haha let me know how that goes

best comment ever ahhahahahhaa

So how would you describe the quaternions? Instead of 1 imaginary part, they have 3.

I say this all the time when talking to physics students

Now i need to find someone that want to discuss complex math hahahaha

I mean. His Hip is hella Sus (Yes, it is 2 years ago the meme probably did not exist)

@@drenz1523 very sus

Don't forget division by zero.

Yeah, why can’t 1/0 be defined as j?

@@notafeesh4138 There are branches of math where the square root of a negative number is useful and has real-life applications. And using i as a number expands math without breaking anything. No one has ever found a real-life application for a number that is the result of dividing a real number by zero (sometimes the LIMIT of such a procedure is useful, though), and such a number breaks math pretty easily were it to exist. (Nearly all "proofs" that 1=2, or whatever obvious nonsense, get there by dividing by zero and pretending they didn't.) 0/0 is a much more complicated case, as in some contexts there's an actual number hidden by that 0/0, but you can't tell what it is just by looking at the 0/0. (After all, the "result" of 0/0 is the x in 0 * x = 0, which could be basically anything.) And sometimes it's just as meaningless as 1/0.

@@notafeesh4138 It's tempting to thin of n/0 as "dividing zero times," but it's actually dividing into 0 parts. I would be tempted to say n/0 = 0, but I'm not a "math-magician" as my physics prof used to say.

It's making a statement. The zero means you're not doing anything with the number. It's still there, alone, standing on its own. Saying nonoperational would have been better than undefined. Using a physical example, such as ohm's law: E = I • R. If resistance (R) becomes 0 in a super conducting circuit, then what does current (I) mean when you get the answer, used under normal conditions, by dividing voltage (E) by resistance (R) when R = 0? Are voltage and current equal?

Rote memorization is a terrible way to learn. Especially today, with nearly infinite access to references (both good and *ahem* less-good quality), I believe that (most) school should be about demonstrating the ability to solve problems rather than the ability to memorize tabular data. In my engineering curriculum most testing was open-notes, and even closed testing sometimes allowed a "crib sheet" of your own design. My core physics and math classes, however, expected memorization of relationships and formulas. Memorization comes with repetition, but let me say that repetition should come with use (Ohm's Law, for example), not studying for hours to memorize equations for the heck of it. The equations that you use all of the time WILL start to stick, and the lesser-used relationships will always be there in your reference books when you need them. It seems to me that I'd rather trust a bridge built by someone who checked their references than someone who is "pretty sure" they remember correctly. Everyone I know has had at least one test where they thought they did well and were later surprised by a poor grade...

I figured out the rotation thing, from playing with my TI-89 graphing calculator. What happens with (-1)^x when x is not an integer? The math expression that it gave me, showed that it rotates along the unit circle on the complex plane, and that -1 and 1 were just special cases of when x is an integer. Thus *_i_* now makes more sense, as it is just the half-180º rotation of multiplying by -1. As shown in (-1)^(1/2) or √(-1) as it is often stated. Half a factor would result in a 90º rotation to where? Somewhere *not on* the real number line? *_i_* and *_j_* ? Exactly how do those relate to the quaternions, which also has a *_k_* ?

I could give you the outline of the magnitude of the problem . Imagine this: 1. my math teacher from high school would made some clumsy comments about hotter girls in my class, and he was in his 50s. 2. I did not understood any new mathematical field we tackled from his explanations, as they were so damn poor and vague. 3. From a friend who took private lessons with him, I heard that his explanations were excelent. My logical deduction - he knew math, yet hesitated to teach us, because he wanted to look smart, especially to the girls. I spent 3 years with him, and I sensed some things. Might not be, but it does not matter... The point is , that It is not only a problem of curiculums, and such. but that we are dealing with very problematic people. This guy made me think that I am stupid for math, and altered my life course onward in a sense of education, because he was satisfying his patological needs. And I am just one of many with that story. We are dealing with problematic teachers who are in need of therapy.

I can try to answer for the "i" and "j" ambiguity.. in the 16th century some Italian mathematicians were competing to solve the cubic equations and they all came up with solutions involving imaginary numbers; they were Scipione dal Ferro, Niccolò Tartaglia, Gerolamo Cardano, Rafael Bombelli. The point is that in Italian "i" and "j" were just two different graphical ways to print the "i" letter. "j" was just a nicer way to print "i". You could write Julius Caesar or Iulius Caesar, it was just a graphical variation. Letter-shapes were standardized later, thanks to the huge spread of Gutenberg's press. I would just use "i" as "imaginary" and forget about the j usage

Yes, the rotation explanation was excellent. Cheers!

same for the irrationals

And for the Big Bang, if I remember rightly

@@upandatom Aren't they called irrational because they can't be expressed as a RATIO of integers?

