as "fake numerals". language is funny.
@@freedomgoddess Yeah that is funny.
My idea: wall numbers, as opposed to floor numbers (the reals). It references the axes of the complex plane and is a bit of a pun, as in "apartment complex"
huh, I never thought of using i the way you used it in the second method. I always saw it as just a notation formality of sorts for negative square roots. That's honestly pretty cool!
Many students haven't seen the square root of a complex number
sqrt(5+12i)=? ruclips.net/video/_sHrvoIjvUM/видео.html
Please sir find all values of 4^x=x^64
I got two 256 and 1.0229
learn about W-Lambert function and values in W_0(x) and W_-1(x). In these the values are real but W(x) can define a lot of complex values.
I love coming back to these kind of math problems as an adult! It just feels like my brain is able to interpret them so much better than I could as a teenager.
Is it -(-3)^2+9=0
Creative answer. But not right for what was asked.
@@petersearls4443can you elaborate?
@@coldheaven8007 x^2+9=0→x^2=-9=+or- 3i As written the answer has to be an imaginary number. But they had a good fix if they were asking how to modify the expression.
@@petersearls4443 thanks
So -3 would give you the same result as 3?
Yes
Squaring a positiv number or a negative number will result in a positive number always. On the other hand the square of i will return -1. Multiplying a postive number with a -1 will result in the negative of said number. So yes (+3i)^2 and (-3i)^2 does end up to be -9 in both cases.
Was already screaming "±3i" when I saw the thumbnail
Screaming at inanimate objects and thumbnails is a sign of madness. Before long you'll be using imaginary numbers too! Let's hope your psychiatrist only charges negative numbers. 😂
@@another3997 Oh boy, I love it when I get paid for the services I'm using! WAAAAAAAAAAAAAAAAAAAAAAAAAA
I wonder how non mathematicians feel when they see a dude start screaming +or-3i 💀
Since I've learned all of this long ago this video felt like a nice rest from what I'm used to doing in my classes
Is that "al-jabbar" there?
Haven’t taken algebra for a while. I was about to object when I actually really read the equation.
Shared it with my family.
This was fun.
You're so good at explaining things intuitively. I knew the answer was ±3i, but I didn't know how to arrive at that answer from the ground up. Now I know two ways!
That takes me way back to school. I completely forgot about imaginary numbers since I don’t use it at my job and I’m feeling nostalgic just remembering what I did in specialist maths and in Engineering maths decades ago
Can "i" solve these?
(1) x^2=-1
(2) sqrt(x)=-1
(3) abs(x)=-1
Solution: ruclips.net/video/NSRcOfAjY9o/видео.html
1. x=±i
2. No solution
3. No solution
@@MC_Transport2 depends on whether or not you're using + or - square root. Technically the negative square root of 1 is in fact -1
why the square of imaginary number is -1?
Because square root of -1 is the imaginary number (i)
That's because the definition of i is the square root of -1. So by squaring both sides, you get i^2 = -1
Since we couldn’t solve square root of -1 with real numbers, we added the complex numbers with i being defined as i = (-1)^1/2, so therefore i^2 = -1 per definition
yours is the only true answer @@meme_ranker_Demet
it's the very definition of i.
Thank you for the lesson, good sir! Nice to learn something new! :)
Secondary school teachers in the UK seem to have a vendetta against negative numbers, saying things like they are the "opposite" of positive numbers. The problem you posed in another video about "-3^2 = -9" now has taken hold whereas 5 years ago the answer would have been given as +9. There are going to be lots of problems for kids doing engineering and physics.
I mean, negative numbers are defined as the additive inverse of positive numbers, no? In that sense negative numbers are intuitively the opposite of positive numbers, though I do agree it’s important to build on that and define negative numbers using more precise terms after students have an intuitive grasp of what they are.
@@rainbowhorsecake4376 If I am testing a piece of electronic equipment and the voltmeter probe reads "-6.32V" that is a value. It is not the "opposite" of anything! Negative numbers are as legitimate as positive ones, nothing subordinate or secondary about them.
