I've been watching you for weeks now under the impression that you had at *least* 50k subscribers, I was surprised that it's not even at 15k! Definitely an underrated channel!
10:13 When I was an adjunct instructor, I would tell my students that complex numbers were used in electrical engineering. Finally I made a promise to myself and future classes that the next time I covered the topic, I would actually learn how it was used. I was essentially able to show the real-world interpretation of addition, subtraction and multiplication by a real number or a pure imaginary number. The only thing I didn't understand is what it meant when you multiplied by a nontrivial complex number, i.e., one where neither a nor b was 0.
When you multiply two complex numbers together, remember that you M the M's, and A the A's. Multiply the magnitudes, and add the angles. As an example, consider (4 + 3*i) multiplied by (12 + 5*i). Using FOIL, we get 4*12 + 12*3*i + 4*5*i + 15*i^2. Simplifying, we get 33 + 56*i. This has a magnitude of 65, which is consistent with multiplying magnitudes of 5 and 13. In an application, what this would mean is combining two electrical filter circuit transfer functions, to find the overall transfer function. It could also mean starting with a phase-shifted waveform, and sending it through a filter circuit.
Thank you for this video! you have no idea how much I’ve needed this! I always say the same to people who try to tell me my best friend is “imaginary”. That is just so hurtful and toxic. Through math they’re now proven wrong
C and R[i] are the same. They are both isomorphic to R^2 as vector spaces, but R^2 is not a field because there is no multiplication between vectors defined.
Thanks for answer. So The complex plane is represented as two real dimensions. It's a coordinate system in RxR, so essentially, it's two dimensional. But when we think of a complex number we don't think in terms of coordinates (x,y) rather each z∈C is just one entry. So how do we order complex numbers
@@Jo-bx6ezwe can’t order complex numbers, or at least not in an order that behaves the way we’d expect it to behave. For example, you know that 5 > 3. That means that if we take a positive number (for example 4) and multiply it by 5 and 3, then 5(4) > 3(4), or simplified, 20 > 12. It’s been proven that you can’t do the same with complex numbers. If you have that a > b, and multiply them by a complex number c that is bigger than 0, then we would expect to get that ac > bc. But it doesn’t (or if it does, then you probably broke another property of we expected to get from ordering them). In other words, you can’t order the complex number in a way that actually feels like you ordered them
@@NahhFam My Bad for ruinin' it chief 🫡. *Sigh* 😮💨 *Deep Breaths* 😮💨 😮💨 😮💨 It appears I have made a severe and continuous lapse in my judgement.... 😔
As a mathematician, computer scientist, and guitarist, it is interesting to see how the symbol i became overloaded. i is used to indicate imaginary number, integer variable, and index finger. Cool explanations!
The letter i is also used to denote electrical current. For that reason, electrical engineers often use j instead of i to denote the imaginary unit. And they have a good reason to. Electrical engineers use imaginary number all the time.
I dont think you should leave out the "5 levels" out of your youtube title. Subscribers might recognize the color thing on the thumbnail but Id imagine, new viewers dont. Its an inviting concept to know beforehand that the video in question explains something in increasing levels of complexity.
@@DrSeanGroathouse the 5 levels thing is good and will catch thr algo sooner rsther than later, if you have a few more of these in the pipeline you're all set
The claim that "imagniary numbers are made up" is right, but for the wrong reason. Most students who exclaim this do so in the belief that the real numbers are, well, more "real" than imagniary numbers. But the thing about that is that all numbers are made up in essence. There is no such thing as a "number" in the universe, they are constructed objects which only exist in our heads. That goes for real numbers as well. So if we are being pedantic, imagniary numbers are "imaginary", but so is every other kind of number in existance as well. I write this without having watched the video yet (it's in my watch later).
I have to disagree with the idea that numbers aren't real. They certainly aren't fictive in the way that stories are: their properties are not made up, but found. There are true things about them that will still be true even if nobody ever finds out.
@@ErreniumThe same logic can be applied to imaginary numbers then, there are true things about them that hold regardless whether we discover it or not. Both are equally real in that sense
"There is no such thing as a "number" in the universe", that's a dubious claim. One can argue that mathematics is a study of structure, and while our mathematical laws of how the universe works are certainly approximations, if humans were to be handed the true law from above, the way it is structured would certainly be an area of study for mathematicians. And for now it looks like numbers do have a place in the overall structure of our universe, complex numbers being absolutely indispensable. You might go the technically correct way and claim something like "There is no way to know if there's such a thing as a number in the universe" or "There is no way to know if the universe is mathematical or not", but those claims are of no more interest than something like "There is no way to know if god exists".
I feel that the term "imaginary" is just a misnomer that stuck. I would prefer the term "lateral", and I believe that for some languages, that is the case.
ah, i was kinda hoping the math major would bring up isomorphism classes of algebras! the thing is, a mathematician doesn't really think any of these definitions of the complex numbers are any more correct than any other. a set of formal symbols, a subalgebra of End(R²), the polynomial ring R[x]/(x²+1), the algebraic closure of R, a real 2D vector space with an automorphism squaring to -id: all of these constructions are "basically the same." the precise definition of this uses abstract algebra. in this context, we define an algebra (over R) to be any vector space over the real numbers with a bilinear product of vectors. examples of this structure are the real numbers themselves, the complex numbers, and the quaternions, but also strange things like R³ with the cross product. we also say that two algebras are isomorphic if there's a linear map between them which doesn't change products; that is, A and A' are isomorphic if there is some bijective linear f: A -> A' so that f(uv) = f(u)f(v). intuitively, this is saying that the only real difference between A and A' is the labels, since you can use f to essentially change the labels without changing the structure of addition or multiplication. now, if you look at the multiple definitions of the complex numbers, they're all different objects. however, they're also all vector spaces over R with a product between elements, so they're all algebras over R! and it turns out that they are all isomorphic: given the formal symbol i, you map it to the matrix ((0, -1), (1, 0)), the polynomial x, a solution to x²+1 = 0, or the automorphism squaring to -id. (also send 1 to 1 or id in each case; because each formal symbol is a+bi, this determines a map on all formal symbols.) this determines an isomorphism in each case, so every representation is isomorphic! in that sense, the thought of picking one representation is a bit silly. the complex numbers are kind of just what they are, and these explicit realisations are just concrete versions of a more general phenomenon. because we know they are all isomorphic, we can switch between them whenever we want and whatever we do with them remains valid.
This is my favorite, quick visualization for why i * i = -1: On a number line, performing -1 * -1 is like starting at 1, then making two 180 degree turns, first to -1, then back to 1, for a full 360 degree turn. That shows how two negatives multiplied make a positive. Similarly, performing i * i is like making two 90 degree turns, starting at 1 then first turning 90 degrees to 'i' on an axis perpendicular to the real number line (the imaginary number axis), then another 90 degree turn to -1, making a 180 degree turn in total. That shows how two imaginary numbers multiplied make a real number.
In a sense, all numbers are imaginary. Mathematics is a modeling system. Just like a drawing of a building is not the building. It’s how the building is imagined.
So, is "imaginary" just a nice way of saying "Hard for a human to comprehend but clearly valid because it solves the equation."? Seems to be my takeaway.
I do not have a best definition. As a physicist, it depends in the application. I did not know the quotient defintion, ie, # 5. Thanks! I went to a French high school in the 1970s. The complex numbers were introduced as "dilations" , ie, a rotation times a streching, your level 3 was my level 1. Now my every day definition is the usual a+bi definition with field properties... like everyone else. That definition makes the complex numbers "real" in a physical sense. You should mention that in Quantum Mechanics, the Schroedinger equations is similar to the wave equation but the appearance of "i" in there is what gives quantum mechanics it weirdness. By the way, I am now preparing a talk where I use quaternions..... they are an extension of complex numbers but not a field.
Most clear explanation of the definition of C = R[X]/(X^2+1) I have ever heard, congratulations! An anedocte: my calculus 2 teacher used to say: I know tho types of numbers, integers and complex numbers. As integers are too complicated, I will talk about complex numbers :)
Gemini 1.5 Pro: The video is about imaginary numbers. It explores what imaginary numbers are and why they are important. The video is divided into five levels. Level 1: Where the name imaginary numbers came from In the 1500s, mathematicians were trying to solve cubic equations. Cardano, a mathematician, came up with a formula to solve these equations. But the formula involved the square root of a negative number. Since negative numbers weren't understood at the time, Cardano called these numbers imaginary. Even though he thought of them as imaginary, he realized that they could be used to solve real problems. Level 2: The usual way we define complex numbers today Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers can be added, subtracted, multiplied, and divided. Level 3: How complex numbers can be defined as matrices Complex numbers can also be defined as matrices. A 2x2 matrix can be used to represent a complex number. This way of defining complex numbers does not require the imaginary unit i. Level 4: Practical applications of complex numbers Complex numbers have many practical applications. They are used in the Fourier transform, which is used to analyze signals and compress data. They are also used in electrical engineering and quantum mechanics. Level 5: An abstract algebra perspective The video talks about complex numbers from the perspective of abstract algebra. It explains that the complex numbers are algebraically closed, which means that any polynomial equation with complex coefficients has a complex root. The video also discusses another way to define complex numbers using polynomial rings.
There are infinite dimensions to numbers, just that most of the time we only work with one, and sometimes with two that’s where complex numbers come in and sometimes with 4 that’s where quaternions come in, and sometimes we need much much more and that’s why we have vectors (although they form different group than complex numbers and quaternions but we can also present complex numbers and quaternions as vectors). So yeah, “imaginary” numbers are as real as they get.
This video was really interesting, you have a rare talent of making abstract ideas accessible. Thank you again. As a developer, I have to say that the definition via matrices is the one I like best. Very Beautiful.
I’m not knowledgeable enough to apply it, but it’s pretty interesting to me that the complex numbers are isomorphic (?) to the geometric algebra scalar plus bivector.
I would argue that any numerical concept that does not occur in nature - such as imaginary numbers, or infinity, are 'imaginary'. I see how they are useful mathematically, but can't help wondering if they aren't some sort of workaround for something we don't quite understand.
I’ve been learning quantum mechanics recently, and that has given me an entirely new appreciation for the complex plane, especially when it comes to phase. The Euler equation is a way to split a number up into complex components without altering its absolute value. This is a very important property when we want to adhere to conservation principles while still allowing interference patterns to emerge under certain circumstances.
