Imaginary Numbers Are Real [Part 2: A Little History]

Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.c....
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Part 2 especially owes a debt to Paul Nahin's excellent book: An Imaginary Tale: The Story of sqrt(-1). Nahin presents a very thorough account of the development of imaginary numbers, which was invaluable in creating this series.

[@amohogride3739]

Imagine a mathematician running around the street carrying a sheet of questions finding an opponent. This is basically Pokemon

[@thegecko1992]

SQUIREROOTLE, I CHOOSE YOU

[@sawada_.]

@@thegecko1992 SquareRootle*

[@stonetop]

It uses derivative on e^x, it wasn't very effective

[@megugu2155]

Millenium Problems gon be like Arceus

[@savagenovelist2983]

No, it would be more like a Pokemon Gym. You go to a designated Math Dueling place and somebody gets you into the next match for your skill level.

[@Technomancr]

Why did you abandon calling them lateral numbers?

[@WelchLabsVideo]

+Anthony Trupiano Yeah - good point. It's just a question of how much I should break with convention here - lateral makes way more sense, but most people don't know what you're talking about when you say "lateral number".

[@KManAbout]

+Welch Labs I thought they are called complex numbers

[@KManAbout]

***** but a+b = i is impossible because no real numbers exist that are equal to i. if you mean that complex numbers included any combination of real numbers with i then I understand. for example if you mean 6+ 2i is a complex number but not 6 or 6i.

[@KManAbout]

okay cool

[@MamboBean343]

Soldier ˙ Wait! The complex numbers include real numbers like √2 and imaginary numbers like i and 6i. However, like many groups of numbers, the term "complex number" may only refer to the non-real, non-strictly imaginary numbers (a+bi, where a and b are real)

[@xxnotmuchxx]

Mathematicians should continue to have duels. That sounds pretty cool.

[@ssunvenomcast4975]

Yea!!

[@rich1051414]

It actually caused a lot of groundbreaking mathematics to be lost to time, due to greed. It basically caused proprietary formulas so the mathematicians could keep winning duels.

[@SlaveToMyStomach]

Kind of like how patents are used today. Not winning duels but ... profit!

[@gabrob27]

Please consider revising your pronunciation of Italian surnames - video is nice but Tartaglia and Del Ferro are suffering in their deathbeds.. :)

[@SiddharthBhatt24]

Ever heard of MIT Integration Bee? It's basically a math duel tournament.

[@HDitzzDH]

3:35 That is not quite correct, the current definition of the square root only accounts for the positive number whose square is the number inside the square root, the only time you inlude +/- is when you're solving an equation. The square root itself is only defined for one positive number.

[@pradyunmore6727]

@ GZA yes you are correct. I don't know why many people in the comments haven't pointed this out!

[@귀여운다람쥐-i8c]

I didnt know some ppl think root9=-3

[@theawezome6699]

This is only for notation purposes, ie it is the pricipal root.

[@Даниил-ц4э5о]

bruh you've said it a month before me so sad:)

[@bdbd7045]

I was looking for this comment!!

[@ryannelson768]

this is a KZbin gold mine

[@anonymoustraveller2254]

Ryan Nelson yeah

[@anarki777]

Holy shit. Math duels? Are you serious? That's pretty hilarious.

[@Nothing_serious]

anarki777 It's still a thing. There's even math Olympic. I think one of Numberphile's host is a champion.

[@steelep5623]

anarki777 Expecto Numerum!

[@msDanielp369]

yeah they were actually math freestyle battles they were pretty dope. only XVIth century kids will remember 😫👌🏻🔥

[@PsyQoBoy]

The Romans did maths duels and the Egyptians played duel monsters you do the maths!

[@timmy18135]

Engard

[@rabeebibrat1805]

A HUGE thumbs up for the historical context, which math texts don't provide. Interesting series! Keep up the good work.

[@sajjadulhaq8914]

Words are just not enough to explain how awesome these series are. Only true mathematician can understand that how difficult it is to prepare such lectures. I have been searching for such kind of study from many years. I am truly great-full for this series.

[@humbledb4jesus]

3920506-13232-39850-23...422-4670-74!!!!! translation: only true mathematicians speak in numbers....