@@karhukivi All of math is made up bro.
-6.32V is an opposite charge relative to a proton's positive charge, no?
@@nahoj.2569 A reading of -6.32 is not the same or the opposite of +6.32. If you are working on electrical equipment, the two numbers are very different!
@@karhukivi it is opposite tho. It's 6.32 below a reference point, while 6.32 is 6.32 above that same reference point.
Thank you for a great video. You gave an easily understood explanation and also reintroduced me to an old friend, Math. It has been too many years since I last spent some time with that friend.😊
oh wow 32 minute ago. i love your channel man!
I said -3i before I clicked. Is that still correct?
When I taught Algebra, I avoided using the descriptor "moved to the other side of the equal sign". I hammered on the students that solving equations can be easy, if you remember that you MUST do the same thing to both side of the equal sign, to keep both sides equal to each other.
I shuddered, when I heard you say to "move the 9 to the other side". NO! You subtract 9 from both sides!
Move to the other side means the same thing. We all understood that.
Hate to break it, but not giving any alternative where you can is a bad way of teaching. Yes, from the "correct" standpoint, there is no such thing as "moving" numbers, but some people can simplify calculations in their head just by doing that exact process
It's the same thing man...
To be honest, you don't need to move numbers or subtract numbers from both sides. Just use quadratic formula.
One of the things that always nakes math difficult for me, is the fact that I forget that I can just do stuff like 3i.
Imaginary numbers are part of the real world . Just because "i" cannot be projected on a 3D graph it sometimes is forgotten that the "i" plane is laterally displaced. It still exists as any electrical/engineering student will confirm.
No, it does not exist, it is a mathematical trick to make the calculations easier.
We need complex numbers because of trigonometry. Sinusoidal stuff creates huge issues and problems that are not easily solved without complex numbers.
Thank you for the simple clear explanation.
Outside of a class, how many people have actually needed to calculate something using imaginary numbers?
Engineers and other scientists.
Real numbers are used in the real life, on cash registers, scales, spreadsheets and calculators. But of course, most of those uses of numbers are just completely made up: the measuring system, the monetary values and etc.
Imaginary numbers are used, fittingly, in basically anything that you can't see directly or that isn't obvious at first sight. You need complex numbers for basically any useful application of math that interacts with the natural, physical world at a significant level of detail. It allows us to use formulas to describe real world events that simply wouldn't be possible without a hidden, imaginary value to _round_ things up. It really adds a whole new _dimension_ to what math can do.
I'm personally an electronic engineer. Complex numbers (imaginary numbers) are very commonly used to denote the phase shift in AC circuit analysis.
It's actually so commonly used that we use a 'j' instead of 'i' in rectangular form, as the lower case 'i' is already used for electric current.
If you ever needed to guesstimate an ETA, you have.
Many people uses complex numbers.
It would be helpful to just plot the graph so people can intuitively see why it can't equal 0 in real space.
I have to keep reminding myself that so so many people sleep their way through school. Took me all of about 3 seconds to know the answers.
This brings back bad memories-- but why am I more interested in this now that I'm out of school?
Because school sucks and the only reason it exists is to ensure a new generation of useful idiots.
Because you don't have to worry about your GPA.
That's easy because I'm good with imaginary numbers. I often imagine I'm a billionaire. Now in the real world it's not true, but in my imagination....... Lol
awesome writeup. I had a feeling it was going to have to involve i in some way. Thank you for this
As an 11 grader, this seems very illegal 😅
How did you not learn this in maths already?
@@Ggdivhjkjldepends on country, I've noticed the more eastward you are the harder it is but the more you know, asia being the hardest, east-south europe being plenty easier and west europe/north america being the easiest or around east-south europe's level depending on school
Overall I wish we had a universal education system
This is like 8th or 9th grade shit. School must actually be getting worse
@@Ivan_Santos-Perez It's the result of a degrading educational system that heavily focuses on memorization instead of applying your knowledge, we study 1st language/history/chemistry more than we study math and that seems to be a common thing in most of the world, now don't get me wrong at least for stuff like chemistry and history there is ways to use your knowledge so it's not a lost cause, but something like 1st language taking priority over math is just sad as of now, unless you are just someone with such a mind the most they should be teaching is what/why/how languages, grammar, vocabulary, and some more, not the entire history of the language including but not limited to literature, figures of importance, etc.
