Imaginary Numbers Are Real [Part 1: Introduction]

[@masterstealth11]

“We’ll be using the term lateral from now on” *continues to say imaginary*

[@WaldoTheWombat]

where are your comments?

[@neh1234]

It's your imagination, dude.

[@donutello_]

@@neh1234it's his lateral* now

[@smallgreen2131]

√anti-apple = banana

[@megalampada2372]

@@smallgreen2131 Wut?

[@aTomallic]

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.

[@hareecionelson5875]

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

[@whatname3676]

Wait, calculators don't respond real numbers?

[@deleted-something]

Rip

[@the_demon149]

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…”

[@john-ic5pz]

​@@the_demon149so sad they didn't digress for a minute. minds are open far before H.S....perhaps more so

[@kswiorek]

In polish it's even worse. Imaginary numbers are "liczby urojone", "uroić" means to hallucinate...

[@pan_czerwony5437]

czyli dokładnie to samo

[@kswiorek]

@@pan_czerwony5437 imaginary to bardziej wymyślone, imagination to wyobraźnia i to nie ma takiego negatywnego znaczenia.

[@pan_czerwony5437]

@@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

[@swinki33]

@@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 :)

[@pan_czerwony5437]

@@swinki33 oh god

[@br1lliantplanets643]

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

[@morgiewthelord8648]

Very cool

[@nvitined]

Well, imaginary numbers in Chinese still has the ‘imaginary’ meaning. It’s called 虛數 I think

[@masterspark9880]

“The Tiananmen Square protests are lateral”

[@user-vv1do1wg1j]

@@masterspark9880 LMAO

[@jasonmaguire7552]

@@masterspark9880 legendary

[@jalfire]

thank you Gauss. It sounds much less awkward to say that I have a lateral girlfriend

[@QED_]

@Jalfire: Me . . . I just keep it to myself and don't mention it to anyone else at all.

[@ShizL]

so original

[@alvarogoenaga3965]

@@QED_ lateral girlfriend= mistress

[@sauceaddict9569]

Lmfao😂😂

[@miguelalvesmacedo]

still awkward though

[@Black_Kakari]

Brah, he pulled a rainbow out of his paper. Drugs

[@legalizearson3346]

damn

[@G4mm4G0bl1n]

LOL!

[@hurm3660]

*Bruh not brah lol

[@jkn8932]

How did he did that?

[@conversationswcocco]

JATIN GANDHI editing

[@billcannon]

You delighted me with the 3D lateral-plane visualization. Well done.

[@arnaldog12]

Me too! How did he do it?

[@neithanm]

After Effects?

[@luizpaulo6535]

+Neithan magic

[@want-diversecontent3887]

William Cannon It may look 3D, but it's 4D. It's explained in part 10-13

[@arquitectronico]

You´re not the only one!!! XDD!!

[@eziola]

Man, this is one of the greatest, mind-opening playlists about the beauty of math that has ever been posted to KZbin. This taught me to see algebra visually. Absolutely incredible. You are so talented. Why did you stop making series like these? They're incredible!

[@SuperPBrady]

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]

Paige Brady Precalc students don’t understand imaginary numbers by that point?

[@xwqkislayer7117]

Ngl this graph is.harder to understand than explaining it normally.

[@UTU49]

@@-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.

[@UTU49]

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.

[@andreeacat7071]

@@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.

[@rinfeast3445]

its all fun and games in math class until the graph starts speaking 3d

[@Email5507]

You will see Fourth Dimension in future, which you will not express or understand in 2d papers like you do 3 dimensional shapes.

[@lindaday884]

be still my heart!

[@vladymartinez1232]

@@Email5507 impossible to understand, impossible to imagine, we can only "speak" about it, i love it!!!!

[@khodis2002]

Imagine classes in vr headsets

[@lukiepoole9254]

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.

[@august_klevberg]

The easiest way to understand negative numbers is by picturing my bank account.. 😔

[@zekzimbappe5311]

And if u don't have any account like me

[@spartanhead8131]

@@zekzimbappe5311 Watch other peoples poor bank accounts.

[@bufdud4]

@@zekzimbappe5311 then that's lateral bank account

[@umniyahirfan5026]

I LOVE YOUUU

[@Nick-ui9dr]

And mine imaginary numbers. ;)

[@DrPG199]

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.

