I’m a tutor and every time I explain imaginary numbers to precalculus students they are so confused. Your graphic here illustrates it perfectly. Thank you for taking the time to do this!
@@-Burb Tons of people take a Calculus course who have very weak algebra. It makes it damn hard for them to pass it, but that's just the way it is. I sometimes wonder if they shouldn't let someone take a Calculus course at all unless they got a B+ in an algebra course... RECENTLY. Letting them take it on the strength of getting a C+ in an Algebra course 4 years ago: that's just setting them up for failure. My brother, my research supervisor, and myself: all 3 of us failed a Calculus course after getting As in high school math. Obviously we just weren't doing any work, but it illustrates that even if you did well in high school math, you can still fail Calculus. Everyone should therefore not even start a Calculus course unless their Algebra is solid.
I've also tutored math. I don't recall helping students with Complex math a lot, but I would just tell them that imaginary numbers are, in a sense, less real than "Real" numbers... but that they are still useful.
@@UTU49 It’s because most of the people taking precalculus are either seniors or juniors, since some schools don’t offer the first level of math in 7th grade option that allows sophomores to take it. Seniors don’t really care about the material and may not even understand it because if they didn’t care to learn it earlier than senior, chances are they’re just doing it for credit.
I remember a time where I was joking around with my algebra 1B teacher; "Hey it's kind of wacky that the calculator responds no real numbers does that imply the existence of imaginary numbers?" "Yes." I cannot describe the internal panic I had at the idea of seemingly non-existent numbers.
Something similar happened to me lol. And then the teacher just breezed right by it! It was mid lesson, and she was just like “Oh yeah numbers that don’t exist exist, but that’s high school stuff, anyway…”
Man, this is one of the greatest, mind-opening playlists about the beauty of math that has ever been posted to RUclips. This taught me to see algebra visually. Absolutely incredible. You are so talented. Why did you stop making series like these? They're incredible!
By the way, imaginary numbers ARE called “lateral numbers” in China. It could just because it’s easier to pronounce(less syllables in the Chinese language), but Gauss would be proud
KommentarKanal I knew I was in for a show the minute the video title mentioned imaginary numbers being real. Better Explained already demonstrated how the number line is really a number plane, and how multiplying by /i/ is like rotating rather then scaling or stretching, but seeing it visualized like that made my day.
Thank you so much! Your 3d animation was apparently what I needed for the imaginary numbers to finally make sense. It's a great feeling discovering the missing piece you need to understand a concept.
I show this first video of the series every single semester that I teach Algebra students about "imaginary" numbers for the first time. Really gets through to them!
@@definesigint2823 tldw, he simply added a new dimension. Basically saying lets add imaginary axis to solve imaginary number. Hence no wonder he didnt get anything but views and clicks
@@juvenileygo (nods, thanks for clarifying) When I first saw these I was looking for a quick answer to the equation. While I didn't regret watching the series, it was a decision I hadn't expected to make when I first clicked.
The easiest way to understand real and imaginary numbers is by using my bank account balance as an example: its real part is $10 and its imaginary part is $10 million dollars.
SWEET! You managed to tell us and show us what you're telling us within the first 2 minutes and used the remaining time to expand upon it. instant subscription!
Math: If I have two apples, and I give you one, I will have one apple left. Finance: If I have two apples, and I give you one, you will have to repay me the apple in full after a set period of time, plus interest which is to be calculated as a percentage rate of the apple divided by the amount of time it took you to repay me the apple in full.
Politics: If I have one apple, and I give you one, everyone will shout & scream that they didnt get one & band together to try to force me to give them apples.
Economics: I have two apples, I give you one, but few people realize that apples are produced in a farm, and are worried that there isn't enough, and not even Apple farmers seem to know where apples come from (except the Bank of England Apples which plainly stated the truth). I'm MMT. A Neoclassical Economist would describe things that I think are not true and responsible for the mess economies are in (because they are run on the assumption that the currency issuer should behave like a currency User, & other things that don't apply anymore to modern money): ruclips.net/video/TDL4c8fMODk/видео.html
Feeling dumb must not be considered as a problem, it's the first step to get a solution, if you are aware that you are dumb ,then only you can become more wise by sorting out and solving the reasons, because only you know what's inside your head, so only you have the ability to make yourself bright. Rather than ignoring dumbness,cure it.
I just happened to think of this amazing series again, and searched to see how the channel is doing now, and discovered that it got active again 2 weeks ago! Lucky me. Lucky us. The world would benefit much from your inspiration! Hope this comment would give you a little boost of encouragement for your upcoming work!
Awesome video! I loved that visualization where you pulled the surface out of the flat paper, that was a big WOW moment! I've worked with imaginary numbers a ton, I studied physics in college, but this video still had an affect on deepening my understanding. Excited to watch the rest!
What I don't understand about that visualization is that after he pulls the surface out, there are an *infinite* number of roots. I thought he just said that there are exactly as many roots as the degree of the polynomial?
frother - There actually only two roots. The "infinite" intersection of the 3d parabola to the imaginary plane is actually just the extension of the whole parabola through 3 dimensions (x, y, i ). Two roots can be seen by taking a different "slice" view point along the new dimension parallel to the coordinate plane (3 units above paper). This will give a new coordinate view of the parabola that does indeed intersect at two points.
That's a good point, but it's only a problem with the visualization. In fact there are only two roots. The problem is that to really plot the function, we would need 4 dimensions, not just 3, since the input of the function requires 2 dimensions (real and imaginary/lateral) and the output is also a complex number so it would also need 2 dimensions to plot properly. In this visualization they simply didn't plot the imaginary part of the output value of the function, only the real part. And there are indeed infinitely many complex numbers whose square's real part is -1. But for most of them there is a nonzero imaginary part (except for the 2 actual roots, i and -i).
This is one of the best explanations I heard about anything. Incredibly well done and "easy" to understand! I wish they could teach at university or school like this :/
Great video. If algebraic concepts were explained this way in all schools, there would be far more students interested in Maths and Science. Keep the good work!
@@xwqkislayer7117 in computer systems, if a number is too big to be stored, it loops back to a negative number example: Let's say we have a binary system that can store 8 numbers: 000, 001, 010, 011, 100, 101, 110, 111 If we want to represent negative numbers, it makes sense to put them before the positive ones, so let's say: 000 = -4 001 = -3 010 = -2 011 = -1 100 = 0 101 = 1 110 = 2 111 = 3 so the biggest number we can represent is 3. If we had another digit, we could have: 1000 = 4 But we don't. So if we tried to ”add 1” to our 3, it would be: 111 + 1 = (1)000 so our system would see 000 and think it is -4 This is integer overflow, when we don't have enough digits to represent big numbers which causes a mistake that turns it negative.