Consider the points in a plane. How can we make a field of these? Well, we need a 0 for addition, so pick a point to be 0. We can define addition of two points using the fourth point in the parallelogram containing the two points and 0. Using Euclid's geometry we can show that this is commutative and associative, so is a good addition. Indeed, the reflection about 0 is the additive inverse. To multiply we need a new point for an identity. Let's pick a new point called 1. We already have addition on this line from above and the point -1. Each point on the plane defines an angle relative to this line. To multiply two points construct the angle that is the sum of the two given angles and scale the distance from 0 to one by the distance from 0 to the other, using the distance to 1 as the unit. For two points A and B not on the line this consists of constructing a triangle similar to 01A with the 01 side on 0B. We then use Euclidean geometry to show multiplication is commutative and associative. Further these two operations distribute, so we have a field. Now construct the perpendicular to the line 01 at 0. Find either point with length 01 on this new line, and call it I. Now the product of I with itself is the point -1. The ability to pick arbitrary points in the plane (or completeness) makes this the complex numbers. Starting only with the points 0 and 1 and a ruler and compass we get the dense subset of constructable numbers.
True. Technically, if we consider imaginary numbers to be imaginary because we can't intuitively see them, we should consider the real numbers to be imaginary as well because we can't actually measure or see any irrational with any instrument because an irrational number is infinitely long and has no repeating/predictable pattern. In any interval larger than 0, there's an infinite number of rational and irrational numbers, so we can't physically distinguish between a rational and irrational number by observation. We just constructed the reals from the rationals because we know that the answer to certain problems cannot be answered without them (e.g. the solution to x^2 = 2, the ratio of a circle's diametre to its circumference, or the solution to the infinite series 1/n!), which is exactly why we constructed the imaginary numbers as well.
Now prove that there is a unique ordered field to refute the claims that the reals aren't real either! (I am half joking but the universe don't lie! If we weren't supposed to use them then why would be so damn special)
Theres no reason to believe imaginary numbers exist in real life. They are useful abstractions that help us do calculations with our current model of math. Our math model and reality are not the same thing.
To be honest for me with immaginary the question was never if they were truely immaginary, but instead if the properities we ascribe to immaginary numbers actually are arising instead out of the fact that we represent it as a scalar.
I completely understand the history and where the original terminology for imaginary came from based on the fact of trying to solve for or understanding what the result of sqrt(-1) is. And since the term imaginary was originally used the character i has been since then the standard notation for imaginary unit vector where i = sqrt(-1). This also later extended to become a fundamental part of the Complex Numbers. However, overtime and with more and more brilliant people working with them we have noticed fundamental properties of them especially with their relationship to rotations, and the trigonometric functions. Here I would like to elaborate on this a little bit, and some of the things I'm about to mention is going to challenge the recent status-quo of everything we've been taught about other specific properties within all of mathematics in regard to specific definitions, theorems, postulates, axioms, etc. I'm going to challenge this. However, with the nomenclature that I'm about to present, if we begin or start to adapt to this more practical use, it ought to help make it more intuitive to understand with a lot more clarity in seeing the apparent relationships and what they actually are. Sure, I'm used to calling them the imaginary numbers and I've been taught this since the mid 80s going back to elementary school. My challenge here isn't against the common terminology in regard to the Complex Numbers. For me this is still very appropriate. However, within the Complex numbers and the Complex field, we associate the expression of a single value such as: 5 + 3i to have a real component the 5, and the "imaginary" component 3i. I'm not trying to change this notation of using i. This is standard and is just fine. However, what I would like to see happen here is for people to start abandoning the use or term of imaginary. Instead of referring to them and teaching them as "imaginary", I think it would be best to call them what they actually are. Now before we can do that, what exactly are they if they're not imaginary? Well, to better understand this, we have to start treating scalar real values as actually being vector quantities. Consider the value of 5. We think of this as being scalar and in some arbitrary sense this is just fine, however, it is actually not just a scalar, but it is also a vector. We've been taught that scalar and scalar operations are not vectors, however, this is not exactly true. Sure, it is still scalar because it is a one-dimensional linear value. However, it still has a signed direction. Here, 5 is implicitly understood by default to be a positive. Its additive inverse would be (-5). Here, (-5) has the unary minus sign attached to it. These two values are identical in magnitude as can be seen from the expression |5| == |-5| = TRUE. In other words, the absolute value of 5 and (-5) are equivalent expressions. Where they differ are in their sign which implies their directions. These two values when added together 5 + (-5) gives us 0. This is what makes them an additive inverse. Here we are performing a linear transformation, a horizontal translation along the x or horizontal axis. These two values are 180 degrees or PI radians of rotation from each other. If we rotate 5 by +/- 180 degrees or +/- PI/radians we will end up getting a value of (-5). This is true for all Real values in R along X. The only exception to this is 0. If we rotate 0 by anything it remains at 0. This is due to the additive identity property such that a + 0 = a for all a. From this point on we only need to use the unitary values (vectors) of +/- 1, and +/- i. Here we can treat 1 as being (1,0), (-1) as being (-1,0). As for +/- i, we'll come back to this but first we must establish a better context of the relationship between 1 and i other than just seeing it as the sqrt(-1). Before we do this, we also need to understand another basic property of linear transformations and how a given set of rotations is also equivalent to a given set of linear translations. If we take the value 1 as the vector (1, 0) and we subtract it by itself twice such as: (1,0) - (1,0) - (1,0) or by adding to it, the following vector (1,0) + (-1,0), + (-1,0) which is also equivalent to (1,0) - (2,0) and (1,0) + (-2,0) respectively. We can see that simple arithmetic expression of 1 - 2 = (-1) is the same thing as by taking the value of 1 and rotating it by +/- 180 degrees or +/- PI radians. Here we did two horizontal translations of itself in the opposing direction and by doing so we literally reflected or mirrored the unit vector about the Y or vertical axis. This will be important a little later. Here the origin of 0 or (0,0) is the point of rotation, the point of reflection, the point of symmetry. Understanding this as well as the relationship between 0 and 1 and understanding that the value or the unit vector is orthogonal, perpendicular, and normal to the value 0 or the zero vector is key to understanding why I challenge the notion of the nomenclature of using the term imaginary for the values of a*i. First let's better understand this relationship of orthogonality. We do know that 90 degrees or PI/2 radian angles are Right angles and that the two vectors or line segments at their intersection is orthogonality and that they are perpendicular. We see this at the origin of any given coordinate system with at least 2 or more axes, and within Right Triangles and other polygons. If we take the vector and rotate it by 180 degrees we end up at the vector . Okay, if we rotate it by half or by 90 degrees where do we end up? If we rotate by 90 degrees or PI/2 radians we end up at the point . These two vectors and their perpendicularity will be extremely important in their relationships to, all linear expressions, the trigonometric functions, as well as i and all other complex values. And once this is properly established, the new methodology or terminology of representing these values will make complete sense and it will also make other areas throughout mathematics more feasible to understand in an intuitive manner such as division by 0, tan(90), and other phenomena throughout mathematics that we've been long taught is "undefined" or an "error". These are the notions that I am challenging. We are taught that division by 0 is undefined, to treat it as an error and that we can't do it. Well, several centuries ago, people in general even the top mathematicians of their times thought the same thing about what we commonly know now as being the imaginary numbers, and even at one point in history some believe the same thing can be said about negative numbers. This is where we need to have an appropriate foundation in order to provide proper context. To illustrate this better we are going to use the slope-intercept form of the line y = mx + b. For the general case we will set b = 0. Which will simplify this form to y = mx. And here all linear expressions or lines will pass through the origin (0,0) either in the XY plane or within the Complex plane. This leaves us with the value of x along the x-axis, the slope or gradient of the line denoted by m, and its output or translational height in Y. The slope of a line is defined to be rise / run which can be calculated from any two points on a line given by the formula m = (y2 - y1) / (x2 - x1) or by deltaY / deltaX. Which is simply the ratio of the rate of change in Y with respect to the rate of change in X. How much vertical displacement or translation is there compared to how much horizontal displacement or translation there is. This is a linear relationship. It is a ratio proportion. (continued...)