Up and Atom No, in the case of “irrational” it’s the non-mathematical meaning that is unfortunate, since the mathematical one is pretty literal and straightforward.

Angular numbers could be a better term (well anything could be a better term as most people say!)

Imagine -i/4 timing

Challenge accepted!! @@RandomCatFromFrance

Acb Thr "j" in electronics

@@davidsonjoseph8991 90° rotation is more important.

People always think of the square root of -1, when hearing i, but really, almost every idea that uses it is interested in the algebraic rotation.

Dvd Ftw Uh... yeah. Thanks. I know. Most software uses i regardless.

Because that's not how imaginary/complex numbers came about historically. Originally, they came up when trying to find a general way of solving cubic equations - if you try to find a solution algebraically, you get intermediate steps involving taking the square roots of negative numbers, even when the final answers are all real numbers. So mathematicians imagined that those values were meaningful, and worked out how to manipulate them without ever treating them as really being numbers. It was a couple of centuries after that before anyone thought of a geometric interpretation of complex numbers as a complex plane. So, tradition, mostly. There is also the point that there are more natural ways to conceptualise rotation - in general, multiplying by a complex number both rotates and scales, so introducing complex numbers as a way of doing rotations immediately raises the question of why you'd invent some weird two-dimensional numbers to represent rotation when you can just continue to use a 1-dimensional angle. The algebraic approach gives you a scenario where complex numbers are actually necessary, not just useful, even if polar form is more convenient for many purposes.

@@AndreaCalaon73 FYI, posting a URL in a comment without any additional info often gets that comment thread suppressed by YT. It might be different since the OP comment got a ❤️ but in general, YT dislikes seemingly random links to external sites.

Yes. Also, that's Plato for you: not a full philosopher, only a dwarf philosopher.

Nice

mathia

Quaternions *Lv100*

Me too thanks

@@The_NSeven haha yes

@@aniofri include me in the screenshot

this is true, numbers as a rule are badly named. I kind of like how the transcendental numbers sound though :)

@@upandatom Surreal numbers also deserve their name ;-)

They're called irrational numbers because they're not ratios, and transcendental numbers because they transcend the algebraic numbers

jfb-1337 It’s the non-mathematical meaning of “irrational” that’s unfortunate.

Surreal numbers seem apt to me

glad you found it useful!

You're an engineer and didn't know this? Are you serious?

Good pun dude

SCREW QUATERNIONS

it is pretty broken

I recommend the 2 videos by Ali Abdaal on evidence-based techniques to study for exams. They're not about how to cheat the system but rather how to learn, understand, and retain new knowledge. It's similar to the way Brilliant works but on your own education!

Try my KZfaq channel mathfullyexplained

drowning someone seems a bit drastic tho

Up and Atom yeah, the Pythagoreans were pretty hardcore, and not in a good way.

@@ericherde1 To this day, Math department meetings haven't changed...

So basically they were upset

Here’s what AC Grayling writes about it:” The discovery of irrational numbers was so traumatic for the Pythagorean’s, legend has it, that the man who made the discovery (or, some of the legends say, the man who revealed it after the order’s members had been sworn to secrecy about it), namely Hippasos of Metapontum, was punished by being drowned.” (History of Philosophy, p. 23)

I would recommend taking calculus and ordinary differential equations as they are used quite a bit in the mathy side of biology.

They come up in the physics of MRI machines... don't "count" them out!

Silt Biotechnology without complex numbers? I have 3 words for you...... Mantis shrimp eyesight

I used to think because I was into art... I'd never need this math. Then I got into computer graphics, ha ha ha ha ha ha

"Since I am studying Biotechnology I will never use complex numbers in my future job." As a biotech engineer I can tell you, I'm always impressed how much from my study I could use again in some way. You'll never know what might turn out to be helpful in the future. After Steve Jobs dropped out of university, he might have thought, this calligraphy course was a waste of time. However, it turned out, that this skill contributed to the outstanding graphics of the Macintosh.

And all this complicated mess just because we couldn't restrict the domain of the function to the original limited domain? Trying to represent 2/4 or 1/2 by counting fingers, is holding a finger out half-way? That violates the digital principle of bi-stable bits. After a while when my finger gets tired, is it now 1/3? Or 2/3? So why is my fancy cheap scientific calculator, unable to calculate 1.5! ? Domain simplistic much? So finally the complex numbers are the complete closed set of numbers that result from all algebraic operations. But then, aren't the quaternions quite cool? But what operation can I do, to a real or complex number, to produce a quaternion, other than simply positing that quaternions ought to exist? And then are the quaternions the ultimate numbers? Can I stop there?