@@xwqkislayer7117 i simplified it a little bit to get the idea across, please keep in mind this is not exactly how computers represent numbers. computers are actually built to represent negative numbers using a thing called two's complement: if you have a positive number, flip all the digits, then add 1, that will be how you represent its negative. This way, we can actually represent 0 as 000 e.g.: 2 is represented as 010 so to get -2, you do 101 + 001 = 110 this way, you can add the individual digits to get 0 back: 2 + (-2) = 0 010 + 110 = (1)000 The maths is easier this way. That also makes it easier to recognise which numbers are negative, as the first digit will be 1 if it's negative, and 0 if it's positive (-2 = 110, +2 = 010)
i'm loving this. gotta bookmark this and watch the rest of the parts. the talk about the negatives being 'unnatural and weird' is extremely relatable - not now of course, but back when i was in school when i was first introduced to it. and then you use this very relatable notion to explain how 'imaginary' numbers are exactly the same in that they may _feel_ unnatural but in truth they aren't (just like how we initially felt about negatives) - that is so good and appreciated. very exciting.
I remember when Mrs. Cerigo introduced imaginary numbers, I just threw up my hands and said, "Well, that does it, they've run out of stuff to teach us. Now they're just making things up." Glad to know I was wrong.
@@-Burb LOL. This makes no sence. It's like to say "we invented letters, but languages are all discovered". Even worst, cause words are always related to something real, but math just don't give a F about reality.
Suddenly remembered watching this whole series as a kid. I barely even knew what imaginary numbers but i really do feel like I've retained a lot from these videos -- must be some of the best science/math explainers on youtube. Also sidenote it's a shame youtube trends have switched from breaking up videos to making one long one because they you never get the feeling of noticing a "part 12" pop up on the side and thinking "well. i've gotten this far. might as well keep going" and when you finally reach that one it's not even the last one.
It's an (uncommon) misconception that Euler "didn't know what to do with negatives". Euler was the most productive mathematician to ever have lived. He dealt with complex numbers and complex functions in full generality, it is simply nonsense to say that he didn't know what to do with negative numbers. (It is true that he assigned negative values to some positive (divergent) series, but that was 100 % intentional.)
52 years after studying the complex numbers in school, I finally understand complex numbers and more!!! Absolutely brilliant!!!! (I'll have to stop now or I'll wear out the keyboard unevenly - too many exclamations :)
When i see wise people like you watching this and enjoying the beauty of acquiring a clarified version of old knowledge with such enthusiasm at such age, that sir makes me feel like who am I with 40+ age to feel down that I feel I wasted parts of my life not continuing to learn things I used to enjoy thinking that I am already old. Thank you sir for giving me hope that I am not alone at enjoying such knowledge. Thanks for sharing the passion to learn.
Here again, after a few years. Just wanted to let you know that, watching this was definitely one of the most memorable moments in my math journey. I got a whole lot more interested in Graphs and Complex Numbers, learnt to accept them as a concept that weirdly works.
Great video! I think back to my years of college when I learned electrical engineering. We were buried in math every year: calculus, differential equations, differential equation with complex variables, and more. During most of those years, the square root of minus one was central to the math. Just days before my friends and I were to graduate, we were relaxing in the cafeteria drinking coffee, when one of my friends suddenly folded his arms and shook his head negatively left and right. "I don't care what they say," he said with a determined look on his face, "I will never believe in the square root of minus one!" We all laughed.
Maths works really well in network analysers. In high frequency work, telecommunications industry etc. You have to work in 3 axis, frequency and time but also voltage
Including Workbook, including Solutions.. You are a true Math Hero!!! Even including Dutch subtitles for bonus comprehension!!! Impressive mate!!! Amazing work!!
Alternately, a better way to think about it is that no mathematical systems are 'real' in that they are necessary to describe physical observations, they're all models we made up, imaginary numbers are just a useful extension to one system of math that allows us to describe a certain system of useful relationships a fairly compact way.
this is truly why a lot of people find math difficult to understand. a lot of the names are grotesquely indescriptive. if they had more intuitive names, people would be able to pick things up much quicker, instead of having to first memorize what it means, in addition to learning how it works.
I can't thank you enough for taking the time to make these imaginary numbers videos. We're about to start the topic in my applied Algebra 2 class, and it's always so hard for me to come across as fully believing in these things. I have been looking for a video like yours for a while. I may actually play them all! Really, this is just so awesome.
Hey, this is really cool. I love the animation for "lateral numbers". I have started seeing some of the impact of imaginary numbers watching 3b1b's series, but he just sort of travels around a unit circle with them. Lets see how the next videos in the series do.
21st century: "let's call them fake numbers" 22nd century: "flat numbers, because Earth is flat, so is everything" 23rd century: "Numbers are individuals too! Each number should have a name! Isn't that right, Richard?; *-3:* _Yes._ "
As someone that still cannot understand imaginary number after 4 years of undergrad study, you put a relatable analogy with negative number. Now I understand why I need to understand it. Able to understand is a different topic.
TootTootMcbumbersnazzle Of course the guy with the anime profile doesn't have a sense of humor while attempting, in vain, to be humorous himself. "oh u"
+Frasafrase Great question! I'll explain in detail as the series progresses, but yes, the function i show does have too many roots. This is because it's the only the real part of f(x) = x^2 +1 for complex x. In part 8 I'll show the real and imaginary parts together, and we'll see exactly 2 roots - I didn't want to overwhelm everyone in part 1. Thanks!
+Frasafrase It's a consequence of 'graphing' a four dimensional relationship in three dimensions. Color weighting takes a bit of getting used to. Remember that mapping a complex number to a complex number (For example (3+2i)^2 = 5+12i) requires two dimensions for both the 'departure' and 'arrival' points.