[@trulyUnAssuming]

sqr(9) = 3 and sqr(9) != -3. At least in the usual definition. Since you define the squareroot as the inverse function of f: [0, infinity] -> [0, infinity]; x -> x². But the solution to x²=9 is x=+/-sqr(9)=+/-3

[@災厄-b9o]

Glad someone pointed it out.

[@danielfloresretamal2471]

the square root of a positive number (or 0) is unique and always positive (or 0)

[@dabzdavid2378]

@@danielfloresretamal2471 unique and nonnegative

[@danielfloresretamal2471]

@@dabzdavid2378 that summarizes it

[@Trucmuch]

You guys get that by your own definition i is not the square root of -1.

[@wiredelectrosphere]

This math duel thing is presented like you can make an anime out of it

[@maxif4950]

You watched dr stone?

[@cristaldark4228]

Yesssss! !!

[@yousiffaris696]

I have been searching for squaring numbers animation or explanation on x,y and I still haven’t found yet all of what I find is 3 squared = 9 and bla bla bla😤

[@yareyaredonut]

Actually that's just what we do in exams

[@kinokonyan]

mmm yes

[@Maceta444]

Nerds before: "I just murdered someone in a math duel and I'm on the 30th page solving this single equation" Nerds now: "Why this python code not work"

[@julesverne6287]

How long did it take to cut out all the continents of the world?

[@WelchLabsVideo]

+Lorcan O'Brien Good question - that didn't actually take too long. Shooting with camera motion, however, takes FOREVER.

[@LiquidModernityTastesLikeUrine]

Nice profile picture

[@beta5770]

This is gold. Never have I seen better VFX used in presentations for such a basic topic.

[@TaigiTWeseFormosanDiplomat]

So that is how...Vertassium got the term of math duel?

[@takeuchi5760]

Yeah, looks like he copied these videos.

[@ExperimentarEnCasa]

beautiful videos! LOVE THEM

[@MilloSteve]

Tu canal es una mierda ahora

[@daver72016]

This is just long winded CRAP, because there ISN'T one definitive correct answer to this !

[@daesmua]

@@MilloSteve Jajjaja si

[@alesdiaz1177]

@@MilloSteve Con lo mitico que era el canal en su momento

[@manjunathnavalgund7266]

wow....this series is addictive....I am binge revising my school algebra

[@mistyseas]

Manjunath Navalgund I hate math and I like this for some reason

[@robertwilsoniii2048]

This playlist is awesome!! I love learning the history of the math while learning the math! This is how it should be taught in general.

[@samuelkoesnadi]

This is very very useful. I like it a lot. Thank you for your hardwork on this video!

[@WelchLabsVideo]

Thanks for watching!

[@WoLF42x]

The square root of 9 is strictly 3. When approaching x^2 = 9, only when you square root do you get the plus-minus in front of the positive root to indicate +-3

[@pipolwes000]

+WoLF42 No. Both 3 and -3 are square roots of 9, since 3*3 = 9 and (-3)*(-3) = (-1)*(-1)*(3)*(3) = 3*3 = 9. Depending on what problem you are solving, different solutions will make sense but both positive and negative values are valid square roots when you have no other constraints.

[@sagarprasad1436]

pipolwes000 this is the general mistake we all make..i know -3 ×-3 is 9 but when taking square root only positive values appers...if u try to use the -ve values then there are ways to prove 1=2 which is not true...i dont know how much maths u have studied and i will also not tell u to believe me but try to gather information by yourself and i assure u 500% u will believe what i am trying to tell u

[@sagarprasad1436]

WoLF42 glad to find someone identified the mistake👍

[@kilianmelcher3811]

Yesss, you are right!

[@junhaowang1834]

pipolwes000 Principle square root is a function, you can't get two answers!

[@Syz_gy]

You make math interesting for a reason so so many math curriculums fail to see. You include the context. It's one thing having formulas thrown at me, and the rules for solving them, and being told to memorize. It's another thing entirely to find out why those formulas were made to begin with.

[@WelchLabsVideo]

Amen!!

[@Syz_gy]

@WelchLabsVideo I want you to know I found you through your atom bomb videos. I'd also like you to know the quality of your videos holds up 6 years later. You should be very very proud of yourself mister professor dad.