Try my KZfaq channel mathfullyexplained.

It also means even if it's free, you can't afford it ;-)

No worries that's awesome you understand them better!

i remember reading that they were called "imaginary" by Euler. Haven't checked it in sources, though

What you described here is _not_ historically accurate. I highly recommend Veritasium's video on how imaginary numbers were invented. As a brief summary (this history takes place over the course of 300 years, but I've managed to reduce it to 4 paragraphs), square roots of negative numbers started out as a _necessary_ intermediate step to find (real number) roots of cubic polynomials. Much like there is a quadratic formula, there is also a cubic formula to find the roots of cubic (degree 3) polynomials. However, depending on the coefficients, plugging numbers into the cubic formula would sometimes give square roots of negative numbers. This was a bit perplexing to mathematicians of the 16th century, since, unlike with quadratic polynomials where some of them had no (real number) roots, _every cubic polynomial has a (real number) root._ So in order to find this real number root, sometimes you _had_ to use square roots of negative numbers. The term "imaginary number" was coined by René Descartes. And this was in line with European thinking of numbers. Because mathematics in Europe had a strong Greek tradition which was rooted in geometry, European mathematicians always thought of numbers as representing geometric ideas, such as length, area, and volume. European mathematicians at the time disliked negative numbers, but they could still kinda make sense of them geometrically if thinking about length/position in a certain direction. However, to European mathematicians, lengths, areas, and volumes were _always_ nonnegative. There was no possible length, even considering directions, which gave a square of negative area. As such, there was no _number_ (length) which squared to (resulted in a square of area) -1. So in order to work with such a number (length), you had to _imagine_ it. That is how Descartes first used the name "imaginary". Euler developed his famous formula (e^(it) = cos(t)+i*sin(t)) in the mid-18th century, but even this, alone, did not make the full connection between complex numbers and rotations that we think about today. At this point, Euler still viewed i as a number which had to be imagined, since it did not represent a valid length. It wasn't until Jean-Robert Argand developed the complex plane at the beginning of the 19th century that a geometric interpretation of complex numbers took shape, where, much like negative numbers represented a _signed_ direction/position, so too did imaginary numbers (within a planar configuration, rather than a linear configuration). Euler's formula could then be applied to view complex number arithmetic as movement within a plane (as opposed to lengths of a square yielding certain areas, as numbers had been previously thought of). (Note: Caspar Wessel also had a geometric understanding of complex numbers, and he did so roughly 10 years before Argand, at the end of the 18th century. However, Wessel's publication went unnoticed for roughly 100 years, but Argand's work was noticed fairly quickly, about 7 years after he disseminated it.) But it was Gauss who really cemented complex numbers' place in the "mathematical canon" so to speak since he showed just how necessary and applicable they were within all sorts of mathematics. Gauss also was the person who coined the term "complex number". But you also have a bit of a misconception about the distinction between the terminology "imaginary number" and "complex number". The term "complex number" did not _replace_ the term "imaginary number". An _imaginary number_ was always a number which squared to a negative number. In other words, an imaginary number is one of the form bi where b is a real number and i^2 = -1. A complex number is any number of the form a+bi where a and b are real numbers and i^2 = -1. As such, the imaginary numbers form a subset of the complex numbers (a+bi where a = 0). (To be fair, the real numbers also form a subset of the complex numbers, a+bi where b = 0.) Gauss disliked the terminology "real" and "imaginary", so he proposed a new naming convention: positive real numbers would be called "direct numbers", negative real numbers would be called "inverse numbers", and imaginary numbers would be called "lateral numbers" (since they moved _laterally_ to the direct/inverse directions). Then, a number compromised both of a direct/inverse component and a lateral component was called a "complex number", where "complex" comes from the meaning of being comprised of multiple parts (much like an apartment complex is comprised of multiple apartments). While the name "complex number" stuck, Gauss's preference for direct/inverse and lateral numbers could not overturn the 200 years of momentum the terms "real" and "imaginary" had built up.

Quaternions = real * (imaginary + joke + kooky)

GMP Studios I was about to argue until it struck me that you are indeed right. 🤔

How's that?

What about noninteger powers of complex numbers?

Try my KZfaq channel mathfullyexplained

Yes I've had a very slow start to the year, but it should (hopefully) be smooth sailing from here!

that sucks. i was given a similar response when i asked what triple integrals were doing