Only the "real" part of the graph is shown. In order to be a root, its real AND imaginary part (graph not shown) must intersect the y=0 plane. That only occurs at x=0+i and x=0-i, the two roots. Have a look at this plot. The two parts of the graph aren't shown together, but the roots are the values that overlap where the real "saddle" crosses the y=0 plane (two parabola shapes) and where the imaginary plot crosses y=0 (a cross-shape of points centered on the origin and aligned with the axes). www.wolframalpha.com/input/?i=plot+z%3D(x%2Biy)%5E2%2B1
It's a plot of the real component of the complex function w=z^2+1 Here, z is the "independent" complex number that used to be x and w is the "dependent" complex number that used to be y. The reason that this function seems to have many zeroes is that it's not looking at all of w, only the real part. If you looked for points which have zeroes in the real AND imaginary parts, you'd find there's only two of those. I have a strong feeling this didn't help much..
Honestly this is the most important video out there explaining imaginary numbers. This has to be archived in museums for generations to come. Thank you very much for the important work!
Here's the thing - add spin with leverage into these equations and you'll find gyrodesics. Temporal torsion in conserved momentum. I really like lateral. I'm voting for them in 2020. Go team.
1:47, it seems the solution to x^2+1=0 is a curve (or two) instead of 2 point (+i and -i). actually x is in 2 dimensional plane, so is f(x). so it requires 4 dimensional to show the function.
This is the video that ignited great interesst in math in me for maybe the first time. As such it was a major contribution to why i am studying mathematics today. Thank you.
Leonardo Aielo Tassi nope, 0!=1 One way of seeing it is by thinking that the factorial function tells us how we can order stuff, A&B can be ordered {AB} and {BA} 2!=2 {A}gives just one "{A}" (1!=1) And the empty set { ø } can be ordered in one way {ø} 0!=1
Fantastic! Humorous, informative, brings further maths to life. I teach this subject, yet I've never once seen the graphic at 1:50 in any maths textbook.
I'm largely self taught in anything beyond grade 8, so I didn't learn about complex numbers until I got into electrical engineering. Learning complex numbers in the context of a practical application like AC anaylsis helped them make a LOT more sense to me. It's too bad most students in school don't have that same benefit. I can't count how many times I stuggled to learn a math concept in school only to discount it as arbitrary nonsense because I couldn't understand it from the rote memorization way we were meant to learn. I find it much easier to learn something if I'm able to connect it to something useful or meaningful so I can understand it from an intuitive point of view, rather than just memorizing that this is how it works and taking their word for it.
Negative numbers have practical applications like debt, which is a concept that even pretty young kids are familiar with. Complex numbers and their practical applications are pretty tough to come by at the moment or become common very late in one's education, which I think contributes a lot to their un-intuitiveness. Complex numbers really do make AC analysis a lot easier to understand, much in the same way that negative numbers make debt easier to quantify. Same could be said about polar coordinates
@MikeProductions1000 I thought so, too. That's one of my favourite aspects of electronics is how often you come across topics which seem totally unrelated, but are closely linked in some very non-obvious way. The other part I like is just seeing all of these seemingly arbitrary concepts in math being put to practical use for the first time ever. Which goes for anything in engineering, really.
One of the coolest parts of my abstract algebra class was when defining field extensions lead to a complete definition of the complex numbers in terms using field extensions on reals. I think what was so cool about this is that, if you look at how other numbers are proven to exist in the context of axiomatic set theory, numbers like the rationals are also defined as equivalence classes of naturals. This basically means if you accept the existence of rationals, you basically have to accept the existence of complex numbers
When you "pull" the graph up and make it three dimensional, then yes it crosses the X axis, but it suddenly looks like it crosses it in a lot more places than just 2... and it should only be 2. So I dont think that 3d model was a good representation
The actual function values would be the outermost edge of the shape, the actual extension of the plotted line, not the interior area. Which would be a _different_ but related function (probably involving calculus). It was filled in only to provide visual context for us viewers.
Actually, this is a prank video by some jerk, cuz for the eq f(x)=x²+1, we are working with only 2 dimensions. Where the hell did you get the 3rd dimension from ? so for every question, just simply add another dimension if can't solve it?
I "learned" imaginary numbers at some point in school, very briefly. But I never truly understood them. You managed to do a better job in 5 minutes of RUclips video than 20+ years of education. Finally I truly get it, and it's not even hard. Completely demystified, like a great cloud has been lifted. You are a legend.
Thank you so much! 'Imaginary' numbers were my big stumbling block in A level maths, and my maths teacher was unable to explain them (because he only got the job for being the headmaster's old chum). This video has a made it clear for the first time to me. If only my maths teacher had explained it as another dimension like this, instead of "You don't need to know how it works, just memorise how to use it to pass your exam". I might have passed that A level and become an astrophysicist as I wanted.
No, don't go there. You were a rational human being before when you couldn't understand imaginary numbers. You actually knew intuitively that it was all a load of garbage and just fantasy. Now you've come to accept them as real when they aren't. Go back to the light.
@@tomjscott They're demonstrably real. Physicists have demonstrated that our most fundamental powerful theories of reality only work when using complex numbers. They're as real as any other number system; to assert otherwise is ignorance
I think the best way to describe the imaginary part is to imagine the world of forgotten things, when you found a thing, this means that it changed the state to real.
Dude I remember watching this video for the first time and understanding absolutely nothing of it. Now that I've had imaginary numbers and that stuff in school, I still don't understand it.
The best way to see is this ..... For ancient people the negatives are imaginary because they were beyond there imagination. And for us imaginary (lateral) numbers are imaginary.....in future, people gonna make videos about us thinking that imaginary are imaginary...
I LIKE YOU I LIKE THE WAY GRAPHIC TO EXPLAIN ..... WHEN FIRST TIME LOOKING AT THE PICTURES OF YOUR MV AND WHEN I SEE YOU MOVING PICTURES ....... I HAVE SOME IDEAS UNDERSTANDING OF IMAGINARY NUMBERS IMMEDIATELY ................
AWESOME : Watched the whole serie! THIS was the best and most intuitive explanation of number theory and complex number ever, where also math newbies could follow and get a deep understanding! Thank you soo much was this highly entertaining and educative masterpiece! ❤️👍🏻💡 that was a tremendous effort of work and brain you put into it! 😇
I remember find this series years ago, it made me think of complex and imaginary numbers as completely natural and not strange at all, i want to thank you for being such a great teacher!
@@WelchLabsVideo Hello, please I think this is an amazing video but I would love if you could include the resources where you found all this info so people can do further reading . Thanks
![Imaginary Numbers Are Real [Part 2: A Little History]](http://i.ytimg.com/vi/2HrSG0fdxLY/mqdefault.jpg)
![Imaginary Numbers Are Real [Part 2: A Little History]](/img/tr.png)
![The most beautiful equation in math, explained visually [Euler’s Formula]](http://i.ytimg.com/vi/f8CXG7dS-D0/mqdefault.jpg)
“We’ll be using the term lateral from now on”
*continues to say imaginary*
where are your comments?