(...continued) So how does this linear relationship in the form of y = mx relate to the rotations, the trigonometric functions, the i, or the complex values? Well, let's simplify this expression of y = mx even a little more by applying the multiplicative identity property of (a * 1 = a) for all (a) to the slope m. Here we can let m = 1 and through this property x is unchanged. This leaves us with the expression y = x. Here this expression alone which can be treated as both assignment and equality. For all values of X in X, Y is also equal to X. Here X remains unchanged. This is equivalent to adding 0 to any and every element in X to get Y through the use of the additive identity property. This expression when plotted by the equivalent pairs { ..., (-1,-1), (0,0), {1, 1}, ... } gives use the line that bisects the XY plane in both the 1st and 3rd quadrants. We know that this line has a 45 degree or PI/4 radian angle both above and below it gradient between X and Y. We know this and that it is self-evident for several reasons. First, we know that the X and Y axes are perpendicular, orthogonal to each other as they create a Right Angle at the Origin which is 90 degrees or PI/2 radians respectively and dividing them by 2 gives us 45 degrees and PI/4 radians. A slope of 1 is 45 degrees or PI/4 radians. We can also see this from evaluating or solving the slope formula for any two points. Here, I'll just use the origin (0,0) and (1,1) to illustrate this in the general case. (y2-y1)/(x2-x1) = deltY/deltaX = (1-0)/(1-0) = 1/1 = 1. From this we can also represent or substitute deltaY for sin(t) and deltaX for cos(t) where t or theta is the angle between the line of y = mx+b and the +x-axis. With these we can also define the slope formula for m as the following: m = sin(t)/cos(t) = tan(t). We use this form of the slope when the value of the angle is known. m = sin(45)/cos(45) = tan(45) = 1. So where does division of 0 come into play? We are about to see this in translational action by evaluating the slopes or gradients of linear expressions with respect to their angle of rotation. We have been taught that when a slope of a line is 0, that it is horizontal or that all horizontal lines have a slope of 0. This true. We can see this by the following: m = sin(0)/cos(0) = 0/1 = tan(0) = 0. We are also taught that division by 0 and tan(90) are undefined. This is one of the notions that I want to challenge as well as the common nomenclature of calling i and multiples of i, the imaginaries and we will see why shortly. But first, we need to understand the relationship between the angles and the slopes. This table should help. range of angles | range of slopes t = 0 | 0 0 < t < 45 | 0 < m < 1 t = 45 | m = 1 45 < t < 90 | m > 1 : limit extends to +infinity within the first quadrant. The mirror or reflection for negative slopes is also true within the 3rd quadrant approaching negative infinity. t = 90 | m = ? : This is where we are taught that it is undefined. This is what I'm going to challenge. We saw that when t = 0, the slope is also 0 through sin(0)/cos(0) = 0/1. We are seeing this as a fraction by division which is appropriate however, when t = 90 and we have sin(90)/cos(90) which gives us the reciprocal of 0/1 being 1/0, now all of a sudden there appears to be an issue and we can't do this? I beg to differ. If we take the fractional values of the slope a/b or deltaY/deltaX and treat them as such that they are the vectors or coordinate pairs (deltaX, deltaY) or (cos(t), sin(t)) everything will begin to make proper sense. Oh, wait a minute, I've seen this notation before. These are the coordinate pairs for the Unit Circle. So why is division by 0 and tan(90) not undefined? The issue here is that they are actually well defined, it's just that they are no longer a 1 to 1 or many to 1 relationship, instead they are a 1 to many relationships and we don't typically like this because we are always looking for an exact precise result due to our hubris and nature of wanting to predict things. And this kind of thinking and teaching really needs to stop. Why am I claiming this? Let's go back to when I previously mentioned rotating the vector by 90 degrees we ended up at the point , here we translated a vector by a 1/2 half step of a linear translation. A rotation by 90 degrees is a 1/2 step horizontal translation. If we translate by another 90 then we have a full linear translation along the same line. Here we can see that the unit vectors and both exist in the XY plane. We also know that by rotating a line around the unit circle doesn't break rotation at intervals of 90 + 180*n where n is an integer. If sin(90)/cos(90) = tan(90) is "undefined" then the clock on your wall wouldn't work and yet it does! How why? Well, we also know that when we multiply any value by i to give us a*i where a is any real value, this is the same thing as rotating a by 90 degrees in the Complex plane. Take the value of rotate it by 90 degrees or multiply it by i and we will end up at . If we look at the powers of i we will see this pattern: i^1 = sqrt(-1) = rotation by 90; i^2 = -1 = rotation by 180 degrees, i^3 = - sqrt(-1), rotation by 270 degrees and i^4 = 1 = rotation by either 0 or by 360 degrees and we've come full circle. And this is also has a modulus operational property. So, when we look at division by 0 and we look at it from within the context of slope, rotating by 90 degrees, multiplying by i, they are all equivalent. This is perpendicularity, well verticality within Perpendicularity to be more precise. Verticality is the reciprocal of Horizontality. Therefore, when we see a fraction or division by 0 in a/b where { a != 0; for all a } And b = 0 we can view these as the vectors of the form where x is located at a. Within the context of slope, here we have 0 translation in X and only translation in Y. With this vector notation of x at a, for all a with no other translation in x but only in y. This is the vertical line at x = a and it is perpendicular to the point or value of . Thus and are perpendicular values. The input value is a at x, and the output is all values of Y at x. This is the 1 to many relationship that most don't typically like. Here we have infinite slope, this is verticality. When we look at the relationship and the similarities between the sine and the cosine functions this again is self-evident. We know that they are continuous rotational, sinusoidal, circular, transcendental, periodic wave functions. We know that both of them have the same exact range and domain, their ranges are [-1,1] and their domains within the Reals is the set of all Reals. I won't directly get into complex composition here, but it can be inferred via the coordinate pair within the complex plane (cos(t), i*sin(t)). We also know that they have same period of 2PI or 360 degrees. We also know they have the same wave form; in other words, their shapes are the same. The only major differences between them are their initial starting positions, the properties that one is an ODD function, and the other is an EVEN function. The starting point for sine is and the sine is an ODD function. The starting point for cosine is and it is an even function. Their wave forms or their graphs are exactly 90-degree or PI/2 radian horizontal translations of each other. So here if we stop calling the "imaginaries" by that naming convention and instead call them what they are, the "orthogonals" with respect to the "reals" or better stated, the "perpendiculars" and if we start to treat ordinary values as vector entities, we can clearly see that every number that exists including 0, 1, all in between and all that extends out to infinity are Circular. The expression 1+1 = 2. Is the unit circle located with its center at the point (1,0). We can see this from the first operand being the vector translating it horizontally in the +x direction by the vector arriving at the new vector with its center at The sum or the addition of the two is the diameter of the circle. Division by 0 is not undefined. When the denominator is 0, it is perpendicular to its corresponding numerator in terms of a fraction, and in terms of division or by repeated subtraction, division by 0 the divisor is 0 and we are subtracting repeatedly by 0 to reduce the dividend to 0 and because of the additive identity property of a + 0 = a, this is the same as a - 0 = a, and there is no change to the dividend, and we can or would perform this subtraction an infinite amount of times never being able to reduce it. This is vertical slope or verticality within perpendicularity with respect to horizontality. So, in hindsight or in essence diving by 0 is almost equivalent to multiplying by i, or by taking a real value and rotating it by 90 degrees not from the X-axis to the Y-axis, but at x into the Y direction for all points in Y at x. This is Well defined! Sure, it might be ambiguous because of the generated infinites but it's not "undefined". Just food for thought. Think critically for yourself instead of being forced into believing something because that's what's written in the textbooks, and we must believe that. Yeah, and they are never always 100% right! They always have some error and or falsehoods either unintentionally or by design. Challenge everything, put it to the test!
I prefer to think of i as the critical necessary part of a useful mathematical mechanism for introducing second dimension to a 1D vector. In other words, I think of the real numbers as a way to position a point on a line, and complex numbers as a way to position a point on plane. And I wish someone could tell me how we can extend this to the third dimension.
@@Niko-dj2vjthank you! I wasn’t expecting the fourth dimension to be the answer. I’ll have to take a look into Hamilton‘s work and also brush up on vector analysis. 🎉
An important clarification: if you just want to position points on the plane, the complex numbers are entirely unnecessary; you can get by with just using vectors. This is something you _can_ generalize to any number of dimensions - you can absolutely set up a number system that allows you to represent points in 3d space without having to go to anything 4-dimensional. The power of complex numbers comes from the fact that they also have a _multiplication_ that carries geometric meaning (scaling/rotating). This is much more powerful and unique property, and this can't be done for most dimensions. If you want to be able to talk about _rotations_ of 3d space (and not just points in the space), then you do need to use the 4d quaternions.
There is no log base 1 of anything, because it involves dividing by zero to produce it. There is a natural log of -1, which is i*pi*(2*k + 1) where k is any integer. There also are logs with bases of negative numbers, that utilize complex log to find. Logarithms are also a transcendental function, that goes beyond algebra, so it is not an operation considered to determine whether a set is algebraically closed. Even so, it is only logs of zero, and log base one, that produces a singularity for logarithms in the domain of complex numbers. For everything other than zero and one, logarithms are closed in the complex field.
I have a doubt regarding multiplicartion by 0, I understand that using common sense any number by 0 is 0, but then, I discovered this 1x0=a, 1=a/0, following this 1x0≠a, so multiplying by 0 would be undetermined, I tried questionin many AI's and googled this but they can't seem to prove where exactly I'm wrong.
This doesn't imply that multiplying by zero is undetermined, only that dividing by zero is undetermined. 1*a = 0 is perfectly valid, which implies that a=0. Rearranging this as 1 = a/0 is where we have a problem, because you are dividing by zero. There are an infinite number of solutions to 0*x = 0, so rearranging to solve for x, doesn't produce any particular one of them. It produces x=0/0, which is indeterminate. You need more information about how the numerator and denominator both approached zero, to find out what 0/0 is, in the limiting case as the top and bottom both approach zero. If we set up 1*a = b, and isolate the 1, we can salvage a correct equation out of this. This gives us 1 = b/a, which will be valid as long as a and b both equal each other, and neither of them equals zero. Immediately to the left, and immediately to the right of this problem point at a=0, you'll get b/a = 1 on the condition that a=b. So this hole at a=0 becomes a removable singularity, where it is a hole in what is otherwise a continuous function.
Sure, you don't _need_ to add more objects in order to get algebraic solutions to more equations, but that doesn't mean you _can't._ What if, instead of ℝ[X]/(X²+1), you instead did ℝ⟨X, Y⟩/(X²+1, Y²+1) (where the angled brackets denote that XY does not necessarily equal YX). Now you have two different objects, X and Y, that can square to -1. Not only that, but if you include an additional component to the modulus of (XY + YX), it's possible to show that (XY)² _also_ equals -1. So by adding 1 additional variable, and a rule for how it interacts with the other variable, we get a fourth object automatically with the same squaring to -1 property of the other new objects. In case you haven't noticed, ℝ⟨X, Y⟩/(X²+1, Y²+1, XY+YX) is actually a definition for the quaternions, which much like how Complex numbers are excellent at making things go spinny, quaternions are excellent for making things go spinny in 3D. There's also nothing saying those are the only things that can go there. While there already exist ℝeal numbers that satisfy x² = 1 and x² = 0, there's nothing stopping you from defining ℝ[X]/(X²-1) or ℝ[X]/(X²), both of which result in interesting cousins of the Complex numbers.
Mathematicians have invented a LOT of similar number systems like the complex numbers. As a group they are called Hypercomplex numbers. quaternions, tessarines, coquaternions, biquaternions, Split-biquaternions,Dual quaternions, octonions, split-complex numbers, dual numbers
the most profound thing is to add directions in space as primitives, INSTEAD of using complex numbers. "i" does not specify the plane of rotation. There are an infinite number of objects that multiply to -1. So, even though i^2 = -1, it does not mean that sqrt[-1] is definitely i. Directions in space as an example: right*right=1 up*up=1 right=-left up=-down right*up = up*left = left*down = down*right you can derive from this that: right*up = -up*right it anti-commutes. this means that multiplication does not commute in general. but note that these objects square to -1: right * up * right * up = (right*up)*right*up = -(up*right)*right*up = -up*(right*right)*up = -up*1*up = -up*up = -1 "i" is a 90 degree rotation in an unspecified plane. (right*up)^2 is a 90 degree rotation in the plane specified by (right*up). You need to be really really careful with 3 directions in space; Because there are 3 separate planes of rotation!
I really think that calling the square roots of negative numbers "imaginary" is a stumbling block for many people trying to learn it. I think we would do our future generations a favor if we decided to globally change the convention on what we call them. They are as extant as negative numbers. True, you can't have i cows, but you also can't have -1 cows. However people are quite willing to accept negative numbers but balk on imaginary. To keep formulas intact, I think we should keep the signifier 'i' and find a new word beginning with i to say it stands for.
Its ineteresting that for some cases the more "difficult" explanation is far easier to understand and for others its the "simple" one. Just shows people understand tasks completly different and may ace certain problems and struggle heavily on others.
So basically, imaginary numbers started as an intermediate step, which gave it the name, but have since shown up more in other implications just like real numbers. Then how does that make it not "imaginary"?
Imaginary is a name coined by a critic. It has nothing to do with them lacking practical application, which is what the term commonly leads people to believe.
I call them "spinny numbers", because they make things go spinny (being rotation matrices). Many objects studied in physics _love_ going spinny, so naturally spinny numbers end up being extremely useful when describing them.
@@angeldude101 Yeah. They are spinny numbers. The relate directly to sinusoidal behavior via euler's relation if you have 1 dimension of spinny numbers. If you work with 3 dimensions of spinny numbers (i, j, k); they allow for rotation in three dimensions without gimbal lock. In principle; they could be used to handle rotation in roughly as many spatial dimensions as you want to consider.