It's embarassing, really, especially since I am stuck in this generation
@fitmotheyap yeah, I know why it's happening, and it's up to you to educate yourself on what school misses out on, but with ai muddying up the internet, you guys are kinda screwed
The marker switching is crazy to me lol
I wanna know who decided imaginary numbers are okay.
Of course mathematican
Mathematicians who wanted to solve certain types of problems, which are relevant to the real world.
@@MirlitronOneWell, if it's OK to use imaginary numbers to explain problems, it's OK to use other imaginary things to explain the world around us. For example, gravity is simply the result of billions of really tiny Smurfs, throwing velcro ropes onto everything physically and pulling it back down. Earthquakes are caused by large rock monsters farting, and the universe was made by "God", who then buggered off to do something more useful. See... imaginary things can explain ANYTHING.
@@another3997 do... do you honestly think that just because they're called imaginary numbers that they don't exist? Do you also eat urinal cakes because they have cake in the name? Surely you realize how silly of a statement this is to make. If you don't like the name "imaginary" call them quadrature numbers, or orthoganal numbers. If you don't understand them, that's fine, don't come out here pretending that everyone else is wrong because of your own personal incredulity. Be better than that, cheers.
Veritasium has a fun video on this question: Epic Math Duel
What happens if you move the x sqaured to the right side of the equation and becomes negative x squared?
Bruh, you misspelled j
that's iota
Electrical engineer found
j and k are reserved for quaternions my guy.
Oi fellow Electrical brethren
i think everyone here failed the sarcasm identification test
This explanation is out of this world.
All you have to do is graph X^2+9 and quickly see that 3 is not the solution
Or do some simple math. 3^2 + 9 = 9 + 9 = 18, and 18 =/= 0
That's because they never claimed that 3 is the answer.
Yeah 3(or -3) isnt the answer. It’s 3i/-3i.
I double checked my work and still somehow got texas as my answer 3 times
I may be dumb but I don't think any math invoking imaginary numbers counts as math basics lol
ok so I knew about imaginary numbers but the factoring bit you showed on the right hand side of the board hat probably wasn't firecrackers or whatnot you just heard, it was my mind being blown away.
Thanks so much I watch this stuff to keep my brain sharp and I found this really interesting.
“What number multiplied times itself and then add 9 comes out to zero?” When you say it like this it really shows how it’s not solvable.
I was almost one of those people who solved it wrong but I got stuck at remebering how any negative number squared becomes positive. Although I was familiar with the concept of imaginary numbers, I never would have solved using them until he reminded me that the square-root of a negative number is imaginary.
bro's obssesed with complex number😭😭
Holy shit this just brought me back to school the moment he introduced the square root I remembered the process and answer
so instead of failing mathematics, i should have made stuff up like imaginary numbers
How about inventing something else, like imaginary letters?
Very well done!
They are not saying -3, because they are looking at it too quickly. The human brain works that way, it's not because they don't know the actual answer.
But.. The actual answer ISN'T that x=-3
I would hade liked to put an exclamation mark after the above to stress what I wrote, but it would immediately follow a mathematical expression, and that would have messed it up completely :-)
That is not how the human brain works.
Amazing. Please keep teaching.
haters will say they had no education and grew up in a poor family (they have several thousand in just their phone)
I've solved your equation like 1^x=2 , without seeing your video. But i got only principal value i.e , without getting any "n" on principal value, could u please tell me, is it necessary to give general solutions?
I would say, that that's up to you. Nobody will hold you responsible for, how you solve an equation you find on youtube. That being said, I would find it more satisfying to find the full set of solutions, if I could, and to formulate a general solution would definitely be nice. In a formal setting, it would depend on the wording of the problem.
My junior year Algebra 2 memories are flooding back. I think I blocked them out or forgot about them, but i understood what was being said
Why not apply directly the quadratic formula?