[@NightmareCourtPictures]

the money is just hidden in a different dimension

[@abcdxx1059]

Lol

[@abcdxx1059]

@@NightmareCourtPictures lol😂

[@Fitnessbydrew]

@@NightmareCourtPictures Im dying laughing!!!

[@ishworshrestha3559]

Ok

[@mab3667]

2:50 I know many of you just wanted to see this

[@rlyrosy]

💖💖

[@juvenileygo]

But a nerd like me wants explanation on how he solved the equation. Sadly he got nothing

[@definesigint2823]

@@juvenileygo Note, this is first in a series of 13 videos (all published here).

[@juvenileygo]

@@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

[@definesigint2823]

@@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.

[@blazeknight2009]

The presentation of math has never been so fun and interesting like this one here. Kudos to thee. 10/10

[@WelchLabsVideo]

Thank you!

[@harmitchhabra989]

@@WelchLabsVideo Keep Making *Great* Videos. And Thank You For Such An *Amazing* Explanation.😀

[@patrickjmt]

This series of videos is really amazing, great job and keep it up!

[@ofentsetshepe]

hey patrick ..best maths teacher/professor/tutor on youtube

[@obsidiansiriusblackheart]

Ayyy Patrick shoutout for being the reason I passed first year maths 👌 👌 👌

[@JunieHaloway]

[@krisfix7982]

Shoutout to forpatricks for also the reason why i passed all my classes lol

[@Madmun357]

Patrick, your videos are my Go-To videos.

[@maxwellscheinfield6612]

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!

[@JoseVega-td3iw]

I just did the same an hour ago.

[@nuklearboysymbiote]

euler: -1 > ∞ He predicted integer overflow

[@xwqkislayer7117]

Can you specify what integer overflow is? I'm sorry I dont know lol.

[@nuklearboysymbiote]

@@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]

@@nuklearboysymbiote Thanks I didnt know that lol

[@nuklearboysymbiote]

@@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)

[@xwqkislayer7117]

@@nuklearboysymbiote ah ok ill keep that in mind. Thanks for the info

[@TheEclipsedLock]

What pulled me in was the 3D graph in the thumbnail :p

[@josepablobermudez6283]

same

[@tgw607]

@@josepablobermudez6283 Me too. I wonder if you can make that with a 3d printer or do you need a 4d?

[@Eric-jb1ym]

Production is top notch

[@g00dvibes47]

so is the explanation. concise, accurate, visually easy to understand. trifecta

[@TJShare]

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.

[@samovarmaker9673]

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.

[@abdulhermiz782]

Very true indeed.

[@jimmybelgium]

How do I always see see you? On every geography now video I've seen ur comment and now on math? Holy crap man

[@pebblepod30]

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.

[@pebblepod30]

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): kzbin.info/www/bejne/inWvZZZum7KCes0

[@igamingmp1526]

Finance is math

[@Norman_Lazarevich]

If you ever feel dumb,just remember at somepoint you can do what Leonhard Euler can't.

[@whycity2057]

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.

[@HighestRank]

couldn’t *

[@afreen5058]

*slowly applaudes * I love this comment. Its perfect.

[@machomachinmachinmachinmac6910]

Was he from Houston?.......The Houston Eulers.

[@hubb8049]

@@machomachinmachinmachinmac6910 He was from Basel

[@wawathulu5637]

Therapist: The square root of -1 can't hurt you, it doesn't exhist. The square root of -1:

[@akuljamwal3085]

_[Imaginary Screams]_

[@Tatiana-jt9hd]

*_[Lateral Screams]_*

[@AnhThuNguyen-zz2hm]

*exist

[@mtlicq]

square root of 1 can't hurt, but square root of -1 hurts!

[@jimr8552]

Neither can division by 0 - oh wait is this the year 2020? You haven't gotten to n based multidimensional mathematics yet.

[@YT-pv8fn]

Mistakes: 1. You can stretch that conic section in any direction to prove your point 2. The equation is 2D, making it as a 3D solution is cheating 3. It was Argand who talked about complex numbers. He has 7 page paper on it. He used it to rotate the numbers around a point or an axis, other than subtraction or addition 4. Imaginary numbers are real imiginary, not a bad name 5. You hypothesis does contradicts with phase shift denotation using complex numbers, thus, violates Euler equation ....and much more

[@Kugelschrei]

I never knew I could have that much fun watching a math video, well done.