[@ojkwame]

@@WelchLabsVideoPlease is your info from history of mathematics by Dan Burton ? If not then can you share your resource(s)?

[@Algebrodadio]

Bonus for deriving the quadratic formula in like 2 seconds.

[@alexanderherbertkurz]

I love the way this mixes history and math. And handwriting and computer animation. Absolutely brilliant. Should be one of my favourite math videos now. Added: It is actually almost addictive ... I am going to watch the rest of the videos even though I know the stuff.

[@tankmaster9212]

LOVE THE FACT THAT YOU BRING MATH'S TO LIVE AND VISUALIZABLE

[@Qebton]

this video is so underrated man its so freaking amazing i'm going to show it to everyone i know

[@TonyAKA30]

Math has never been this interesting!

[@EduardoHerrera-fr6bd]

But it's interesting to answer your comment.

[@zlatan4467]

@@EduardoHerrera-fr6bd it's interesting too reply yours too.

[@DarkGourmand]

@@zlatan4467 yours too

[@saadparekh]

@3:35 only criticism of this excellent series of videos. Square root of 9 is just +3.

[@clips9294]

Just watched Veritasium

[@tdubmorris5757]

This is pretty much the older shorter and better version of the veratassium video 6 years later

[@toastyarmor6858]

3:19 the 2nd line is -(15^2)/27 the 3rd line should be -225/27 (or simplified), which is not -121 what gives

[@rabeebibrat1805]

That's a good point

[@David-mm6nx]

He wrote it wrong. It is supposed to be 15^3 as in the 1st line, c^3, which yields 15^3/27 AKA 125

[@strangelongcat7830]

I'm literally in love with this channel

[@simplecastic]

you make algebra look fun and love the history i never knew

[@angelabakloyvovtchaikovsky1609]

Math is useless in real life

[@yashwantmente9293]

That's how maths is supposed to be taught, combining it with history. Most of the times we don't even know why certain things came into existence, and everything seems like fiction deduced just for students to write exams. Everything makes perfect sense now.

[@ASLUHLUHC3]

Same with science

[@geekonomist]

Math is supposed to integrate arithmetic, algebra, calculus, history, and LOGIC. Show me any other dimension beyond the 3 we know of; use a measuring tape while doing so. There are no results from the sqrt of a negative. Once you integrate this, you can add to the math history lesson that men have often fallen prey to ridiculous contradictions as a matter of popular ideas. IE God, the Minimum Wage, Quantitative Easing, Altruism.... and the square root of -1

[@tifforo1]

x^3 = 15x + 4 has three real-valued solutions, the simplest of which is 4. 4^3 = 64. 15*4 + 4 = 64. Apparently, you need imaginary numbers to solve it, even though the solution is real. Interesting.

[@antonxuiz]

tifforo1 I was trying to approach to the solution with Bolzano's theorem and found it out too XD Also, what about using Ruffini 1 0 -15 -4 0 4 16 4 ---------------- 1 4 1 0 x^2+4x+1 x= (-4+-(16-4)^1/2)/2 = 2+-[2•(3)^1/2] = 2• [1+-(3)^1/2] There's the other two Maybe it's just that their formulas were useless xD

[@niteshsingh6375]

I just watched this video and got amazed with the concept, everyone who has studied complex numbers has thought about its practical use but got no answers here I got something new.

[@Fo2shstein]

Just wanted to pop in & compliment the effort in these videos, love the math/history/stickman commentary combo. Hope to see more!

[@randyekrer431]

It really helps to think of negative numbers and imaginary numbers as not numbers, but operations/operators. They don't exist is real life; they help as intermediaries in problem solving.

[@1959Berre]

That is true for all numbers, they do not exist in real life. Any number is but a representation of something. You can say: give me one apple, but you cannot say give me a one.