Hi , I have a question that i didn't find an answer for , it would be nice if you can explain/answer it 😅 What is ∞⁰ equal to? We just started taking limits (grade 12) and our teacher told us to treat ∞ as a number (just like when multiplying a negative number by a positive number we get a negative outcome and so on) so i was wondering if ∞⁰=1 ? I asked her but she told me that she has never encountered this expression then went on and asked the maths supervisor but the supervisor simply said: there is nothing such as ∞⁰ , where did you get this from? 😃 It would be nice if you can answer it 😊 apologies if this isn't clear , English isn't my first language 🙏🏻
You can indeed multiply infinities like that but as you probably know some limits are indeterminate forms! ∞⁰ is also such a form. Intuitively speaking, the "infinite" part tries to "push" the limit and make it infinite. The "zero" part tries to push the limit and make it 1. What "wins" is how "big" the "infinite" part is and how infinitesimally close to 0 the "zero" part is. For example, as x tends to positive infinity, the limit of x^0, which is a ∞⁰ form, is 1. However, as x tends to positive infinity, the limit of x^(1/lnx), which is also a ∞⁰ form, is e! Another notable indeterminate is 1^∞, an example being e = (x tends to infinity, lim (1+1/x)^x)
I always felt like imaginary number is a mathematical tool. I can feel the physical significance of the number '1' like '1 apple'. But how can I feel the physical significance of 'i' like in 'i apple'?
@@willnewman9783 But I can have '-1 degree Celsius'. Negative numbers then mean something is lower than a reference. Here, we can get a physical meaning too.
@@fahminrahman3543 I am not saying there is such a "physical interpretation" that you claim, but you picked a bad example. Temperature scales are messed up. The -1 you speak of makes sense because the scale is bad. It would be like if I invented a new unit of length, "tores", so that -1 tores = 2 feet. When using a good scale like Kelvin, negative temperatures make a lot less sense. (I think negative Kelvin can still exist, but from what I can tell, it is somewhat of a trick.)
As a physics major with strong mathematical inclinations at heart, I’m always frustrated with people from the physics community-very bright guys indeed-thinking ostensibly poorly about mathematical concepts and how very “real” they are. Their explanations of how “i is just a piece of notation we invent so we can do some real world calculations” or the more atrocious belief “1+2+3+…=-1/12, only if you could actually sum up to infinity” is utterly infuriating 😑 Glad to see eloquent mathematicians like you educating us on how to view maths as a coherent, logically consistent and not at all esoteric discipline 👌🏻🫶🏻
so what is the square root of negative one? its like saying Schrodinger's cat is dead and alive at the same time...its not logical or consistent with the mundane real world of sensory perception. It seems offensive that the world of quantum mechanics can't be made logical or functional or consistent just by using real numbers, the world can only make 'sense' if we use the antisense notation of i....which begs the question of how a particle can be both present and absent in the same location at the same time
@@mrosskne well, if I recall correctly, it was Gauss, one of the greatest mathematicians of all time, who proposed the name “lateral numbers”. And I think it does have some importance, because a lot of people think that mathematicians just waste their time studying imaginary stuff with no value to society, so the name “imaginary numbers” doesn’t help. And sadly, the people that decide study plans and government budgets are not mathematicians. So public perception of what mathematicians do and what mathematics is, is important
imaginary number are really useful electro-magnetics BUT all can be done without. I still consider imaginary pure mathematical abstraction that is very handy - not real
Complex numbers can describe real world things, for example positions and rotations. There’s no reason why the only way of accepting numbers as “existing” is if they can count apples. That’s an arbitrary decision. If we think of numbers as positions in a grid, additions as combining the arrows that point from the origin to those positions, and multiplications as rotating and scaling arrows, then complex numbers are perfect to describe positions and arrows in a plane in the real world. And while it’s true that this is a very specific setup to represent complex numbers and that we’re not actually seeing complex numbers… we do the same thing with real numbers: I’ve seen 7 apples, but I’ve never seen a 7 itself. We can use apples to represent natural numbers, or positions to represent complex numbers. But ultimately both are just models to represent numbers, and no model is inherently more “correct” than another. In other words, I think we shouldn’t say “these types of numbers do exist, and these don’t exist”. Instead, I’d argue that both real and complex numbers are equally valid in existing. Either both exist, or both are invented.
It depends entirely on how you view our system of numbers. If you feel that our system should describe the real world, then imaginary numbers are truly imaginary, because you can't really point to anything in the real world that is represented by an imaginary quantity. Most other elements of our number system are easy to demonstrate: I can count apples, I can add apples, I can divide apples in half, I can make a 5x5 square of apples; I can even almost show a negative number by taking one apple away. That last one's a little bit abstract but it's a pretty small leap. Now, show me imaginary apples ... you really can't. By that basis, imaginary numbers are imaginary. If on the other hand you view our number system as an abstraction that has many useful applications including modeling the real world, then imaginary numbers are just fine, they are no more and no less abstract than anything else. If we go that route, I'd favor calling them "orthogonal" numbers, but that's just me.
Square roots stretch your analogy. Yes, if I have a square lattice of n apples I know the square root of n by counting the apples along one axis. So since we can barely show square roots and we can barely show negative integers, the "numbers represent the real world" position becomes weak and is frankly unable to explain a lot of math that we cannot readily represent in simple apple terms. I think the better apple analogy with apples for complex numbers would be that apples in a line left and right are real apples but those not on that line are imaginary apples. But no matter what we do, we are pushing the abstraction to unreasonable limits that don't really help anyone, and we are really getting hungry for apple pie.
@@Qermaq I admit you have to split the apples into fractional apples if you want to show that the square root of 10 is about 3 1/6. But if you don't like apples for this purpose, how about a volume of sand you can form into a square; then the lengths of the sides show a square root.
Complex numbers can describe real world things, for example positions and rotations. There’s no reason why the only way of accepting numbers as “existing” is if they can count apples. That’s an arbitrary decision. If we think of numbers as positions in a grid, additions as combining the arrows that point from the origin to those positions, and multiplications as rotating and scaling arrows, then complex numbers are perfect to describe positions and arrows in a plane in the real world. And while it’s true that this is a very specific setup to represent complex numbers and that we’re not actually seeing complex numbers… we do the same thing with real numbers: I’ve seen 7 apples, but I’ve never seen a 7 itself. In other words, I think we shouldn’t say “these types of numbers do exist, and these don’t exist”. Instead, I’d argue that both real and complex numbers are equally valid in existing. Either both exist, or both are invented.
@@AndresFirte Show me a length of 12 real inches; it's easily done with a ruler. Now show me a length of 12 imaginary inches. Rotating your ruler doesn't make it NOT real inches. Just as you've seen seven real apples, you've never seen seven imaginary apples. You keep going back to "well if you PRETEND that real phenomena are imaginary phenomena" ... the fact that we have to pretend is what makes them imaginary.
@@kingbeauregard my point is that there’s no reason why numbers should be necessarily associated with length. Is there imaginary length? No. Also no negative length. I’m not arguing that. I’m saying that the *model* to represent numbers doesn’t necessarily have to be length: it can be something else. For example, positions in a plane. There, we can visualize and distinguish 12 from 12i. I hope I’ve explained myself better this time: no model is inherently more valid than other models. And no model shows us the numbers themselves, I’ve seen rope of 30cm of length, but I haven’t seen 30 on itself. Since no model is inherently “correct”, I feel imaginary and real numbers have the same right to exist. (And I have no opinion on if they exist or not. Just in that they have the same right to exist)
I like the field extension version. That was the first one where I felt like they weren't just pulled out of thin air. You can use field extensions to compute the discrete Fourier transform in a finite field. That helped me finally wrap my head around Fourier transforms. With the Reals, you can compute an arbitrary primitive nth root of unity. In a finite field, it depends on the factors of |F| - 1. So you can compute a 16-element DFT in F_17.
I think the way our number system works is flawed in a way where i is considered imaginary, or special. I don't think a perfect mathematical number system can also be a perfect practical number system. Just a thought
To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/DrSean . You’ll also get 20% off an annual premium subscription.
I've been watching you for weeks now under the impression that you had at *least* 50k subscribers, I was surprised that it's not even at 15k! Definitely an underrated channel!
I just subscribed!
Thanks so much, I really appreciate that!
Ah yes. Nobody has 15k! subscribers as that is far far more than the world population
10:13 When I was an adjunct instructor, I would tell my students that complex numbers were used in electrical engineering. Finally I made a promise to myself and future classes that the next time I covered the topic, I would actually learn how it was used. I was essentially able to show the real-world interpretation of addition, subtraction and multiplication by a real number or a pure imaginary number. The only thing I didn't understand is what it meant when you multiplied by a nontrivial complex number, i.e., one where neither a nor b was 0.
When you multiply two complex numbers together, remember that you M the M's, and A the A's. Multiply the magnitudes, and add the angles.
As an example, consider (4 + 3*i) multiplied by (12 + 5*i). Using FOIL, we get 4*12 + 12*3*i + 4*5*i + 15*i^2. Simplifying, we get 33 + 56*i. This has a magnitude of 65, which is consistent with multiplying magnitudes of 5 and 13.
In an application, what this would mean is combining two electrical filter circuit transfer functions, to find the overall transfer function. It could also mean starting with a phase-shifted waveform, and sending it through a filter circuit.
Thank you for this video! you have no idea how much I’ve needed this! I always say the same to people who try to tell me my best friend is “imaginary”. That is just so hurtful and toxic. Through math they’re now proven wrong
At 11:45, what's the difference between R[i] and the set of Complex numbers C? And isn't C just R^2?
C and R[i] are the same. They are both isomorphic to R^2 as vector spaces, but R^2 is not a field because there is no multiplication between vectors defined.
Thanks for answer.
So The complex plane is represented as two real dimensions. It's a coordinate system in RxR, so essentially, it's two dimensional. But when we think of a complex number we don't think in terms of coordinates (x,y) rather each z∈C is just one entry. So how do we order complex numbers
@@Jo-bx6ezwe can’t order complex numbers, or at least not in an order that behaves the way we’d expect it to behave.
For example, you know that 5 > 3. That means that if we take a positive number (for example 4) and multiply it by 5 and 3, then 5(4) > 3(4), or simplified, 20 > 12.
It’s been proven that you can’t do the same with complex numbers. If you have that a > b, and multiply them by a complex number c that is bigger than 0, then we would expect to get that ac > bc. But it doesn’t (or if it does, then you probably broke another property of we expected to get from ordering them).
In other words, you can’t order the complex number in a way that actually feels like you ordered them
_"Get real."_
_"Get rational."_
"Get integer."
"Get natural."
@@roihemed5632First one dossnt really work out. Second one should be 'Be Natural' imo.
@@antagonisticalex401 NO FUN ALLOWED
Get sexagintaquaternion!
@@NahhFam My Bad for ruinin' it chief 🫡.