[@StormCougarTypeZero]

Standupmaths mang

[@TomCatFromMA]

More real world applications would've been nice for us noobs. So, thumbs down.

[@theviniso]

Numberphile has some cool video too

[@DoomRater]

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.

[@iOSMinecraft120]

Mathologer is cool too :)

[@thevirtuousghost9480]

The anti-apple

[@sergiotheg13]

Strikes again.

[@moioyoyo848]

Do u guy like android?

[@stanislaslaurent970]

@@moioyoyo848 Everybody does

[@moistnapkin1599]

Title: Mathematicians: Well yes but actually no

[@Goosnav]

Reddit moment!

[@cmswrD]

@@Goosnav Goosnav

[@139-b7j]

@@Goosnav destruction 100 holy shit you destroyed him dude you're breathtaking wholesome big chungus

[@CarbonRollerCaco]

Naw, negative numbers are the real Schrödinger's numbers.

[@AlineSanchezRamirezBaruc-xm6qb]

Every number is a representation, just like signs

[@papaluskask999]

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

[@jeremybuckets]

"From here on, let's let lateral mean imaginary." Continues to use "imaginary" through the rest of the video.

[@tibschris]

"Imaginary numbers are real" Oh u

[@tibschris]

***** I wonder if you understand humor...

[@jcraig6431]

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"

[@bluedroid1541]

That's the same as saying "There are more than two genders".

[@tibschris]

Don't.

[@reeshav4955]

whats wrong with anime -_- this aint an insult to math, dont get triggered

[@poposterous236]

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.

[@sonalidasgupta3562]

...but...you were RIGHT....they did "make-it-up".....😈 ....God made the Natural numbers; everthing else is "made-up" 😆 ..(misquoting Kronecker)

[@simonO712]

I mean, _all_ of mathematics is "made up". That doesn't make it any less useful though.

[@-Burb]

simonO712 No, all of math is discovered. The symbols we make are made up, but math itself if completely real and all discovered.

[@arya6085]

@BeetleBUMxX you're just calling everything in this comment section cute. Pretty cute ngl :)

[@АлёшаИнкогнитов]

@@-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.

[@newtonlkh]

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!

[@WelchLabsVideo]

Thank you!

[@mathewleonlopez]

Me: *hates math* Also me: *Watches this video because it was recommended*

[@AnkitYadav-zg5zd]

even your name contains math bro!

[@mathewleonlopez]

Bruh so true lmao

[@dearalzghoul4760]

Car

[@user-ly3st7pz7n]

Dear Alzghoul bus

[@wilma7612]

@@dearalzghoul4760 bike

[@JamesR624]

4:39-4:51 Ahh, of course. 2 Apples - 3 Apples = 1 Microsoft.

[@ultimatesans2175]

You forgot to square the two terms on the left.

[@Pushed2InsanityYT]

@@ultimatesans2175 then that would be a google

[@piotrmachowski]

@@ultimatesans2175 Then it would be -5 (2^2-3^2=4-9)...

[@ninjaslash52_98]

thank you now it makes sense

[@jcespinoza]

3:41 I'm more concerned with what happened to Australia in this map

[@jess6685]

OMG

[@zaurxzx]

how about Antarctica

[@flowerwithamachinegun2692]

It became lateral

[@ИмяФамилия-й8н6ы]

And Greenland

[@clorx1]

makes sense since australia was invented in the late 20th century.

[@Dejawolfs]

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.

[@naviddavanikabir]

for those who just started watching this, make absolutely sure you watch all the way to part 13. prepare to be blooooowwnnn. awesome series.

[@WelchLabsVideo]

haha, thanks!

[@tigriscallidus4477]

13 parts?... Who has time for that....

[@surin925]

Where's the next video though?

[@thorstambaugh1520]

Don't spoil the ending where the dragons die!

[@maurer7279]

blo...blown? O.o uh no thanks

[@MartinMadsen92]

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.)

[@dalitas]

you should change the text of "0!" to just "0" or "0." since 0!=1

[@WelchLabsVideo]

Word.