[@jacopoparenti9939]

Mario says: Zero is just a placeholder Luigi says: Numbers are lame, let's invade something

[@htoodoh5770]

Lol

[@neverrip6809]

Zero is the capital letter in every equation ie, 0+6/2(0+3) = 0+1. What y'all are doing here is not maths... it's not an equation... it's a function... it's code... Binary code... FX function is not about working out the area of part of a circle. FX function is about drawing circles and curves using pixels! Remember FX Graphics in 1990?? In binary code, we don't use a capital 0 to start equations and we do not recognise brackets at all... 6/2(3) = 6/2*3 = 9. Maths is all but lost, binary code is the new religion. A religion where the devil has tricked u into believing there are infinite universes, black holes, wormholes, faster than light travel, an abundance of life in our galaxy, at the same time convincing u that he doesn't exist, and that even in infinite universes, there is not a god in any of them o.0

[@stephenzhao5809]

3:11 Let's take the innocent-looking equation x^3 = 15x + 4. When we plug into Cardan's formula, we get a result that involves the square root of negative numbers. The square root of a negative number created enough of a problem to stop Cardan in his tracks. Square roots ask us to find a number, that multiplied by itself, yield the number inside the root sign. The square root of 9 is 3 because 3 times 3 is 9. Importantly, the square root of 9 is also negative 3 because negative 3 times negative 3 also positive 9. 3:38 ... 4:10 However, in this case, we know there's at least one solution because of the way cubics are shaped. Regardless of their coefficients, cubic functions will always cross the x-axis at least once, meaning that our equation x^3 = 15x + 4 will have at least one real solution. So what we have here is a problem that must have an answer, and the formula that has been proven to work. But when we put these together and try to solve the problem with our formula, we quickly arrive at what appears to be the impossible; the square roots of negative numbers. Sometimes when things break in math and science, it means jsut that; they're broken. But there're other more interesting situations in which broken mathematics gives us the keays to unlock new insights. 4:47 The way in which Cardan's formula was broken turned out to beincredibly important to mathematics and science, and that's what we'll discuss next time.

[@dactylntrochee]

Whoa! This is a shocker to me, right from the start. I was no great student in 1965, but numbers always came easily to me, and I remember a happy 98% on our (New York) statewide final exam in algebra. My takeaway from that time is that quadratics were solved on inspection, and that there was no mechanism to find the roots. The concept of a universal formula, as I recall, was actively denied. I wonder if this is senility! What could possibly have made me think that? Well, it's a delight to see it now, all these years later.

[@waselu7578]

3:35 No, the square root is a stricly positive function, it is defined as the unique real positive number which, when multiplied by himself, gives X It can be confusing because, when switching a square to the other side of the function, we don't end up with only the square root but also its negative counterpart, so for x² = 9 it would give 3 and -3 But the square root of a number is only positive

[@chielvooijs2689]

I believe you're thinking of the _principal_ square root. Every number has two square roots, but usually when we say "the square root of a number", we mean "the _principal_ square root of a number", which is strictly positive.

[@waselu7578]

@@chielvooijs2689 That is right indeed, the principal square root is denoted with the radical sign and is always a positive number, the other square root of a number is a negative radix The problem is that in the video he wrote only the principal one and said it was also equal to a negative value, which is false

[@rassilontdavros3004]

3:18 I’m confused, think I’m missing something... how does (15^2/27) equal 125?

[@jacekborecki9171]

I was also confused, but there is small mistake. Should be 15^3/27.

[@jackd6269]

thank you this series! im an electrical engineering student and use "imaginary" numbers all the time, but i never understood where they came from or why they're called imaginary if they are part of solutions to real problems

[@pulkitsingla7421]

Umm I've been taught that square root of 9 is just 3.. Calling it a negative 4 is actually a wrong concept.. negative square root of 9 gives us negative 3..

[@insertname956]

sqrt(9) = 3 or (-3), is how I learned it though

[@juyeongnam8600]

고마워! 우리 교수님이 수업을 너무 못하셔서 수학 포기할까 했는데, 당신 너무 잘 가르치는거같아. 한국어 번역까지 있어서 너무 행복하다. 고맙다. 당신의 채널 구독하고 좋아요 눌렀다.

[@arekolek]

In Italian "the letter pair gl, if followed by an i or an e, represents a sound similar to ll in million.", so basically, the g in Tartaglia is silent.

[@ZombiesSlaier]

arekolek nnnnnnnope.

[@1violalass]

@@ZombiesSlaier He's right. It's pronounced Tartalya.