It's your imagination, dude.
@@neh1234it's his lateral* now
√anti-apple = banana
@@smallgreen2131 Wut?
I’m a tutor and every time I explain imaginary numbers to precalculus students they are so confused. Your graphic here illustrates it perfectly. Thank you for taking the time to do this!
Paige Brady
Precalc students don’t understand imaginary numbers by that point?
Ngl this graph is.harder to understand than explaining it normally.
@@-Burb
Tons of people take a Calculus course who have very weak algebra. It makes it damn hard for them to pass it, but that's just the way it is.
I sometimes wonder if they shouldn't let someone take a Calculus course at all unless they got a B+ in an algebra course... RECENTLY.
Letting them take it on the strength of getting a C+ in an Algebra course 4 years ago: that's just setting them up for failure.
My brother, my research supervisor, and myself: all 3 of us failed a Calculus course after getting As in high school math. Obviously we just weren't doing any work, but it illustrates that even if you did well in high school math, you can still fail Calculus. Everyone should therefore not even start a Calculus course unless their Algebra is solid.
I've also tutored math. I don't recall helping students with Complex math a lot, but I would just tell them that imaginary numbers are, in a sense, less real than "Real" numbers... but that they are still useful.
@@UTU49 It’s because most of the people taking precalculus are either seniors or juniors, since some schools don’t offer the first level of math in 7th grade option that allows sophomores to take it. Seniors don’t really care about the material and may not even understand it because if they didn’t care to learn it earlier than senior, chances are they’re just doing it for credit.
You delighted me with the 3D lateral-plane visualization. Well done.
Me too! How did he do it?
After Effects?
+Neithan magic
William Cannon It may look 3D, but it's 4D. It's explained in part 10-13
You´re not the only one!!! XDD!!
I remember a time where I was joking around with my algebra 1B teacher;
"Hey it's kind of wacky that the calculator responds no real numbers does that imply the existence of imaginary numbers?"
"Yes."
I cannot describe the internal panic I had at the idea of seemingly non-existent numbers.
Yo, that's a kick in the discovery, I wish I had thought of that before when I was taught about the set of all real numbers
Wait, calculators don't respond real numbers?
Rip
Something similar happened to me lol. And then the teacher just breezed right by it! It was mid lesson, and she was just like “Oh yeah numbers that don’t exist exist, but that’s high school stuff, anyway…”
@@the_demon149so sad they didn't digress for a minute. minds are open far before H.S....perhaps more so
In polish it's even worse. Imaginary numbers are "liczby urojone", "uroić" means to hallucinate...
czyli dokładnie to samo
@@pan_czerwony5437 imaginary to bardziej wymyślone, imagination to wyobraźnia i to nie ma takiego negatywnego znaczenia.
@@kswiorek Ale to synonimy,a w Angielskim jest dość mały zasób słów w porównaniu z Polskim wiec można uznać to za to samo,ale rozumiem tok myślenia
@@pan_czerwony5437
What did π say to i (the square of -1) ? Get real (ie. not imaginary)!
What did i say back? Be rational
Evergreen math joke :)
@@swinki33 oh god
Brah, he pulled a rainbow out of his paper.
Drugs
damn
LOL!
*Bruh not brah lol
How did he did that?
JATIN GANDHI editing
thank you Gauss. It sounds much less awkward to say that I have a lateral girlfriend
@Jalfire: Me . . . I just keep it to myself and don't mention it to anyone else at all.
so original
@@QED_ lateral girlfriend= mistress
Lmfao😂😂
still awkward though
Man, this is one of the greatest, mind-opening playlists about the beauty of math that has ever been posted to RUclips. This taught me to see algebra visually. Absolutely incredible. You are so talented. Why did you stop making series like these? They're incredible!
This series of videos is really amazing, great job and keep it up!
hey patrick ..best maths teacher/professor/tutor on youtube
Ayyy Patrick shoutout for being the reason I passed first year maths 👌 👌 👌
Shoutout to forpatricks for also the reason why i passed all my classes lol
Patrick, your videos are my Go-To videos.
The presentation of math has never been so fun and interesting like this one here. Kudos to thee. 10/10
Thank you!
@@WelchLabsVideo Keep Making *Great* Videos. And Thank You For Such An *Amazing* Explanation.😀
its all fun and games in math class until the graph starts speaking 3d
You will see Fourth Dimension in future, which you will not express or understand in 2d papers like you do 3 dimensional shapes.
be still my heart!
@@Email5507 impossible to understand, impossible to imagine, we can only "speak" about it, i love it!!!!
Imagine classes in vr headsets
fourth dimension is rotate in 3d space. It would have a pitch, roll, and yaw. It's quaternion.
Fun fact about bi-nion and quaternion. They are MATRIX.
By the way, imaginary numbers ARE called “lateral numbers” in China.
It could just because it’s easier to pronounce(less syllables in the Chinese language), but Gauss would be proud
Very cool
Well, imaginary numbers in Chinese still has the ‘imaginary’ meaning. It’s called 虛數 I think
“The Tiananmen Square protests are lateral”
@@masterspark9880 LMAO
@@masterspark9880 legendary
The easiest way to understand negative numbers is by picturing my bank account.. 😔
And if u don't have any account like me
@@zekzimbappe5311 Watch other peoples poor bank accounts.
@@zekzimbappe5311 then that's lateral bank account
I LOVE YOUUU
And mine imaginary numbers. ;)
I never knew I could have that much fun watching a math video, well done.
Standupmaths mang
More real world applications would've been nice for us noobs. So, thumbs down.
Numberphile has some cool video too
KommentarKanal I knew I was in for a show the minute the video title mentioned imaginary numbers being real. Better Explained already demonstrated how the number line is really a number plane, and how multiplying by /i/ is like rotating rather then scaling or stretching, but seeing it visualized like that made my day.
Mathologer is cool too :)
Thank you so much! Your 3d animation was apparently what I needed for the imaginary numbers to finally make sense. It's a great feeling discovering the missing piece you need to understand a concept.
I show this first video of the series every single semester that I teach Algebra students about "imaginary" numbers for the first time. Really gets through to them!
I just did the same an hour ago.