*Sigh* 😮💨
*Deep Breaths*
😮💨
😮💨
😮💨
It appears I have made a severe and continuous lapse in my judgement.... 😔
As a mathematician, computer scientist, and guitarist, it is interesting to see how the symbol i became overloaded. i is used to indicate imaginary number, integer variable, and index finger. Cool explanations!
It’s also used as a summation index and for Apple’s phone or tablet OS. ;)
The letter i is also used to denote electrical current. For that reason, electrical engineers often use j instead of i to denote the imaginary unit.
And they have a good reason to. Electrical engineers use imaginary number all the time.
Complex numbers are the real numbers. "Real" numbers are a feature limited demo.
I dont think you should leave out the "5 levels" out of your youtube title. Subscribers might recognize the color thing on the thumbnail but Id imagine, new viewers dont. Its an inviting concept to know beforehand that the video in question explains something in increasing levels of complexity.
Thanks! That's very helpful. I'll edit the title to add that piece in
@@DrSeanGroathouse the 5 levels thing is good and will catch thr algo sooner rsther than later, if you have a few more of these in the pipeline you're all set
@@DrSeanGroathouse I see what you did there with the title... 5 levels of COMPLEXITY in a video on complex numbers? Hue hue well played.
Every number is imaginary if you are a formalist.
The claim that "imagniary numbers are made up" is right, but for the wrong reason. Most students who exclaim this do so in the belief that the real numbers are, well, more "real" than imagniary numbers. But the thing about that is that all numbers are made up in essence. There is no such thing as a "number" in the universe, they are constructed objects which only exist in our heads. That goes for real numbers as well. So if we are being pedantic, imagniary numbers are "imaginary", but so is every other kind of number in existance as well.
I write this without having watched the video yet (it's in my watch later).
I have to disagree with the idea that numbers aren't real. They certainly aren't fictive in the way that stories are: their properties are not made up, but found. There are true things about them that will still be true even if nobody ever finds out.
@@ErreniumThe same logic can be applied to imaginary numbers then, there are true things about them that hold regardless whether we discover it or not. Both are equally real in that sense
"There is no such thing as a "number" in the universe", that's a dubious claim. One can argue that mathematics is a study of structure, and while our mathematical laws of how the universe works are certainly approximations, if humans were to be handed the true law from above, the way it is structured would certainly be an area of study for mathematicians. And for now it looks like numbers do have a place in the overall structure of our universe, complex numbers being absolutely indispensable.
You might go the technically correct way and claim something like "There is no way to know if there's such a thing as a number in the universe" or "There is no way to know if the universe is mathematical or not", but those claims are of no more interest than something like "There is no way to know if god exists".
@@aryansinha3629Duh?
@@YKS_efehanWrong
Even real numbers are imaginary :-)
No... But real numbers are COMPLEX!
Remember, a complex number has a REAL part and an IMAGINARY part!
@@JJ_TheGreat all numbers are based on imagination. But I know what you meant :-)
I feel that the term "imaginary" is just a misnomer that stuck. I would prefer the term "lateral", and I believe that for some languages, that is the case.
ah, i was kinda hoping the math major would bring up isomorphism classes of algebras! the thing is, a mathematician doesn't really think any of these definitions of the complex numbers are any more correct than any other. a set of formal symbols, a subalgebra of End(R²), the polynomial ring R[x]/(x²+1), the algebraic closure of R, a real 2D vector space with an automorphism squaring to -id: all of these constructions are "basically the same."
the precise definition of this uses abstract algebra. in this context, we define an algebra (over R) to be any vector space over the real numbers with a bilinear product of vectors. examples of this structure are the real numbers themselves, the complex numbers, and the quaternions, but also strange things like R³ with the cross product. we also say that two algebras are isomorphic if there's a linear map between them which doesn't change products; that is, A and A' are isomorphic if there is some bijective linear f: A -> A' so that f(uv) = f(u)f(v). intuitively, this is saying that the only real difference between A and A' is the labels, since you can use f to essentially change the labels without changing the structure of addition or multiplication.
now, if you look at the multiple definitions of the complex numbers, they're all different objects. however, they're also all vector spaces over R with a product between elements, so they're all algebras over R! and it turns out that they are all isomorphic: given the formal symbol i, you map it to the matrix ((0, -1), (1, 0)), the polynomial x, a solution to x²+1 = 0, or the automorphism squaring to -id. (also send 1 to 1 or id in each case; because each formal symbol is a+bi, this determines a map on all formal symbols.) this determines an isomorphism in each case, so every representation is isomorphic!
in that sense, the thought of picking one representation is a bit silly. the complex numbers are kind of just what they are, and these explicit realisations are just concrete versions of a more general phenomenon. because we know they are all isomorphic, we can switch between them whenever we want and whatever we do with them remains valid.
This is my favorite, quick visualization for why i * i = -1:
On a number line, performing -1 * -1 is like starting at 1, then making two 180 degree turns, first to -1, then back to 1, for a full 360 degree turn. That shows how two negatives multiplied make a positive.
Similarly, performing i * i is like making two 90 degree turns, starting at 1 then first turning 90 degrees to 'i' on an axis perpendicular to the real number line (the imaginary number axis), then another 90 degree turn to -1, making a 180 degree turn in total. That shows how two imaginary numbers multiplied make a real number.
another great video! probably the best summary one can get in 10 minutes.
thanks dr Sean!
This content is incredibly high quality. Your channel will definitely blow up soon.
All numbers are imaginary
If I draw the number 1 then it seems pretty real to me
@@jamie31415 1+0i
@@jamie31415 one apple and one week are very different ones
oh really? Then explain how i have 5 fingers on my hand
@@overlord3481 Rotate your hand 90 degrees, now it's 5i fingers. Don't rotate 180 degrees, or you'll owe 5 fingers.
I found a grammatical mistake in the thumbnail. “I exists!” is correct!
I feel so bad that the first thought that came to my mind when I opened the video is that the guy looks so nerdy.
I really like that this channel focuses on trying to speak at everyone’s level in the same respectful way. Good work as always Sean!
In a sense, all numbers are imaginary. Mathematics is a modeling system. Just like a drawing of a building is not the building. It’s how the building is imagined.
Yes they are. Don't make things up.
So, is "imaginary" just a nice way of saying "Hard for a human to comprehend but clearly valid because it solves the equation."? Seems to be my takeaway.
I guess I'm implying that it "correctly solves the equation" since I guess you could divide by 0 to screw up all sorts of things.
I love your videos you deserve millions of subscribers
Thanks so much, I'm glad you liked it!
So cool matrix isomorphism! Saw it a long time ago in an algebra book and didnt remember, thanks!
I do not have a best definition. As a physicist, it depends in the application. I did not know the quotient defintion, ie, # 5. Thanks!
I went to a French high school in the 1970s. The complex numbers were introduced as "dilations" , ie, a rotation times a streching, your level 3 was my level 1. Now my every day definition is the usual a+bi definition with field properties... like everyone else.
That definition makes the complex numbers "real" in a physical sense.
You should mention that in Quantum Mechanics, the Schroedinger equations is similar to the wave equation but the appearance of "i" in there is what gives quantum mechanics it weirdness.
By the way, I am now preparing a talk where I use quaternions..... they are an extension of complex numbers but not a field.
Most clear explanation of the definition of C = R[X]/(X^2+1) I have ever heard, congratulations! An anedocte: my calculus 2 teacher used to say: I know tho types of numbers, integers and complex numbers. As integers are too complicated, I will talk about complex numbers :)
Gemini 1.5 Pro: The video is about imaginary numbers. It explores what imaginary numbers are and why they are important. The video is divided into five levels.
Level 1: Where the name imaginary numbers came from
In the 1500s, mathematicians were trying to solve cubic equations. Cardano, a mathematician, came up with a formula to solve these equations. But the formula involved the square root of a negative number. Since negative numbers weren't understood at the time, Cardano called these numbers imaginary. Even though he thought of them as imaginary, he realized that they could be used to solve real problems.
Level 2: The usual way we define complex numbers today
Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers can be added, subtracted, multiplied, and divided.
Level 3: How complex numbers can be defined as matrices
Complex numbers can also be defined as matrices. A 2x2 matrix can be used to represent a complex number. This way of defining complex numbers does not require the imaginary unit i.
Level 4: Practical applications of complex numbers
Complex numbers have many practical applications. They are used in the Fourier transform, which is used to analyze signals and compress data. They are also used in electrical engineering and quantum mechanics.
Level 5: An abstract algebra perspective
The video talks about complex numbers from the perspective of abstract algebra. It explains that the complex numbers are algebraically closed, which means that any polynomial equation with complex coefficients has a complex root. The video also discusses another way to define complex numbers using polynomial rings.
There are infinite dimensions to numbers, just that most of the time we only work with one, and sometimes with two that’s where complex numbers come in and sometimes with 4 that’s where quaternions come in, and sometimes we need much much more and that’s why we have vectors (although they form different group than complex numbers and quaternions but we can also present complex numbers and quaternions as vectors). So yeah, “imaginary” numbers are as real as they get.
This video was really interesting, you have a rare talent of making abstract ideas accessible. Thank you again.
As a developer, I have to say that the definition via matrices is the one I like best. Very Beautiful.
I’m not knowledgeable enough to apply it, but it’s pretty interesting to me that the complex numbers are isomorphic (?) to the geometric algebra scalar plus bivector.
One of my favorite RUclipsr’s!
I would argue that any numerical concept that does not occur in nature - such as imaginary numbers, or infinity, are 'imaginary'. I see how they are useful mathematically, but can't help wondering if they aren't some sort of workaround for something we don't quite understand.
This was in no way illuminating. It started at explanation level 100 and finished at 1000. I gave up after 5 minutes.
great video, I knew a lot about complex numbers but I still learned!
I’ve been learning quantum mechanics recently, and that has given me an entirely new appreciation for the complex plane, especially when it comes to phase. The Euler equation is a way to split a number up into complex components without altering its absolute value. This is a very important property when we want to adhere to conservation principles while still allowing interference patterns to emerge under certain circumstances.
Consider the points in a plane. How can we make a field of these?
Well, we need a 0 for addition, so pick a point to be 0. We can define addition of two points using the fourth point in the parallelogram containing the two points and 0.
Using Euclid's geometry we can show that this is commutative and associative, so is a good addition. Indeed, the reflection about 0 is the additive inverse.
To multiply we need a new point for an identity. Let's pick a new point called 1. We already have addition on this line from above and the point -1. Each point on the plane defines an angle relative to this line.
To multiply two points construct the angle that is the sum of the two given angles and scale the distance from 0 to one by the distance from 0 to the other, using the distance to 1 as the unit. For two points A and B not on the line this consists of constructing a triangle similar to 01A with the 01 side on 0B.
We then use Euclidean geometry to show multiplication is commutative and associative. Further these two operations distribute, so we have a field.