[@ZombiesSlaier]

@@1violalass helllll no, it's something like Tartaja, if you know a little bit of Spanish, Italian "GL" is like Spanish "LL"

[@gruggi02]

I am italian and nope, you're ABSOLUTELY wrong. Its sound is distinguished in "hard" or "sweet", depending on the vowel at the end. "Glabro" sounds "hard, "consiglio" sounds sweet. To hear how it sounds, please check any glottology vocabulary. In NO case, the sound of G is silent.

[@kevingarner7231]

Couldn't stop with the first video. Can't stop with the second. Gotta see the third!

[@geminithetwins]

I wish I had a math teacher like this at school :(

[@cmuller1441]

at 0:44 you don't need the determinent if you know that the roots are integers (if you don't the method just fails). For a=1, just list all the numbers that have a product equal to c=2 (1×2 2×1 -1×-2 -2×-1) then select the pair with a sum equal to b=3 (N1=1 N2=2) You get x2+bx+c=(x+N1)(x+N2) so the roots are -N1 and -N2...

[@SoWe1]

I thought the root of a number, your example square root of 9, was defined as the _positive_ solution to the equation "x^2 = 9"? While -3 is also a solution of that it is not the square root of 9? That's how I learned it.

[@r3g1t]

+SoWeMeetAgain This is correct. A negative number can never be a square root by definition.

[@PaveDearce]

+SoWeMeetAgain I also agree. -3 is not a square root of 9, though it is a solution to the equation x^2 = 9.

[@TheRedclaw101]

+SoWeMeetAgain *"In mathematics, a square root of a number a is a number y such that y^2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a.[1] For example, 4 and −4 are square roots of 16 because 4^2 = (−4)^2 = 16."* - Wikipedia I think you're talking about principal square roots.

[@SoWe1]

ฟ้าสรร ฮอว์ส yeah, that seems to be what the english wikipedia says. German wikipedia says it's defined as the positive number. See de.wikipedia.org/wiki/Wurzel_%28Mathematik%29 the formula in the subsection "Wurzel aus negativen Zahlen"

[@hectorjoseberrios503]

From Algebra 1: For any real number n, the sqrt of n^2 = |n|. Proof available upon request.

[@nabeelakhter5799]

One of the best video I have seen on KZbin,It's awesome .Thanks WelchLabs

[@WelchLabsVideo]

Thank you!

[@serdarcangul8391]

3:22 2nd and 3th lines 15*15/27=125 wtf????? it equals something like 8,33

[@michaelosborne7554]

It's supposed to be 15^3, not 15^2, and 15^3/27 does equal 125.

[@TheStrGuitar]

Veritasium actually ripped off your script, nice finding the source!

[@rifatkhan192]

how can you let 3uv+c = 0??

[@calculus3d130]

Good video. Minor critique: when you use the radical symbol for the square root of a number, it always stands for the positive root, never the negative.

[@hawthornroot]

i was watching US car crash vids & somehow i ended up here... now, for part 3, bye for now :)

[@EDUARDO12348]

So glad you made these videos...! you're like a master chef here, adding just the right amount of history spice.

[@JohnGetchel]

These are great! Love math history! If I may make a request. Can you leave the footnotes up longer? I was not able to read them. Or perhaps you could annotate them in the description.

[@barrycavin]

Sqrt(x^2)=abs(x) So, we do not have plus minus results for square root. However it is true that if x^2=y then x=pm(sqrt(y)) where pm stands for plus minus sign.

[@valentinachapovalova5470]

The square root of 9 is not "also negative 3", the square root is positive by definition!

[@ImmortalisPeregrinu]

Negative 3 *is* also a square root of 9. A square root isn't positive by definition; though the positive root is also known as the "Principal root," and that _is_ what he used the symbol for in this video. So he's still kinda wrong. But, we knew what he meant.

[@AuroraNora3]

+ImmortalisPeregrinu You're wrong. The *equation* has two solutions, but the *square root* of something is *always positive*. x^2=9 ** x=3 and x=(-3). But *sqrt(9)=3*. The square root is positive by definition. You should solve this equation knowing that *sqrt(x^2)=abs(x)*. x^2=9 sqrt(x^2)=sqrt(9) abs(x) =*3* x=3 and x=(-3)

[@ImmortalisPeregrinu]

Hoo Dini The square root *IS NOT POSITIVE BY DEFINITION.* Please read: mathworld.wolfram.com/SquareRoot.html "Note that any positive real number has two square roots, one positive and one negative." Square root is a noun, Taking the principle square root is what you are referring to. Extra: *Solutions of a polynomial are also referred to as roots;* if there were only positive ones, unless all quadratics had multiplicities, only having roots greater than zero would break the Fundamental Theorem of Algebra for equations as simple as x^2-1=0.