I was never taught to use i ever. How do you know when to use it? Just when there's a negative?
whenever you need to do square roots of negative numbers
Is uhh -3i and -i3 the same number if yes then -3i and 3i if bot then theres another -i3 it's the "zeros" of the equation
The i is a new concept for me. Thanks for the info!
Guy is a boss at switching marker colors 😊
wait isn’t it 3i. Cause it’s the square root of negative 9, which is basically the square root of 9 x -1. Turn that square root of -1 to i, then square root that 3. You get 3i
You forgot -3i.
@ yeah ur right I keep on forgetting that :c
I never took math this advanced, so this was very interesting to me. Thank you.
You've never taken 9th grade math or higher?
@@HenshinFanatic They don't teach negative roots and imaginary numbers in 9th grade US math. They do go up to pre-calculus as an elective, but mandatory math classes top out at basic geometry here.
So maybe this is "9th grade math" where you're from, but it's not for me.
I almost forgot about the i. Brilliant!
I would just write in: A squared integer can’t be negative, therefore the equation is wrong.
You’re a full blown adult, and you still don’t know how to solve high school maths 😭
@@countesselizabeth Complex or imaginary numbers aren't high-school math.
@ they literally are, but you might have a poor education system where you live.
My mind is very simple 😐 where dose the 3 in the 3i come from??
I'm more fascinated with the rapid pen changes!!
Honestly I'm just glad it turned out you needed imaginary numbers to solve this. I was looking at this and thinking wait, this makes no sense, you can't simply solve that.
I will say I am unable to solve this, mainly because it's been so long I don't even remember if i was exposed to 'i = -1', and thus unable to parse through the square roots needed to solve this one. (I went through a very truncated calculus course after getting speed-blitzed from being unable to algebra correctly to 'competent at trig.' Many things were skipped or briefly gone over but otherwise not focused on) But I at LEAST knew that the answer was not '3'. because I recognize cancelling out +9, meant X^2 = -9. And you don't get a negative nine out of 3*3.
People need to be taught how to sanity check, even if they don't know how to finish a problem. And the biggest sanity check in math is: Anything you do forwards, can be done backwards to where you started.
I learned about imaginary numbers yesterday in algebra 2 and I was surprised I was actually able to apply it to something lol
In engineering, we don't like calling it imaginary, since it is quite useful in very real differential equations and simplification via Laplace and Fourier transformation. Rather, we call it complex. You'll also see a "j" used in place of the "i" to discourage calling it imaginary.
What is a practical use for an equation like this? I enjoy math, but I’ve never pursued anything much beyond basic algebra. As fun as these can be, I like to know what, if any, practical purpose it has. Or is it just fun math stuff?
All electronics use them(voltage, resistance and current cannot be determined without them in the design process) and who doesn’t electronics?
Anything with oscillation, imaginary numbers tend to be used as they're easier to use in a lot of cases.
As peter here has mentioned, electronics is the most common example, if you have an AC circuit you can model the oscillating voltage as a complex number with Eulers identity, and the impedance (like AC resistance) with a complex number, doing this simplifies your problem significantly.
it is in fact possible to do this without imaginary numbers, but that requires solving differential equations, which is possible but can become progressively tedious especially with larger and more complex circuits/systems.
Like everything in math, they're a tool, they just make our lives easier.
The only thing I’ve encountered in mathematics so far that i cannot think or find much practical use for is ‘volumes of a revolution’. i however, is quite important.
"...not in the real world." I'm sorry, what? Imaginary numbers are actual numbers that have that name because they aren't on the "Real number line". Imaginary numbers are used to represent very real world measurements. Orthogonal motion or phase representation in wave phenomena (EM, sound, water, etc.) would be a big one.
How does he switch between the markers so smoothly
I was initially disheartened but it let to me being sickened with an abject fear to the extreme extent that it had me repair to the vomitorium after I had read the title.
Scary indeed. Whether this says something about the relevant EDUCATOR or about the person saying "It is only 3" is imho the question.
This video reminded me why I love math