Now construct the perpendicular to the line 01 at 0. Find either point with length 01 on this new line, and call it I. Now the product of I with itself is the point -1.
The ability to pick arbitrary points in the plane (or completeness) makes this the complex numbers. Starting only with the points 0 and 1 and a ruler and compass we get the dense subset of constructable numbers.
True. Technically, if we consider imaginary numbers to be imaginary because we can't intuitively see them, we should consider the real numbers to be imaginary as well because we can't actually measure or see any irrational with any instrument because an irrational number is infinitely long and has no repeating/predictable pattern. In any interval larger than 0, there's an infinite number of rational and irrational numbers, so we can't physically distinguish between a rational and irrational number by observation. We just constructed the reals from the rationals because we know that the answer to certain problems cannot be answered without them (e.g. the solution to x^2 = 2, the ratio of a circle's diametre to its circumference, or the solution to the infinite series 1/n!), which is exactly why we constructed the imaginary numbers as well.
Now prove that there is a unique ordered field to refute the claims that the reals aren't real either! (I am half joking but the universe don't lie! If we weren't supposed to use them then why would be so damn special)
Muy interesante el tema. Pero creo que es una creación de IA. Si estoy equivocado me disculpo, pero el tema es OK.
This dude is a quintessential nerd. He cannot possibly look or sound more nerdy
Theres no reason to believe imaginary numbers exist in real life. They are useful abstractions that help us do calculations with our current model of math.
Our math model and reality are not the same thing.
taking a summer class on complex variables rn, so the abstract alg review is appreciated. banger vid as always
I'm glad you liked it!
To be honest for me with immaginary the question was never if they were truely immaginary, but instead if the properities we ascribe to immaginary numbers actually are arising instead out of the fact that we represent it as a scalar.
Hi Sean, do you have any book/resource recommendations for starting a maths degree by any chance? Thank you
I love ur videos!!!! You should have at least 100k followers!
I'll say more: "i" is imaginary, because every number is.
I completely understand the history and where the original terminology for imaginary came from based on the fact of trying to solve for or understanding what the result of sqrt(-1) is. And since the term imaginary was originally used the character i has been since then the standard notation for imaginary unit vector where i = sqrt(-1).
This also later extended to become a fundamental part of the Complex Numbers. However, overtime and with more and more brilliant people working with them we have noticed fundamental properties of them especially with their relationship to rotations, and the trigonometric functions.
Here I would like to elaborate on this a little bit, and some of the things I'm about to mention is going to challenge the recent status-quo of everything we've been taught about other specific properties within all of mathematics in regard to specific definitions, theorems, postulates, axioms, etc. I'm going to challenge this. However, with the nomenclature that I'm about to present, if we begin or start to adapt to this more practical use, it ought to help make it more intuitive to understand with a lot more clarity in seeing the apparent relationships and what they actually are.
Sure, I'm used to calling them the imaginary numbers and I've been taught this since the mid 80s going back to elementary school. My challenge here isn't against the common terminology in regard to the Complex Numbers. For me this is still very appropriate. However, within the Complex numbers and the Complex field, we associate the expression of a single value such as: 5 + 3i to have a real component the 5, and the "imaginary" component 3i. I'm not trying to change this notation of using i. This is standard and is just fine.
However, what I would like to see happen here is for people to start abandoning the use or term of imaginary. Instead of referring to them and teaching them as "imaginary", I think it would be best to call them what they actually are. Now before we can do that, what exactly are they if they're not imaginary?
Well, to better understand this, we have to start treating scalar real values as actually being vector quantities.
Consider the value of 5. We think of this as being scalar and in some arbitrary sense this is just fine, however, it is actually not just a scalar, but it is also a vector. We've been taught that scalar and scalar operations are not vectors, however, this is not exactly true. Sure, it is still scalar because it is a one-dimensional linear value. However, it still has a signed direction. Here, 5 is implicitly understood by default to be a positive. Its additive inverse would be (-5). Here, (-5) has the unary minus sign attached to it. These two values are identical in magnitude as can be seen from the expression |5| == |-5| = TRUE. In other words, the absolute value of 5 and (-5) are equivalent expressions. Where they differ are in their sign which implies their directions. These two values when added together 5 + (-5) gives us 0. This is what makes them an additive inverse. Here we are performing a linear transformation, a horizontal translation along the x or horizontal axis. These two values are 180 degrees or PI radians of rotation from each other. If we rotate 5 by +/- 180 degrees or +/- PI/radians we will end up getting a value of (-5). This is true for all Real values in R along X. The only exception to this is 0. If we rotate 0 by anything it remains at 0. This is due to the additive identity property such that a + 0 = a for all a.
From this point on we only need to use the unitary values (vectors) of +/- 1, and +/- i.
Here we can treat 1 as being (1,0), (-1) as being (-1,0). As for +/- i, we'll come back to this but first we must establish a better context of the relationship between 1 and i other than just seeing it as the sqrt(-1). Before we do this, we also need to understand another basic property of linear transformations and how a given set of rotations is also equivalent to a given set of linear translations.
If we take the value 1 as the vector (1, 0) and we subtract it by itself twice such as: (1,0) - (1,0) - (1,0) or by adding to it, the following vector (1,0) + (-1,0), + (-1,0) which is also equivalent to (1,0) - (2,0) and (1,0) + (-2,0) respectively. We can see that simple arithmetic expression of 1 - 2 = (-1) is the same thing as by taking the value of 1 and rotating it by +/- 180 degrees or +/- PI radians.
Here we did two horizontal translations of itself in the opposing direction and by doing so we literally reflected or mirrored the unit vector about the Y or vertical axis. This will be important a little later.
Here the origin of 0 or (0,0) is the point of rotation, the point of reflection, the point of symmetry. Understanding this as well as the relationship between 0 and 1 and understanding that the value or the unit vector is orthogonal, perpendicular, and normal to the value 0 or the zero vector is key to understanding why I challenge the notion of the nomenclature of using the term imaginary for the values of a*i.
First let's better understand this relationship of orthogonality. We do know that 90 degrees or PI/2 radian angles are Right angles and that the two vectors or line segments at their intersection is orthogonality and that they are perpendicular. We see this at the origin of any given coordinate system with at least 2 or more axes, and within Right Triangles and other polygons.
If we take the vector and rotate it by 180 degrees we end up at the vector . Okay, if we rotate it by half or by 90 degrees where do we end up? If we rotate by 90 degrees or PI/2 radians we end up at the point . These two vectors and their perpendicularity will be extremely important in their relationships to, all linear expressions, the trigonometric functions, as well as i and all other complex values. And once this is properly established, the new methodology or terminology of representing these values will make complete sense and it will also make other areas throughout mathematics more feasible to understand in an intuitive manner such as division by 0, tan(90), and other phenomena throughout mathematics that we've been long taught is "undefined" or an "error". These are the notions that I am challenging. We are taught that division by 0 is undefined, to treat it as an error and that we can't do it. Well, several centuries ago, people in general even the top mathematicians of their times thought the same thing about what we commonly know now as being the imaginary numbers, and even at one point in history some believe the same thing can be said about negative numbers.
This is where we need to have an appropriate foundation in order to provide proper context. To illustrate this better we are going to use the slope-intercept form of the line y = mx + b. For the general case we will set b = 0. Which will simplify this form to y = mx. And here all linear expressions or lines will pass through the origin (0,0) either in the XY plane or within the Complex plane.
This leaves us with the value of x along the x-axis, the slope or gradient of the line denoted by m, and its output or translational height in Y. The slope of a line is defined to be rise / run which can be calculated from any two points on a line given by the formula m = (y2 - y1) / (x2 - x1) or by deltaY / deltaX. Which is simply the ratio of the rate of change in Y with respect to the rate of change in X. How much vertical displacement or translation is there compared to how much horizontal displacement or translation there is. This is a linear relationship. It is a ratio proportion.
(continued...)
(...continued)
So how does this linear relationship in the form of y = mx relate to the rotations, the trigonometric functions, the i, or the complex values?
Well, let's simplify this expression of y = mx even a little more by applying the multiplicative identity property of (a * 1 = a) for all (a) to the slope m. Here we can let m = 1 and through this property x is unchanged. This leaves us with the expression y = x.
Here this expression alone which can be treated as both assignment and equality. For all values of X in X, Y is also equal to X. Here X remains unchanged. This is equivalent to adding 0 to any and every element in X to get Y through the use of the additive identity property. This expression when plotted by the equivalent pairs { ..., (-1,-1), (0,0), {1, 1}, ... } gives use the line that bisects the XY plane in both the 1st and 3rd quadrants. We know that this line has a 45 degree or PI/4 radian angle both above and below it gradient between X and Y. We know this and that it is self-evident for several reasons. First, we know that the X and Y axes are perpendicular, orthogonal to each other as they create a Right Angle at the Origin which is 90 degrees or PI/2 radians respectively and dividing them by 2 gives us 45 degrees and PI/4 radians. A slope of 1 is 45 degrees or PI/4 radians. We can also see this from evaluating or solving the slope formula for any two points. Here, I'll just use the origin (0,0) and (1,1) to illustrate this in the general case.
(y2-y1)/(x2-x1) = deltY/deltaX = (1-0)/(1-0) = 1/1 = 1.
From this we can also represent or substitute deltaY for sin(t) and deltaX for cos(t) where t or theta is the angle between the line of y = mx+b and the +x-axis. With these we can also define the slope formula for m as the following: m = sin(t)/cos(t) = tan(t).
We use this form of the slope when the value of the angle is known. m = sin(45)/cos(45) = tan(45) = 1.
So where does division of 0 come into play? We are about to see this in translational action by evaluating the slopes or gradients of linear expressions with respect to their angle of rotation.
We have been taught that when a slope of a line is 0, that it is horizontal or that all horizontal lines have a slope of 0. This true. We can see this by the following:
m = sin(0)/cos(0) = 0/1 = tan(0) = 0.
We are also taught that division by 0 and tan(90) are undefined. This is one of the notions that I want to challenge as well as the common nomenclature of calling i and multiples of i, the imaginaries and we will see why shortly. But first, we need to understand the relationship between the angles and the slopes. This table should help.
range of angles | range of slopes
t = 0 | 0
0 < t < 45 | 0 < m < 1
t = 45 | m = 1
45 < t < 90 | m > 1 : limit extends to +infinity within the first quadrant. The mirror or reflection for negative slopes is also true within the 3rd quadrant approaching negative infinity.
t = 90 | m = ? : This is where we are taught that it is undefined. This is what I'm going to challenge.
We saw that when t = 0, the slope is also 0 through sin(0)/cos(0) = 0/1. We are seeing this as a fraction by division which is appropriate however, when t = 90 and we have sin(90)/cos(90) which gives us the reciprocal of 0/1 being 1/0, now all of a sudden there appears to be an issue and we can't do this? I beg to differ.