[@AuroraNora3]

Also, try putting sqrt(x^2) in a graphing calculator, CAS, or Google. It will show you the result |x| and graph it for you.

[@ImmortalisPeregrinu]

Hoo Dini I'm not insinuating that the range of the sqrt function includes negative numbers. It is considered a function because it it limited to show all square roots greater than zero. *BUT THE RANGE OF THIS SQUARE ROOT FUNCTION IS NOT THE FULL SET OF SQUARE ROOTS.* 'All' square roots will still include negatives. Again, please read what I linked you. It explains this. And for the love of god, stop wasting my time.

[@-_Nuke_-]

You are a hero! This video deserves a graham's number of likes!

[@diabl2master]

It's Fior not Foir -.- you even said it right but wrote it "Foir"

[@JM-lh8rl]

Davy Ker It is right in the subtitles, though

[@DanKop2]

Davy Ker no it's foir

[@vitakyo982]

Yes : Antonio Maria Del Fiore

[@blownspeakersss]

_This_ is how math should be taught. Engaging and narrative-driven.

[@obrod7080]

3:40 is factually wrong, square roots cannot be negative by definition.

[@maxwellsequation4887]

Linear- Normal-ish Quadratic- Bit faster Cubic- They call me the fastest man alive

[@steinal1169]

...so imaginary numbers are real but I can't do a/0? ahuh, i'm onto you mathematicians.

[@gamefoun]

i think it should just equal 0, if you divide something into zero pieces you get nothing

[@Crouchasauris]

you're mixing it up. what you said would make sense if it was "0/a", not "a/0". (and 0/a does equal 0 like you believe it should)

[@gamefoun]

Grant Davis i know that 0/x = 0

[@Crouchasauris]

heheh, sorry. What I was saying was that a/0 wouldn't be like dividing something into zero pieces--it would be more like seeing how many 0's can fit into a. Even after infinitely many zeros added together, you still wind up with zero. Hence the "undefined" as opposed to something like "infinity"

[@gamefoun]

Grant Davis ok, thanks for explanation, I just thought it was dividing in pieces :D

[@createtheengineerinyou6921]

I literally clicked the pdf for download accidentally but its a great treasure. Thank you for spreading math.

[@inchicago]

You're wrong at 3:50, with the principal square root definition, every square root has only a positive answer. There are no negative answers to square roots.

[@nicksm7980]

+David MacCumber, you don't really understand what square root is, do you?

[@inchicago]

+Nick Sm Rarely do we consider the negative solution to square roots. Except for perhaps in the quadratic formula. Which is why when you type in sqrt(9) in any calculator you only get 3, not +/- 3.

[@inchicago]

+Nick Sm Just scroll down and you will find others discussing this same idea. Although +/- 3 is a solution to x^2 = 9, it is not a solution to sqrt(9). For square roots, it's just the positive answer. Do YOU understand what a square root is??

[@nicksm7980]

David MacCumber , √ sign is used for denoting only positive number. So, it's not really a square root sign, this is a so called principal (or arithmetic) square root sign. Square root is denoted by ±√. Thus, yes, his notation isn't accurate but his words are correct.

[@inchicago]

+Nick Sm true

[@coach_tae_]

I literally have a physics degree and I've used the number i very often in my field, especially in quantum mechanics (which tripped me out the most) and I'm only just starting to understand that i isn't complete bullshit. It always felt like a tool rather than something that manifested in reality so I couldn't fathom why it appears ubiquitously in quantum mechanics. But that cubic formula and solution example made me realize that it might not be complete bullshit and not just a mathematical tool but something that just might manifest itself in reality. I still can't quite grasp what multiplication means if you can end with a negative after it, but this is progress and I appreciate you for this.