2:50 I know many of you just wanted to see this
💖💖
But a nerd like me wants explanation on how he solved the equation. Sadly he got nothing
@@juvenileygo Note, this is first in a series of 13 videos (all published here).
@@definesigint2823 tldw, he simply added a new dimension. Basically saying lets add imaginary axis to solve imaginary number. Hence no wonder he didnt get anything but views and clicks
@@juvenileygo (nods, thanks for clarifying) When I first saw these I was looking for a quick answer to the equation. While I didn't regret watching the series, it was a decision I hadn't expected to make when I first clicked.
The easiest way to understand real and imaginary numbers is by using my bank account balance as an example: its real part is $10 and its imaginary part is $10 million dollars.
the money is just hidden in a different dimension
Lol
@@NightmareCourtPictures lol😂
@@NightmareCourtPictures Im dying laughing!!!
Ok
Production is top notch
so is the explanation. concise, accurate, visually easy to understand. trifecta
SWEET! You managed to tell us and show us what you're telling us within the first 2 minutes and used the remaining time to expand upon it. instant subscription!
Math: If I have two apples, and I give you one, I will have one apple left.
Finance: If I have two apples, and I give you one, you will have to repay me the apple in full after a set period of time, plus interest which is to be calculated as a percentage rate of the apple divided by the amount of time it took you to repay me the apple in full.
Very true indeed.
How do I always see see you? On every geography now video I've seen ur comment and now on math? Holy crap man
Politics: If I have one apple, and I give you one, everyone will shout & scream that they didnt get one & band together to try to force me to give them apples.
Economics: I have two apples, I give you one, but few people realize that apples are produced in a farm, and are worried that there isn't enough, and not even Apple farmers seem to know where apples come from (except the Bank of England Apples which plainly stated the truth).
I'm MMT. A Neoclassical Economist would describe things that I think are not true and responsible for the mess economies are in (because they are run on the assumption that the currency issuer should behave like a currency User, & other things that don't apply anymore to modern money):
ruclips.net/video/TDL4c8fMODk/видео.html
Finance is math
Watched the whole series, honestly, for the first time in my life, i actually understood what was going on in my math class! Great Job dude!
What pulled me in was the 3D graph in the thumbnail :p
same
@@josepablobermudez6283 Me too. I wonder if you can make that with a 3d printer or do you need a 4d?
Your efforts for making a whole playlist on imaginary numbers is worth of appreciation. Very few can explain in details.
If you ever feel dumb,just remember at somepoint you can do what Leonhard Euler can't.
Feeling dumb must not be considered as a problem, it's the first step to get a solution, if you are aware that you are dumb ,then only you can become more wise by sorting out and solving the reasons, because only you know what's inside your head, so only you have the ability to make yourself bright.
Rather than ignoring dumbness,cure it.
couldn’t *
*slowly applaudes *
I love this comment. Its perfect.
Was he from Houston?.......The Houston Eulers.
@@machomachinmachinmachinmac6910 He was from Basel
Therapist: The square root of -1 can't hurt you, it doesn't exhist.
The square root of -1:
_[Imaginary Screams]_
*_[Lateral Screams]_*
*exist
square root of 1 can't hurt, but square root of -1 hurts!
Neither can division by 0 - oh wait is this the year 2020? You haven't gotten to n based multidimensional mathematics yet.
This is by far the best-presented video on mathematics I saw in my life. 10/10 for your pedagogical skills.
I just happened to think of this amazing series again, and searched to see how the channel is doing now, and discovered that it got active again 2 weeks ago! Lucky me. Lucky us. The world would benefit much from your inspiration! Hope this comment would give you a little boost of encouragement for your upcoming work!
Thank you!
Title:
Mathematicians: Well yes but actually no
Reddit moment!
@@Goosnav Goosnav
@@Goosnav destruction 100
holy shit you destroyed him dude
you're breathtaking
wholesome big chungus
Naw, negative numbers are the real Schrödinger's numbers.
Every number is a representation, just like signs
Awesome video! I loved that visualization where you pulled the surface out of the flat paper, that was a big WOW moment! I've worked with imaginary numbers a ton, I studied physics in college, but this video still had an affect on deepening my understanding. Excited to watch the rest!
A picture is worth a thousand words
What I don't understand about that visualization is that after he pulls the surface out, there are an *infinite* number of roots. I thought he just said that there are exactly as many roots as the degree of the polynomial?
frother - There actually only two roots. The "infinite" intersection of the 3d parabola to the imaginary plane is actually just the extension of the whole parabola through 3 dimensions (x, y, i ). Two roots can be seen by taking a different "slice" view point along the new dimension parallel to the coordinate plane (3 units above paper). This will give a new coordinate view of the parabola that does indeed intersect at two points.
That's a good point, but it's only a problem with the visualization. In fact there are only two roots.
The problem is that to really plot the function, we would need 4 dimensions, not just 3, since the input of the function requires 2 dimensions (real and imaginary/lateral) and the output is also a complex number so it would also need 2 dimensions to plot properly. In this visualization they simply didn't plot the imaginary part of the output value of the function, only the real part. And there are indeed infinitely many complex numbers whose square's real part is -1. But for most of them there is a nonzero imaginary part (except for the 2 actual roots, i and -i).
Thanks, I never expected to get such a clear and helpful answer from the youtube comments!
This is one of the best explanations I heard about anything. Incredibly well done and "easy" to understand! I wish they could teach at university or school like this :/
Great video. If algebraic concepts were explained this way in all schools, there would be far more students interested in Maths and Science. Keep the good work!
"From here on, let's let lateral mean imaginary."
Continues to use "imaginary" through the rest of the video.
for those who just started watching this, make absolutely sure you watch all the way to part 13.
prepare to be blooooowwnnn.
awesome series.
haha, thanks!
13 parts?... Who has time for that....
Where's the next video though?
Don't spoil the ending where the dragons die!
blo...blown? O.o uh no thanks
euler: -1 > ∞
He predicted integer overflow
Can you specify what integer overflow is? I'm sorry I dont know lol.
@@xwqkislayer7117 in computer systems, if a number is too big to be stored, it loops back to a negative number
example: Let's say we have a binary system that can store 8 numbers: 000, 001, 010, 011, 100, 101, 110, 111
If we want to represent negative numbers, it makes sense to put them before the positive ones, so let's say:
000 = -4
001 = -3
010 = -2
011 = -1
100 = 0
101 = 1
110 = 2
111 = 3
so the biggest number we can represent is 3. If we had another digit, we could have:
1000 = 4
But we don't. So if we tried to ”add 1” to our 3, it would be:
111 + 1 = (1)000
so our system would see 000 and think it is -4
This is integer overflow, when we don't have enough digits to represent big numbers which causes a mistake that turns it negative.