If we take the fractional values of the slope a/b or deltaY/deltaX and treat them as such that they are the vectors or coordinate pairs (deltaX, deltaY) or (cos(t), sin(t)) everything will begin to make proper sense. Oh, wait a minute, I've seen this notation before. These are the coordinate pairs for the Unit Circle.
So why is division by 0 and tan(90) not undefined? The issue here is that they are actually well defined, it's just that they are no longer a 1 to 1 or many to 1 relationship, instead they are a 1 to many relationships and we don't typically like this because we are always looking for an exact precise result due to our hubris and nature of wanting to predict things. And this kind of thinking and teaching really needs to stop.
Why am I claiming this? Let's go back to when I previously mentioned rotating the vector by 90 degrees we ended up at the point , here we translated a vector by a 1/2 half step of a linear translation. A rotation by 90 degrees is a 1/2 step horizontal translation. If we translate by another 90 then we have a full linear translation along the same line.
Here we can see that the unit vectors and both exist in the XY plane. We also know that by rotating a line around the unit circle doesn't break rotation at intervals of 90 + 180*n where n is an integer. If sin(90)/cos(90) = tan(90) is "undefined" then the clock on your wall wouldn't work and yet it does!
How why? Well, we also know that when we multiply any value by i to give us a*i where a is any real value, this is the same thing as rotating a by 90 degrees in the Complex plane. Take the value of rotate it by 90 degrees or multiply it by i and we will end up at .
If we look at the powers of i we will see this pattern: i^1 = sqrt(-1) = rotation by 90; i^2 = -1 = rotation by 180 degrees, i^3 = - sqrt(-1), rotation by 270 degrees and i^4 = 1 = rotation by either 0 or by 360 degrees and we've come full circle. And this is also has a modulus operational property.
So, when we look at division by 0 and we look at it from within the context of slope, rotating by 90 degrees, multiplying by i, they are all equivalent. This is perpendicularity, well verticality within Perpendicularity to be more precise. Verticality is the reciprocal of Horizontality.
Therefore, when we see a fraction or division by 0 in a/b where { a != 0; for all a } And b = 0 we can view these as the vectors of the form where x is located at a. Within the context of slope, here we have 0 translation in X and only translation in Y. With this vector notation of x at a, for all a with no other translation in x but only in y. This is the vertical line at x = a and it is perpendicular to the point or value of . Thus and are perpendicular values. The input value is a at x, and the output is all values of Y at x. This is the 1 to many relationship that most don't typically like. Here we have infinite slope, this is verticality.
When we look at the relationship and the similarities between the sine and the cosine functions this again is self-evident. We know that they are continuous rotational, sinusoidal, circular, transcendental, periodic wave functions. We know that both of them have the same exact range and domain, their ranges are [-1,1] and their domains within the Reals is the set of all Reals. I won't directly get into complex composition here, but it can be inferred via the coordinate pair within the complex plane (cos(t), i*sin(t)). We also know that they have same period of 2PI or 360 degrees. We also know they have the same wave form; in other words, their shapes are the same. The only major differences between them are their initial starting positions, the properties that one is an ODD function, and the other is an EVEN function. The starting point for sine is and the sine is an ODD function. The starting point for cosine is and it is an even function. Their wave forms or their graphs are exactly 90-degree or PI/2 radian horizontal translations of each other.
So here if we stop calling the "imaginaries" by that naming convention and instead call them what they are, the "orthogonals" with respect to the "reals" or better stated, the "perpendiculars" and if we start to treat ordinary values as vector entities, we can clearly see that every number that exists including 0, 1, all in between and all that extends out to infinity are Circular.
The expression 1+1 = 2. Is the unit circle located with its center at the point (1,0). We can see this from the first operand being the vector translating it horizontally in the +x direction by the vector arriving at the new vector with its center at The sum or the addition of the two is the diameter of the circle. Division by 0 is not undefined. When the denominator is 0, it is perpendicular to its corresponding numerator in terms of a fraction, and in terms of division or by repeated subtraction, division by 0 the divisor is 0 and we are subtracting repeatedly by 0 to reduce the dividend to 0 and because of the additive identity property of a + 0 = a, this is the same as a - 0 = a, and there is no change to the dividend, and we can or would perform this subtraction an infinite amount of times never being able to reduce it. This is vertical slope or verticality within perpendicularity with respect to horizontality. So, in hindsight or in essence diving by 0 is almost equivalent to multiplying by i, or by taking a real value and rotating it by 90 degrees not from the X-axis to the Y-axis, but at x into the Y direction for all points in Y at x.
This is Well defined! Sure, it might be ambiguous because of the generated infinites but it's not "undefined".
Just food for thought. Think critically for yourself instead of being forced into believing something because that's what's written in the textbooks, and we must believe that. Yeah, and they are never always 100% right! They always have some error and or falsehoods either unintentionally or by design. Challenge everything, put it to the test!
Hmmm i dont understanf. My english is not enought for this
How about the philosophical perspective ?
Finally, a genius mathematician at just the right time.
I prefer to think of i as the critical necessary part of a useful mathematical mechanism for introducing second dimension to a 1D vector. In other words, I think of the real numbers as a way to position a point on a line, and complex numbers as a way to position a point on plane. And I wish someone could tell me how we can extend this to the third dimension.
quaternions. It's a 4 dimensional number system that is, however, used in 3 dimensional fields.
@@Niko-dj2vjthank you! I wasn’t expecting the fourth dimension to be the answer. I’ll have to take a look into Hamilton‘s work and also brush up on vector analysis. 🎉
Complex numbers aren't "necessary" or "critical" to define vectors.
It can actually be shown to be impossible to create 3d numbers (that satisfy similar conditions to the reals and complex numbers)!
An important clarification: if you just want to position points on the plane, the complex numbers are entirely unnecessary; you can get by with just using vectors. This is something you _can_ generalize to any number of dimensions - you can absolutely set up a number system that allows you to represent points in 3d space without having to go to anything 4-dimensional.
The power of complex numbers comes from the fact that they also have a _multiplication_ that carries geometric meaning (scaling/rotating). This is much more powerful and unique property, and this can't be done for most dimensions. If you want to be able to talk about _rotations_ of 3d space (and not just points in the space), then you do need to use the 4d quaternions.
10:30 The square root of i is i^(1/2).
I’m not so sure the complex numbers are alebraically closed like what is log1 of -1
There is no log base 1 of anything, because it involves dividing by zero to produce it.
There is a natural log of -1, which is i*pi*(2*k + 1) where k is any integer. There also are logs with bases of negative numbers, that utilize complex log to find.
Logarithms are also a transcendental function, that goes beyond algebra, so it is not an operation considered to determine whether a set is algebraically closed. Even so, it is only logs of zero, and log base one, that produces a singularity for logarithms in the domain of complex numbers. For everything other than zero and one, logarithms are closed in the complex field.
Keep on going mate. You will get big.
Are all numbers imaginary?
I have a doubt regarding multiplicartion by 0, I understand that using common sense any number by 0 is 0, but then, I discovered this 1x0=a, 1=a/0, following this 1x0≠a, so multiplying by 0 would be undetermined, I tried questionin many AI's and googled this but they can't seem to prove where exactly I'm wrong.
This doesn't imply that multiplying by zero is undetermined, only that dividing by zero is undetermined.
1*a = 0 is perfectly valid, which implies that a=0. Rearranging this as 1 = a/0 is where we have a problem, because you are dividing by zero.
There are an infinite number of solutions to 0*x = 0, so rearranging to solve for x, doesn't produce any particular one of them. It produces x=0/0, which is indeterminate. You need more information about how the numerator and denominator both approached zero, to find out what 0/0 is, in the limiting case as the top and bottom both approach zero.
If we set up 1*a = b, and isolate the 1, we can salvage a correct equation out of this. This gives us 1 = b/a, which will be valid as long as a and b both equal each other, and neither of them equals zero. Immediately to the left, and immediately to the right of this problem point at a=0, you'll get b/a = 1 on the condition that a=b. So this hole at a=0 becomes a removable singularity, where it is a hole in what is otherwise a continuous function.
@@carultch 1x0=a or a=1x0, a/0=1, so then multiplying by 0 isn't equals 0 or if it is then dividing by 0 would also be 0
Sure, you don't _need_ to add more objects in order to get algebraic solutions to more equations, but that doesn't mean you _can't._ What if, instead of ℝ[X]/(X²+1), you instead did ℝ⟨X, Y⟩/(X²+1, Y²+1) (where the angled brackets denote that XY does not necessarily equal YX). Now you have two different objects, X and Y, that can square to -1. Not only that, but if you include an additional component to the modulus of (XY + YX), it's possible to show that (XY)² _also_ equals -1. So by adding 1 additional variable, and a rule for how it interacts with the other variable, we get a fourth object automatically with the same squaring to -1 property of the other new objects.
In case you haven't noticed, ℝ⟨X, Y⟩/(X²+1, Y²+1, XY+YX) is actually a definition for the quaternions, which much like how Complex numbers are excellent at making things go spinny, quaternions are excellent for making things go spinny in 3D.
There's also nothing saying those are the only things that can go there. While there already exist ℝeal numbers that satisfy x² = 1 and x² = 0, there's nothing stopping you from defining ℝ[X]/(X²-1) or ℝ[X]/(X²), both of which result in interesting cousins of the Complex numbers.
11:46 quaternions be like
love the concept of the 5 levels
Mathematicians have invented a LOT of similar number systems like the complex numbers. As a group they are called Hypercomplex numbers.
quaternions, tessarines, coquaternions, biquaternions, Split-biquaternions,Dual quaternions, octonions, split-complex numbers, dual numbers
the most profound thing is to add directions in space as primitives, INSTEAD of using complex numbers. "i" does not specify the plane of rotation. There are an infinite number of objects that multiply to -1. So, even though i^2 = -1, it does not mean that sqrt[-1] is definitely i. Directions in space as an example:
right*right=1
up*up=1
right=-left
up=-down
right*up = up*left = left*down = down*right
you can derive from this that:
right*up = -up*right
it anti-commutes. this means that multiplication does not commute in general. but note that these objects square to -1:
right * up * right * up
=
(right*up)*right*up
=
-(up*right)*right*up
=
-up*(right*right)*up
=
-up*1*up
=
-up*up
=
-1
"i" is a 90 degree rotation in an unspecified plane. (right*up)^2 is a 90 degree rotation in the plane specified by (right*up).
You need to be really really careful with 3 directions in space; Because there are 3 separate planes of rotation!
I really think that calling the square roots of negative numbers "imaginary" is a stumbling block for many people trying to learn it. I think we would do our future generations a favor if we decided to globally change the convention on what we call them. They are as extant as negative numbers. True, you can't have i cows, but you also can't have -1 cows. However people are quite willing to accept negative numbers but balk on imaginary. To keep formulas intact, I think we should keep the signifier 'i' and find a new word beginning with i to say it stands for.