[@leonardoaielotassi1114]

Welch Labs 0!=0, 1!=1, 2!=2, 3!=6; no?

[@dalitas]

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

[@grahamlyons8522]

Dalitas D WOW! A totally unexpected but revelatory and logical answer.

[@want-diversecontent3887]

Graham Lyons x! = x * (x-1)! If 0! = 0 Then 1! = 1 * 0! = 1 * 0 = 0

[@datsmeyall]

Fifteen years ago, little me would have been laughing to the thought of her grown self watching math-videos deep into the night

[@priyavartrajain2166]

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!

[@GroovingPict]

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

[@chocolatethunderific]

yeah, that's what i was thinking

[@xheralt]

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.

[@johnstotz3800]

Wait for the last part. He explains this specific issue.

[@kuladeeparun]

The exact point is mentioned in the workbook, take a look at it.

[@nikhilprasad3947]

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?

[@jonkrieger5271]

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!

[@theviniso]

A picture is worth a thousand words

[@frother]

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?

[@EthanGarcia-n1ov7zi]

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.

[@qorilla]

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).

[@frother]

Thanks, I never expected to get such a clear and helpful answer from the youtube comments!

[@somerandompianoguy3279]

Welch Labs:- "Imaginary numbers are real." His math teacher:- "So you have chosen..... *DEATH* ."

[@jllebrun1]

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 :)

[@golinelli1]

jllebrun1 same feeling but i'm 56!

[@semphony100]

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.

[@pebblepod30]

Hossam Zayed Yeah I feel the same way Hossam, I have wasted parts of my life.

[@TtttTt-ub5xb]

Hossam Zayed حلو

[@QsHsNation1]

It's such a good feeling isin't it :)

[@_Killkor]

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._ "

[@gdash6925]

Every number has already an own name. So your theoretical statement makes no sense.

[@fractal5764]

25 Century: numbers get to choose their gender.

[@_Killkor]

@@gdash6925 No, my 3 is called Richard. Your 3 is called, I believe, Timothy. Your statement is so numberist.

[@gdash6925]

@@_Killkor my 69 is called..... wait

[@want-diversecontent3887]

Microsoft Hites 26 century: Numbers become humans.

[@dominikscherer1882]

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 :/

[@IsomerSoma]

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.

[@the_linguist_ll]

This an 3blue1brown’s video on the inscribed square problem for me (not pursuing math myself, but not because of a lack of interest)

[@filipmarkovic2121]

This is by far the best-presented video on mathematics I saw in my life. 10/10 for your pedagogical skills.

[@Migueldeservantes]

I'm really interested to know how did he make that 2D graphic on to a 3D.. that was awesome...

[@faizanm1563]

Migueldeservantes possibly wolfram alpha

[@優さん-n7m]

he is actually an alien. aliens have such advanced technology

[@user-zu1ix3yq2w]

Migueldeservantes cunts never wanna give away their 'secrets'

[@ns4k_tv]

Probably after effects

[@KittyGata314]

Migueldeservantes math :)

[@NeilSalemMAUSA]

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.

[@itsiz9738]

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.

[@frasafrase]

What is that plot at 2:00? Because it has way more roots than it should.

[@WelchLabsVideo]

+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!

[@thatsmath_yt]

+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.

[@tibschris]

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

[@zinebzinouba2469]

+Alan Mullenix how were they discob vered

[@antoniolewis1016]

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..

[@Epoch11]

No one ever seems to answer the question..."Why do we need imaginary/complex number?" Why can't we simply use Trigonometry instead of what seems yo me to be much more complex and less intuitive? If you could either answer this or post a video about "WHY" these numbers work so much better, I would truly appreciate it.

[@WelchLabsVideo]

+Mark G Great question! This should be become more clear towards the end of the series. Specifically, I'm planning an "applications" video - that will hopefully shed some light on this great question.

[@schwartz478]

+Mark G Its needed to understand areas in physics. In fact, if you want to understand how the world works, you need a good philosophical/mathematical foundation to comprehend the logic of the observations, laws, and models. You need to master math as language and as a philosophy to understand physics. Imaginary numbers are used to model complex waves such as light, electrons, and quantum physics. No one uses physics, is wrong. Understanding things like this brings lots of confidence and insight on a problem than someone who didn't cared.