@@nuklearboysymbiote Thanks I didnt know that lol
@@xwqkislayer7117 i simplified it a little bit to get the idea across, please keep in mind this is not exactly how computers represent numbers. computers are actually built to represent negative numbers using a thing called two's complement: if you have a positive number, flip all the digits, then add 1, that will be how you represent its negative.
This way, we can actually represent 0 as 000
e.g.: 2 is represented as 010
so to get -2, you do 101 + 001 = 110
this way, you can add the individual digits to get 0 back:
2 + (-2) = 0
010 + 110 = (1)000
The maths is easier this way. That also makes it easier to recognise which numbers are negative, as the first digit will be 1 if it's negative, and 0 if it's positive (-2 = 110, +2 = 010)
@@nuklearboysymbiote ah ok ill keep that in mind. Thanks for the info
i'm loving this. gotta bookmark this and watch the rest of the parts.
the talk about the negatives being 'unnatural and weird' is extremely relatable - not now of course, but back when i was in school when i was first introduced to it.
and then you use this very relatable notion to explain how 'imaginary' numbers are exactly the same in that they may _feel_ unnatural but in truth they aren't (just like how we initially felt about negatives) - that is so good and appreciated. very exciting.
never stop doing these videos they are the best out there. thank you so much for taking the time to share them. with us.
I remember when Mrs. Cerigo introduced imaginary numbers, I just threw up my hands and said, "Well, that does it, they've run out of stuff to teach us. Now they're just making things up."
Glad to know I was wrong.
...but...you were RIGHT....they did "make-it-up".....😈
....God made the Natural numbers; everthing else is "made-up" 😆 ..(misquoting Kronecker)
I mean, _all_ of mathematics is "made up". That doesn't make it any less useful though.
simonO712
No, all of math is discovered.
The symbols we make are made up, but math itself if completely real and all discovered.
@BeetleBUMxX you're just calling everything in this comment section cute.
Pretty cute ngl :)
@@-Burb LOL. This makes no sence.
It's like to say "we invented letters, but languages are all discovered". Even worst, cause words are always related to something real, but math just don't give a F about reality.
The anti-apple
Strikes again.
Do u guy like android?
@@moioyoyo848 Everybody does
Suddenly remembered watching this whole series as a kid. I barely even knew what imaginary numbers but i really do feel like I've retained a lot from these videos -- must be some of the best science/math explainers on youtube. Also sidenote it's a shame youtube trends have switched from breaking up videos to making one long one because they you never get the feeling of noticing a "part 12" pop up on the side and thinking "well. i've gotten this far. might as well keep going" and when you finally reach that one it's not even the last one.
Me: *hates math*
Also me: *Watches this video because it was recommended*
even your name contains math bro!
Bruh so true lmao
Car
Dear Alzghoul bus
@@dearalzghoul4760 bike
It's an (uncommon) misconception that Euler "didn't know what to do with negatives". Euler was the most productive mathematician to ever have lived. He dealt with complex numbers and complex functions in full generality, it is simply nonsense to say that he didn't know what to do with negative numbers. (It is true that he assigned negative values to some positive (divergent) series, but that was 100 % intentional.)
52 years after studying the complex numbers in school, I finally understand complex numbers and more!!! Absolutely brilliant!!!! (I'll have to stop now or I'll wear out the keyboard unevenly - too many exclamations :)
jllebrun1 same feeling but i'm 56!
When i see wise people like you watching this and enjoying the beauty of acquiring a clarified version of old knowledge with such enthusiasm at such age, that sir makes me feel like who am I with 40+ age to feel down that I feel I wasted parts of my life not continuing to learn things I used to enjoy thinking that I am already old. Thank you sir for giving me hope that I am not alone at enjoying such knowledge. Thanks for sharing the passion to learn.
Hossam Zayed
Yeah I feel the same way Hossam, I have wasted parts of my life.
Hossam Zayed
حلو
It's such a good feeling isin't it :)
Here again, after a few years. Just wanted to let you know that, watching this was definitely one of the most memorable moments in my math journey. I got a whole lot more interested in Graphs and Complex Numbers, learnt to accept them as a concept that weirdly works.
Great video!
I think back to my years of college when I learned electrical engineering. We were buried in math every year: calculus, differential equations, differential equation with complex variables, and more. During most of those years, the square root of minus one was central to the math.
Just days before my friends and I were to graduate, we were relaxing in the cafeteria drinking coffee, when one of my friends suddenly folded his arms and shook his head negatively left and right. "I don't care what they say," he said with a determined look on his face, "I will never believe in the square root of minus one!" We all laughed.
I'm an engineering student so all of this is extremely interesting to me!! Instant sub! Phenomenal work.
Maths works really well in network analysers. In high frequency work, telecommunications industry etc. You have to work in 3 axis, frequency and time but also voltage
Including Workbook, including Solutions.. You are a true Math Hero!!! Even including Dutch subtitles for bonus comprehension!!! Impressive mate!!! Amazing work!!
The word "lateral" explains everything!
I hadn't been understanding "imaginary" numbers for years! You've discovered this secret for me. Thank you!
i²=-1 démystifié :
ruclips.net/video/2GwSUDm_Rg8/видео.htmlm43s
4:39-4:51 Ahh, of course.
2 Apples - 3 Apples = 1 Microsoft.
You forgot to square the two terms on the left.
@@ultimatesans2175 then that would be a google
@@ultimatesans2175 Then it would be -5 (2^2-3^2=4-9)...
thank you now it makes sense
i loved this video! normally i'm pretty slow at understanding these type of things but you made it real simple and enjoyable, thank you
Alternately, a better way to think about it is that no mathematical systems are 'real' in that they are necessary to describe physical observations, they're all models we made up, imaginary numbers are just a useful extension to one system of math that allows us to describe a certain system of useful relationships a fairly compact way.
Absolutely amazing. Great presentation, the best I've ever seen for a video about imaginary numbers and one of the best ever math videos.
this is truly why a lot of people find math difficult to understand. a lot of the names are grotesquely indescriptive. if they had more intuitive names, people would be able to pick things up much quicker, instead of having to first memorize what it means, in addition to learning how it works.