How about Italian, to reflect their history?
@@carultch I like it!
Its ineteresting that for some cases the more "difficult" explanation is far easier to understand and for others its the "simple" one. Just shows people understand tasks completly different and may ace certain problems and struggle heavily on others.
Cl0,1(R) go brrr
Great video. I'd like to see you explaining what axioms are, until you reach ZFC axiom.
Thanks. This is a great video and a really clear overview of the complex numbers.
👏 well done
Thanks for detailed explanations. Please keep doing this great work.
Great content!
Thank you!
pancake
insightful lol
I exist!
Imaginary numbers are imaginary... just like any numbers
i exist
was the double entendre intentional? (in the title lol)
So basically, imaginary numbers started as an intermediate step, which gave it the name, but have since shown up more in other implications just like real numbers.
Then how does that make it not "imaginary"?
Imaginary is a name coined by a critic. It has nothing to do with them lacking practical application, which is what the term commonly leads people to believe.
They exist but ate not "real"
Yes 'cause they complex
What happened to 11 at 1:53?
Imaginary numbers should really probably be called oscillation numbers; or numbers containing a reciprocating dimension.
I call them "spinny numbers", because they make things go spinny (being rotation matrices). Many objects studied in physics _love_ going spinny, so naturally spinny numbers end up being extremely useful when describing them.
@@angeldude101 Yeah. They are spinny numbers. The relate directly to sinusoidal behavior via euler's relation if you have 1 dimension of spinny numbers. If you work with 3 dimensions of spinny numbers (i, j, k); they allow for rotation in three dimensions without gimbal lock. In principle; they could be used to handle rotation in roughly as many spatial dimensions as you want to consider.
Hi , I have a question that i didn't find an answer for , it would be nice if you can explain/answer it 😅
What is ∞⁰ equal to?
We just started taking limits (grade 12) and our teacher told us to treat ∞ as a number (just like when multiplying a negative number by a positive number we get a negative outcome and so on) so i was wondering if ∞⁰=1 ?
I asked her but she told me that she has never encountered this expression then went on and asked the maths supervisor but the supervisor simply said: there is nothing such as ∞⁰ , where did you get this from? 😃
It would be nice if you can answer it 😊
apologies if this isn't clear , English isn't my first language 🙏🏻
You can indeed multiply infinities like that but as you probably know some limits are indeterminate forms! ∞⁰ is also such a form. Intuitively speaking, the "infinite" part tries to "push" the limit and make it infinite. The "zero" part tries to push the limit and make it 1. What "wins" is how "big" the "infinite" part is and how infinitesimally close to 0 the "zero" part is.
For example, as x tends to positive infinity, the limit of x^0, which is a ∞⁰ form, is 1.
However, as x tends to positive infinity, the limit of x^(1/lnx), which is also a ∞⁰ form, is e!
Another notable indeterminate is 1^∞, an example being e = (x tends to infinity, lim (1+1/x)^x)
@@lemonandgaming6013 Thanks for the explanation
@@QamarMayar If you have any other questions let me know!
demonic realm's representative in math.
Unicorn realm's representative*
What textbook talks about the matrix representation?
Any abstract algebra textbook
@@ClumpypooCP Are you sure? I can't find that in mine.
2 nd
I always felt like imaginary number is a mathematical tool. I can feel the physical significance of the number '1' like '1 apple'. But how can I feel the physical significance of 'i' like in 'i apple'?
You can't have -1 of an apple either. But do you also view negative numbers as "just a tool"?
@@willnewman9783 But I can have '-1 degree Celsius'. Negative numbers then mean something is lower than a reference. Here, we can get a physical meaning too.
@@fahminrahman3543 I am not saying there is such a "physical interpretation" that you claim, but you picked a bad example. Temperature scales are messed up. The -1 you speak of makes sense because the scale is bad. It would be like if I invented a new unit of length, "tores", so that -1 tores = 2 feet.
When using a good scale like Kelvin, negative temperatures make a lot less sense. (I think negative Kelvin can still exist, but from what I can tell, it is somewhat of a trick.)
You can have an impedance of i in an AC circuit
The physical interpretation of i is 90°.
As a physics major with strong mathematical inclinations at heart, I’m always frustrated with people from the physics community-very bright guys indeed-thinking ostensibly poorly about mathematical concepts and how very “real” they are. Their explanations of how “i is just a piece of notation we invent so we can do some real world calculations” or the more atrocious belief “1+2+3+…=-1/12, only if you could actually sum up to infinity” is utterly infuriating 😑 Glad to see eloquent mathematicians like you educating us on how to view maths as a coherent, logically consistent and not at all esoteric discipline 👌🏻🫶🏻
Lol do they not realize basically ALL of math is "just a piece of notation" we use to solve real world problems??
so what is the square root of negative one? its like saying Schrodinger's cat is dead and alive at the same time...its not logical or consistent with the mundane real world of sensory perception. It seems offensive that the world of quantum mechanics can't be made logical or functional or consistent just by using real numbers, the world can only make 'sense' if we use the antisense notation of i....which begs the question of how a particle can be both present and absent in the same location at the same time
I prefer "every number is imaginary" since math is a human creation
Yes they are. They're called imaginary, so they are.
That’s why people have proposed to name them “lateral numbers”. Because the name “imaginary numbers” is a bit misleading in the opinion of some people
@@AndresFirte Hardly. It's not any concern of mathematicians that some people are too stupid to understand that words have more than one definition.
@@mrosskne well, if I recall correctly, it was Gauss, one of the greatest mathematicians of all time, who proposed the name “lateral numbers”.
And I think it does have some importance, because a lot of people think that mathematicians just waste their time studying imaginary stuff with no value to society, so the name “imaginary numbers” doesn’t help. And sadly, the people that decide study plans and government budgets are not mathematicians. So public perception of what mathematicians do and what mathematics is, is important
@@AndresFirte I don't care who proposed it. It isn't needed.
@@mrosskne alright, I disagree for the reasons I mentioned, but I guess I won’t convince you. Have a good day
imaginary number are really useful electro-magnetics BUT all can be done without. I still consider imaginary pure mathematical abstraction that is very handy - not real
All of math is made up. Even the real numbers are abstract constructs.
Complex numbers can describe real world things, for example positions and rotations.
There’s no reason why the only way of accepting numbers as “existing” is if they can count apples. That’s an arbitrary decision.
If we think of numbers as positions in a grid, additions as combining the arrows that point from the origin to those positions, and multiplications as rotating and scaling arrows, then complex numbers are perfect to describe positions and arrows in a plane in the real world.
And while it’s true that this is a very specific setup to represent complex numbers and that we’re not actually seeing complex numbers… we do the same thing with real numbers: I’ve seen 7 apples, but I’ve never seen a 7 itself. We can use apples to represent natural numbers, or positions to represent complex numbers. But ultimately both are just models to represent numbers, and no model is inherently more “correct” than another.
In other words, I think we shouldn’t say “these types of numbers do exist, and these don’t exist”. Instead, I’d argue that both real and complex numbers are equally valid in existing. Either both exist, or both are invented.
It depends entirely on how you view our system of numbers. If you feel that our system should describe the real world, then imaginary numbers are truly imaginary, because you can't really point to anything in the real world that is represented by an imaginary quantity. Most other elements of our number system are easy to demonstrate: I can count apples, I can add apples, I can divide apples in half, I can make a 5x5 square of apples; I can even almost show a negative number by taking one apple away. That last one's a little bit abstract but it's a pretty small leap. Now, show me imaginary apples ... you really can't. By that basis, imaginary numbers are imaginary.
If on the other hand you view our number system as an abstraction that has many useful applications including modeling the real world, then imaginary numbers are just fine, they are no more and no less abstract than anything else. If we go that route, I'd favor calling them "orthogonal" numbers, but that's just me.
Square roots stretch your analogy. Yes, if I have a square lattice of n apples I know the square root of n by counting the apples along one axis. So since we can barely show square roots and we can barely show negative integers, the "numbers represent the real world" position becomes weak and is frankly unable to explain a lot of math that we cannot readily represent in simple apple terms.
I think the better apple analogy with apples for complex numbers would be that apples in a line left and right are real apples but those not on that line are imaginary apples. But no matter what we do, we are pushing the abstraction to unreasonable limits that don't really help anyone, and we are really getting hungry for apple pie.
@@Qermaq I admit you have to split the apples into fractional apples if you want to show that the square root of 10 is about 3 1/6. But if you don't like apples for this purpose, how about a volume of sand you can form into a square; then the lengths of the sides show a square root.
Complex numbers can describe real world things, for example positions and rotations.
There’s no reason why the only way of accepting numbers as “existing” is if they can count apples. That’s an arbitrary decision.
If we think of numbers as positions in a grid, additions as combining the arrows that point from the origin to those positions, and multiplications as rotating and scaling arrows, then complex numbers are perfect to describe positions and arrows in a plane in the real world.
And while it’s true that this is a very specific setup to represent complex numbers and that we’re not actually seeing complex numbers… we do the same thing with real numbers: I’ve seen 7 apples, but I’ve never seen a 7 itself.
In other words, I think we shouldn’t say “these types of numbers do exist, and these don’t exist”. Instead, I’d argue that both real and complex numbers are equally valid in existing. Either both exist, or both are invented.
@@AndresFirte Show me a length of 12 real inches; it's easily done with a ruler. Now show me a length of 12 imaginary inches. Rotating your ruler doesn't make it NOT real inches. Just as you've seen seven real apples, you've never seen seven imaginary apples.
You keep going back to "well if you PRETEND that real phenomena are imaginary phenomena" ... the fact that we have to pretend is what makes them imaginary.
@@kingbeauregard my point is that there’s no reason why numbers should be necessarily associated with length.
Is there imaginary length? No. Also no negative length. I’m not arguing that.
I’m saying that the *model* to represent numbers doesn’t necessarily have to be length: it can be something else. For example, positions in a plane. There, we can visualize and distinguish 12 from 12i.
I hope I’ve explained myself better this time: no model is inherently more valid than other models. And no model shows us the numbers themselves, I’ve seen rope of 30cm of length, but I haven’t seen 30 on itself. Since no model is inherently “correct”, I feel imaginary and real numbers have the same right to exist. (And I have no opinion on if they exist or not. Just in that they have the same right to exist)
I like the field extension version. That was the first one where I felt like they weren't just pulled out of thin air.
You can use field extensions to compute the discrete Fourier transform in a finite field. That helped me finally wrap my head around Fourier transforms. With the Reals, you can compute an arbitrary primitive nth root of unity. In a finite field, it depends on the factors of |F| - 1. So you can compute a 16-element DFT in F_17.
I think the way our number system works is flawed in a way where i is considered imaginary, or special. I don't think a perfect mathematical number system can also be a perfect practical number system. Just a thought