[@Amaglionil]

+Mark G In physics you actually don't really need them. It is just so much more practical than "not using them" that everybody does it. But in the end every physical quantity is a real one.

[@paulestrada961]

Well Mark just think about when you use triginometry. We use trig alot for real numbers and real valued inputs. Buy why should we limit ourselves to just applying triginometry to the real numbers? This video juat explaines how we can create our perspective with a set of numbers that we can't see or represent physically at this moment in our modern time. Maybe one day in the future somewill will show how the sqruare root of negative one can be expressed in physical terms. But for now we have no idea what that type of number represents physically unless we introduce this new concecpt of complex numbers. In our world and physical reality, When the imaginary numbers and real numbers are placed down on the complex plane, some arreas in physics such as electricity, magnetism. If you already know how to use trig with the real numbers, then it should follow that you can apply trig to a plane with two sets of numbers. So working with a complex plane rather than a real valued plane should feel no different other the symbols used such as "i"and a new definion of what a complex number is. Where we are halted with where the real numbers can take us, we can trascend our observations, and calculations with new results from this new set of complex numbers where they arise when dealing with physical applications. We need these numbers to deal with the consequences of how the real numbers have been defined. The areas where real numbers no longer become real to us are where the imaginary numbers are staring us right in the face. We just created these extra definitions so we could apply our previous math techniques and syltes to this new set of numbers that we practically igonored because we originally did not think tha imaginary numbers were numbers like the real numbers. Without getting involved with anything that has to do with our real world and applications; then do you need these numbers, no, not even really the real numbers, but just keep in mind that these complex numbers have as much existence as our real valued numbers. Accepting them to be neccesary in this reality as we do for our supposed "Real Numbers" should be with the same logic behind why we use real valued numbers. Are there numbers out there that we cannot physcally see that represent a model for our some systems in our physical world; yes, they are the complex numbers.

[@Mark16v15]

+Mark G You don't realize it, but your question is sort of like asking why we need numbers to begin with. Before men used symbols such as "5", they would tie five knots in a string to help them remember for instance how many cattle they had. If they owed a man three cows, they could give those to him and undo three of the knots to keep track. If they got themselves into a real bind and owed another man three cows, they could undo the remaining two knots, but then had a problem how to keep track of things after that. Eventually to overcome such problems and make life simpler on everyone, someone came up with symbols to represent numbers, and even numbers like 0 and -1 to aid in solving less straightforward problems. Complex numbers came about when people were scratching their heads trying to solve equations like x^2 + 1 = 0. At first they were just abstract thoughts without much use. However, as the sciences increased, they became useful, again, making life simpler and solving less straightforward problems. I don't remember exactly what the engineering problem in college was (I think it involved an electrical circuit), but the professor solved the problem in two ways, with and without using complex numbers. The first way involved advanced calculus. It was a laborious process, but he finally got to the answer. The second way utilized complex numbers. With some quick addition and multiplication, he arrived at the exact same answer. Everyone was sold on the use of those numbers. Just like people no longer tie and untie knots in strings because writing numbers like 5 -3 = 2 is so much faster, people no longer rely solely upon the real numbers in solving problems because often the use of complex numbers is so much faster.

[@j.a.l.m.9464]

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!

[@diamondblackford5048]

The word "lateral" explains everything! I hadn't been understanding "imaginary" numbers for years! You've discovered this secret for me. Thank you!

[@bertrand3055]

i²=-1 démystifié : kzbin.info/www/bejne/aHjahIh6osSFnZom43s

[@MrPuff1026]

1:54 OH! well that explains that! *goes to look in right dimension*

[@WelchLabsVideo]

Hahahaha

[@weili9349]

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.

[@krishnavekariya9534]

This video's title also had two part Real:What is imaginary number Imaginary:this series gone legendary

[@nehalkalita1]

Your efforts for making a whole playlist on imaginary numbers is worth of appreciation. Very few can explain in details.

[@pewpewdragon4483]

@5:15 we needed students to know things like negative numbers so they can understand what debt is

[@jimhalpert9898]

My therapist "Imaginary numbers aren't real they can't hurt you" Imaginary numbers:

[@ttxxxxxxxxxxxxxxt]

they aren't real but can still hurt you

[@kentGrey]

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!