Fifteen years ago, little me would have been laughing to the thought of her grown self watching math-videos deep into the night
This video's title also had two part
Real:What is imaginary number
Imaginary:this series gone legendary
This comparison to negative numbers is actually so good
I can't thank you enough for taking the time to make these imaginary numbers videos. We're about to start the topic in my applied Algebra 2 class, and it's always so hard for me to come across as fully believing in these things. I have been looking for a video like yours for a while. I may actually play them all! Really, this is just so awesome.
Why the hell RUclips is recommending such an informative video after 4 whole good years?😯
This is the first time "imaginary numbers" (lateral is SO MUCH BETTER) have ever actually made sense to me in a physical context. Thank you.
3:41 I'm more concerned with what happened to Australia in this map
OMG
how about Antarctica
It became lateral
And Greenland
makes sense since australia was invented in the late 20th century.
Hey, this is really cool. I love the animation for "lateral numbers". I have started seeing some of the impact of imaginary numbers watching 3b1b's series, but he just sort of travels around a unit circle with them. Lets see how the next videos in the series do.
21st century: "let's call them fake numbers"
22nd century: "flat numbers, because Earth is flat, so is everything"
23rd century: "Numbers are individuals too! Each number should have a name! Isn't that right, Richard?; *-3:* _Yes._ "
Every number has already an own name. So your theoretical statement makes no sense.
25 Century: numbers get to choose their gender.
@@gdash6925 No, my 3 is called Richard. Your 3 is called, I believe, Timothy. Your statement is so numberist.
@@_Killkor my 69 is called..... wait
Microsoft Hites
26 century: Numbers become humans.
As someone that still cannot understand imaginary number after 4 years of undergrad study, you put a relatable analogy with negative number. Now I understand why I need to understand it. Able to understand is a different topic.
"Imaginary numbers are real"
Oh u
***** I wonder if you understand humor...
TootTootMcbumbersnazzle Of course the guy with the anime profile doesn't have a sense of humor while attempting, in vain, to be humorous himself.
"oh u"
That's the same as saying "There are more than two genders".
Don't.
whats wrong with anime -_-
this aint an insult to math, dont get triggered
What is that plot at 2:00? Because it has way more roots than it should.
+Frasafrase Great question! I'll explain in detail as the series progresses, but yes, the function i show does have too many roots. This is because it's the only the real part of f(x) = x^2 +1 for complex x. In part 8 I'll show the real and imaginary parts together, and we'll see exactly 2 roots - I didn't want to overwhelm everyone in part 1. Thanks!
+Frasafrase It's a consequence of 'graphing' a four dimensional relationship in three dimensions. Color weighting takes a bit of getting used to. Remember that mapping a complex number to a complex number (For example (3+2i)^2 = 5+12i) requires two dimensions for both the 'departure' and 'arrival' points.
Only the "real" part of the graph is shown. In order to be a root, its real AND imaginary part (graph not shown) must intersect the y=0 plane. That only occurs at x=0+i and x=0-i, the two roots.
Have a look at this plot. The two parts of the graph aren't shown together, but the roots are the values that overlap where the real "saddle" crosses the y=0 plane (two parabola shapes) and where the imaginary plot crosses y=0 (a cross-shape of points centered on the origin and aligned with the axes).
www.wolframalpha.com/input/?i=plot+z%3D(x%2Biy)%5E2%2B1
+Alan Mullenix how were they discob
vered
It's a plot of the real component of the complex function w=z^2+1
Here, z is the "independent" complex number that used to be x and w is the "dependent" complex number that used to be y.
The reason that this function seems to have many zeroes is that it's not looking at all of w, only the real part. If you looked for points which have zeroes in the real AND imaginary parts, you'd find there's only two of those.
I have a strong feeling this didn't help much..
I'm really interested to know how did he make that 2D graphic on to a 3D.. that was awesome...
Migueldeservantes possibly wolfram alpha
he is actually an alien. aliens have such advanced technology
Migueldeservantes cunts never wanna give away their 'secrets'
Probably after effects
Migueldeservantes math :)
Honestly this is the most important video out there explaining imaginary numbers. This has to be archived in museums for generations to come. Thank you very much for the important work!
Glad you enjoyed it!
My subscription list has just earned a new Channel.
Very nice illustration.
Woohoo!
Here's the thing - add spin with leverage into these equations and you'll find gyrodesics. Temporal torsion in conserved momentum.
I really like lateral. I'm voting for them in 2020. Go team.
1:47, it seems the solution to x^2+1=0 is a curve (or two) instead of 2 point (+i and -i). actually x is in 2 dimensional plane, so is f(x). so it requires 4 dimensional to show the function.
This is the video that ignited great interesst in math in me for maybe the first time. As such it was a major contribution to why i am studying mathematics today. Thank you.
This an 3blue1brown’s video on the inscribed square problem for me (not pursuing math myself, but not because of a lack of interest)
you should change the text of "0!" to just "0" or "0." since 0!=1
Word.
Welch Labs 0!=0, 1!=1, 2!=2, 3!=6; no?
Leonardo Aielo Tassi nope, 0!=1
One way of seeing it is by thinking that the factorial function tells us how we can order stuff, A&B can be ordered {AB} and {BA} 2!=2
{A}gives just one "{A}" (1!=1)
And the empty set { ø } can be ordered in one way {ø} 0!=1
Dalitas D
WOW! A totally unexpected but revelatory and logical answer.
Graham Lyons
x! = x * (x-1)!
If 0! = 0
Then
1! = 1 * 0! = 1 * 0 = 0
Mr. Welch, you have awesome presentation skills. Thanks for the video. :)
Fantastic! Humorous, informative, brings further maths to life. I teach this subject, yet I've never once seen the graphic at 1:50 in any maths textbook.
Me too, so cool.
I'm largely self taught in anything beyond grade 8, so I didn't learn about complex numbers until I got into electrical engineering. Learning complex numbers in the context of a practical application like AC anaylsis helped them make a LOT more sense to me.
It's too bad most students in school don't have that same benefit. I can't count how many times I stuggled to learn a math concept in school only to discount it as arbitrary nonsense because I couldn't understand it from the rote memorization way we were meant to learn. I find it much easier to learn something if I'm able to connect it to something useful or meaningful so I can understand it from an intuitive point of view, rather than just memorizing that this is how it works and taking their word for it.
Negative numbers have practical applications like debt, which is a concept that even pretty young kids are familiar with. Complex numbers and their practical applications are pretty tough to come by at the moment or become common very late in one's education, which I think contributes a lot to their un-intuitiveness. Complex numbers really do make AC analysis a lot easier to understand, much in the same way that negative numbers make debt easier to quantify. Same could be said about polar coordinates
@MikeProductions1000 I thought so, too. That's one of my favourite aspects of electronics is how often you come across topics which seem totally unrelated, but are closely linked in some very non-obvious way. The other part I like is just seeing all of these seemingly arbitrary concepts in math being put to practical use for the first time ever. Which goes for anything in engineering, really.
This stuff Really really helps me learn math. Even just reading this Gauss quote helps understanding something I really struggle with.
1 minute in, can't wait for the next part!
Nicely Done :D
One of the coolest parts of my abstract algebra class was when defining field extensions lead to a complete definition of the complex numbers in terms using field extensions on reals. I think what was so cool about this is that, if you look at how other numbers are proven to exist in the context of axiomatic set theory, numbers like the rationals are also defined as equivalence classes of naturals. This basically means if you accept the existence of rationals, you basically have to accept the existence of complex numbers
"Numbers are lame. Let's invade something!" - LMAO! Subscribed! =D
@5:15 we needed students to know things like negative numbers so they can understand what debt is
Pulling that graph out of paper is Awesome...
Its in my recommendation..
And i can surely say that i am not disappointed..👌👌👍👍
I could happily be studying for this now!
When you "pull" the graph up and make it three dimensional, then yes it crosses the X axis, but it suddenly looks like it crosses it in a lot more places than just 2... and it should only be 2. So I dont think that 3d model was a good representation
yeah, that's what i was thinking
The actual function values would be the outermost edge of the shape, the actual extension of the plotted line, not the interior area. Which would be a _different_ but related function (probably involving calculus). It was filled in only to provide visual context for us viewers.
Wait for the last part. He explains this specific issue.
The exact point is mentioned in the workbook, take a look at it.
Actually, this is a prank video by some jerk, cuz for the eq f(x)=x²+1, we are working with only 2 dimensions. Where the hell did you get the 3rd dimension from ? so for every question, just simply add another dimension if can't solve it?
At 5:20 - Negative numbers are absolutely connected to things in the real world, just look at my checking account.
I’m just joking, my entire checking account is imaginary.
I "learned" imaginary numbers at some point in school, very briefly. But I never truly understood them. You managed to do a better job in 5 minutes of RUclips video than 20+ years of education. Finally I truly get it, and it's not even hard. Completely demystified, like a great cloud has been lifted. You are a legend.
Woohoo!!
Thank you so much! 'Imaginary' numbers were my big stumbling block in A level maths, and my maths teacher was unable to explain them (because he only got the job for being the headmaster's old chum). This video has a made it clear for the first time to me. If only my maths teacher had explained it as another dimension like this, instead of "You don't need to know how it works, just memorise how to use it to pass your exam". I might have passed that A level and become an astrophysicist as I wanted.
No, don't go there. You were a rational human being before when you couldn't understand imaginary numbers. You actually knew intuitively that it was all a load of garbage and just fantasy. Now you've come to accept them as real when they aren't. Go back to the light.
@@tomjscott They're demonstrably real. Physicists have demonstrated that our most fundamental powerful theories of reality only work when using complex numbers. They're as real as any other number system; to assert otherwise is ignorance
This video completely bought me. Guess I'm a sub now.
me too.
Nice vid.
1:54 OH! well that explains that! *goes to look in right dimension*
Hahahaha
Thank you for this series, a really mind bogglingly "complex" topic beautifully and simply explained
I think the best way to describe the imaginary part is to imagine the world of forgotten things, when you found a thing, this means that it changed the state to real.
Cool idea
I'm watching this for fun
You are not alone :-)
Dude I remember watching this video for the first time and understanding absolutely nothing of it. Now that I've had imaginary numbers and that stuff in school, I still don't understand it.
@@Matheus_Braz same
@@Matheus_Braz not gonna lie you got me in the first half
I envy you dude I love math and i think it’s so fascinating but there’s just some parts of it that I do not understand
"why would we need a number for nothing?" LOL a mathematician with a sense of humour...whats that about.
Greek mathematicians murdered others for accepting the existence of fractions or something so
@@luskarian I thought that story was for the square root of 2 being irrational? Fractions were fine.
@@Brawler_1337
Yeah it's kindof like that
@@luskarian It was Pythagoras who sentenced Hippasus to death by drowning for proving square root of 2 irrational
The best way to see is this .....
For ancient people the negatives are imaginary because they were beyond there imagination. And for us imaginary (lateral) numbers are imaginary.....in future, people gonna make videos about us thinking that imaginary are imaginary...
Ancient man had imaginary numbers.
"how many unicorns do you have in your herd?"
"i have -1"
...this comment is underappreciated
You mean 0. And -1 is not an imaginary number
Slayer
That would mean that he is in debt and owes someone a unicorn
@@livethefuture2492 if he had 1i unicorns he would be in debt of one flat unicorn
I LIKE YOU I LIKE THE WAY GRAPHIC TO EXPLAIN ..... WHEN FIRST TIME LOOKING AT THE PICTURES OF YOUR MV AND WHEN I SEE YOU MOVING PICTURES ....... I HAVE SOME IDEAS UNDERSTANDING OF IMAGINARY NUMBERS IMMEDIATELY ................
AWESOME : Watched the whole serie! THIS was the best and most intuitive explanation of number theory and complex number ever, where also math newbies could follow and get a deep understanding! Thank you soo much was this highly entertaining and educative masterpiece! ❤️👍🏻💡 that was a tremendous effort of work and brain you put into it! 😇
I remember find this series years ago, it made me think of complex and imaginary numbers as completely natural and not strange at all, i want to thank you for being such a great teacher!
Thank you!!
@@WelchLabsVideo Hello, please I think this is an amazing video but I would love if you could include the resources where you found all this info so people can do further reading . Thanks
"Numbers are lame. Let's invade something!" Romans 3:50
boi I thought that was a verse from the Bible lol
Absolutely loved that lol
@@mryup6100 That would be the Qu'ran.
@@captainoblivious_yt Research things before saying it in public or otherwise people will say you a dumb uneducated
@@captainoblivious_yt arrr... Swedenistan. Well, Islam developed the mathematic in the Goldena Age.