[@ElVerdaderoAbejorro]

"Numbers are lame. Let's invade something!" - LMAO! Subscribed! =D

[@mwm48]

At 5:20 - Negative numbers are absolutely connected to things in the real world, just look at my checking account.

[@mwm48]

I’m just joking, my entire checking account is imaginary.

[@takyc7883]

This comparison to negative numbers is actually so good

[@speedsystem4582]

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.

[@the_eternal_student]

the truth of the statement "imaginary numbers are real." makes the name "imaginary" quite profound.

[@IIRomoII]

I'm watching this for fun

[@dragoncurveenthusiast]

You are not alone :-)

[@Matheus_Braz]

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.

[@slashlightning3261]

@@Matheus_Braz same

[@stanislaslaurent970]

@@Matheus_Braz not gonna lie you got me in the first half

[@christianjoseph6502]

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

[@VoidHalo]

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.

[@twentylush]

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

[@VoidHalo]

@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.

[@adilisimon6184]

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!

[@WelchLabsVideo]

Glad you enjoyed it!

[@planetmchanic6299]

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.

[@dhruvshinde6089]

Why the hell KZbin is recommending such an informative video after 4 whole good years?😯

[@MrJanes-cl5sj]

"why would we need a number for nothing?" LOL a mathematician with a sense of humour...whats that about.

[@luskarian4055]

Greek mathematicians murdered others for accepting the existence of fractions or something so

[@Brawler_1337]

@@luskarian4055 I thought that story was for the square root of 2 being irrational? Fractions were fine.

[@motazfawzi2504]

@@Brawler_1337 Yeah it's kindof like that

[@akshatmodi8607]

@@luskarian4055 It was Pythagoras who sentenced Hippasus to death by drowning for proving square root of 2 irrational

[@googleyoutubechannel8554]

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.

[@ernestboston7707]

Ancient man had imaginary numbers. "how many unicorns do you have in your herd?" "i have -1"

[@sonalidasgupta3562]

...this comment is underappreciated

[@valthiriansunstrider2540]

You mean 0. And -1 is not an imaginary number

[@sumeetnagpal6247]

Slayer

[@livethefuture2492]

That would mean that he is in debt and owes someone a unicorn

[@jorgepeterbarton]

@@livethefuture2492 if he had 1i unicorns he would be in debt of one flat unicorn

[@quantumsmith371]

never stop doing these videos they are the best out there. thank you so much for taking the time to share them. with us.

[@PauloSilva-zr8vk]

"Numbers are lame. Let's invade something!" Romans 3:50

[@mryup6100]

boi I thought that was a verse from the Bible lol

[@andreaemanuele7293]

Absolutely loved that lol

[@captainoblivious_yt]

@@mryup6100 That would be the Qu'ran.

[@overlordprincekhan]

@@captainoblivious_yt Research things before saying it in public or otherwise people will say you a dumb uneducated

[@Acalmujannahmalaysia]

@@captainoblivious_yt arrr... Swedenistan. Well, Islam developed the mathematic in the Goldena Age.

[@vanshsingh7270]

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...

[@JabrHawr]

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.

[@masterhidsert1043]

I'm an engineering student so all of this is extremely interesting to me!! Instant sub! Phenomenal work.

[@TheoWerewolf]

EXCEPT.... when you describe the real world with math, the math has to translate into something in the real world. Negative speed, for example as a scalar really doesn't mean anything.. but as a vector (speed and direction), it does. In the same way, while imaginary numbers are 'real' in the sense that they are an internally consistent system that can be used to do useful work, when it comes to real (as in physical) applications, they don't map to anything. One of the most famous cases is going faster than the speed of light, which involves imaginary numbers. The scaling factor is (1 - v^2/c^) ^ 1/2 or the square root of one minus the velocity squared over the speed of light squared. Time is affected by the product of this (viewer's time = local time multiplied by that equation where v is the relative speed). As you approach the speed of light, your local time looks to the viewer as if it's completely stopped, but if you could go just a little faster... you'd be going at a speed that now has an imaginary term. The problem is, there's no physical element that that imaginary term gets mapped to. There is only one temporal dimension. And you can't map it to one of the other three spatial dimensions because they too all now have an imaginary component... so you'd need four more dimensions - four 'real' and four 'imaginary'... and they don't exist. So, we say 'you can't go faster than light because the math shows that you can't because your time and space becomes 'imaginary''.

[@THaWoM]

"when it comes to real (as in physical) applications, they don't map to anything." That's not true; complex numbers were independently invented/discovered by a surveyor named Caspar Wessel who was using them for surveying. They're just 2D vectors with rotation and scaling built into the multiplication operation.

[@MichaelPohoreski]

The Werewolf Nonsense. In AC power imaginary numbers map to phase. In unit quaternions imaginary numbers map to a 720° system that can be interpreted as +1/2 spin and -1/2 spin used by all elementary particles. Scalar values don't convey enough information. i.e. Scalar speed vs Vector velocity. The imaginary unit _i_ can be visualized as a 90° rotation -- that works in ANY dimension. It is up to the context to determine if this has any physical mapping we can relate to.

[@sunkruhmhalaci2592]

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.

[@GraveUypo]

you were really bad at sticking to your guns. lateral numbers lasted like 30 seconds

[@ent8411]

Where my doom fans at?

[@JoseAngelMorente]

Dude, no maths in the Medieval Europe? What about Fibonacci and Oresme?

[@billkillernic]

Fibonacci was counting breeding rabbits

[@brianjohns4912]

Or possibly he was breeding counting rabbits... ;-)

[@NourMuhammad]

You need to read more about the history of the civilizations in this era and see where was the math around the world around that time and before!!!

[@billkillernic]

Muhammad Nour Elmogy and you need to learn better english before you comment on something... the guy asks about european mathematicians in the middle ages...

[@NourMuhammad]

Bill Killernic What's wrong with my English !!!!, My Answer was more generic for his exclamation, have you read my comment properly?! or you just commented without even bother reading it!! have you even read his comment!! you need to have more wide open sight for other's comments, I am sure you misunderstood my comment or his or both! and BTW I was confirming the phrase and emphasizing it, medieval Europe didn't have much until it's late years and after that a lot have changed, read the history for your own good and you will know (if you were searching and reading in the right places) how far was Europe in this era and before from anything related to something called science!

[@docblue2776]

Before the invention of negative numbers, did people have... no debt? 1:42 So well done

[@docblue2776]

Aquaified You can think whatever makes you feel better about yourself.

[@fullmetalflix5195]

Hanlu Cao i think your funny but man you got alot of haters

[@IAmNumber4000]

“Euler once wrote that negatives are greater than infinity” Integer overflow moment

[@SkullCrossGaming]

4:13 voice cracked a portal to an "imaginary " dimension

[@jstdoit5218]

I laughed for way too long at this

[@factsheet4930]

Welp... negative numbers appear in a lot of formulas in physics, that DO describe the real world :P the earliest you would see negatives is with charges, we use the negative sign to describe the opposite charge to what we call the positive charge c: same with complex numbers in quantum mechanics!

[@Levfomin]

I could happily be studying for this now!

[@hersirirminsul]

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.

[@tomjscott]

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.

[@epicmarschmallow5049]

​@@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

[@Tomyb15]

This video completely bought me. Guess I'm a sub now.

[@exequielvarela4960]

me too. Nice vid.

[@nouai]

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.

[@who7644]

Cool idea

[@Apreche]

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 KZbin 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.

[@WelchLabsVideo]

Woohoo!!

[@AlexandraMiah]

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

[@finnam841]

Remember guys : They live in a Whole new dimension 1:27

[@NourMuhammad]

My subscription list has just earned a new Channel. Very nice illustration.

[@WelchLabsVideo]

Woohoo!

[@alialtica6791]

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.

[@anieldayyanelday1771]

Funny, I've always known 'i' and all other such numbers as 'complex numbers', not imaginary. But I think 'lateral' sounds better. Also, I wonder if 1/x will ever have a non-limit solution where x=0

[@nathangrant1824]

1/x where x=0 is undefined and not permitted in all cases, i think.

[@jonathanschossig1276]

Daniel Olowu complex number z: a+bi. Imaginary: n*sqrt (-1)

[@joeirish1000]

Absolutely amazing. Great presentation, the best I've ever seen for a video about imaginary numbers and one of the best ever math videos.