[@awseomgyhero]

What about cube root of negatives??? get rekt maths

[@Friek555]

the cube root of any negative real number is just another negative real number. 2³=8, and (-2)³=-8

[@olayinkaanifowose5099]

get rekt scrub.

[@zairaner1489]

Like there are two square root for every nonzero real number, there are three cube roots for every nonzero real number. One of the is real and the otehr two are complex (the three cube roots form an equilateral triangle in the complex plane).

[@ShankyBady]

Lmao you thought you're a mathematician, you're just a gamer

[@viktor7536]

This comment made my day

[@abc_cba]

Nobody can explain any math concept like you. Salute !

[@MatteoBrucato]

Is it really that hard to try to pronounce Italian names correctly?... -__-

[@ripsumrall8018]

Is it hard for Italians to pronounce English names correctly? I'm guessing at times it is. Oh my bad, you said 'try to pronounce' not actually do it correctly. I'm sure they 'tried' and failed.

[@Vitoria-ji6tq]

yes

[@FREAKSLICER]

These videos have shown me how much fun a regular math class has sucked out of learning math. I’m actually interested in math for once in my life

[@rossboyd9801]

Good video, yet again! I like this series but please can we have a brief recap at the end of each episode?

[@WelchLabsVideo]

+Ross Boyd Great idea, I'll see if I can integrate these with future videos.

[@Kratos2772]

You have a mistake on the third line of del Ferros solution for the cubic formula, where v^3, in (u^2+2uv+v^3)(u+v)+c(u+v)=d, should be v^2.

[@matheussilvacarvalhodeoliv7339]

Hello! At 2:59 the correct formula to substitute x for y-b/3 would be x³+bx²+cx+d=0 not ax³... Correct me if i'm wrong :) Nice video!!

[@Даниил-ц4э5о]

that's not really true at some point, √9 = 3 and not -3 because of the definition of the arithmetic square root

[@stevesynan3910]

Awesome series! I had basically no intuition for imaginary (or lateral) numbers before watching this, and now I still have almost no intuition, but it's not nothing, which is something!

[@Expecto_proteinum]

This is my first time on your channel. I subscribed right after I finished watching the first video because I saw the ad at the VERY END of the video! #Respect 🙏

[@juliepowers6037]

Great history lesson for myself and for my students! Thanks for sharing this!

[@anandkeshri3786]

Bro I am from India I love your concept to make videos in this topic I fully support u... Go ahead

[@WelchLabsVideo]

Thanks for watching!

[@AB-et6nj]

Rome: Who cares, let's conquer stuff Italy: Who cares, let's duel

[@cancer1164]

The square root of 9 is +3 ( √9 = 3 ) because the square root of a positive number is always greater than or equal to zero ( √9 ≥ 0 ). Therefore, the square root of 9 is not -3. Another very different thing is this: x² = 9 √x² = √9 |x| = 3 x = ±3

[@glenmartin2437]

Math duels? Much more interesting than dueling with swords or pistols!

[@georgios_5342]

3:50 I've been taught at school that the square root of a number is the positive number that when squared equals the one under the root. So √9=3, not -3. Why??

[@georgios_5342]

@María H I know that signs cancel, but I've only ever been taught (at school) that √a is only the positive number which when squared is a : √|a|^2=a. And when for example I'm doing an equation with x^2=9 I have to then write x=+or-√a

[@emilpysenisoncrack420]

Yes, square root by it self only takes the principal root into consideration. Else it couldn't be called a one-valued function.

[@emilpysenisoncrack420]

But having ± actually more reflects where the square root comes from.

[@steelep5623]

as a french student, "foir" is pronounced "fwahr" but besides that, these videos are beautifully simplistic. it's as if the ELI5 subreddit made a youtube channel

[@bagrankus8589]

Çevirenlere çok teşekkürlerimi sunuyorummm Perfect video :)

[@sagarprasad1436]

square root of 9 is only 3 and not -3...but if say x^2 =9 then x has 2 values i.e 3 and -3...bcoz when u remove the square from x then a mode appers over x....when only the value √9 is given its only solution is 3. if u try to use -3 as a solution too then there is a way u can prove 1=2...and to u guys i am no noob just randomly commenting....i am a maths student .....sorry if i am not able to convey my message properly

[@r.e.d.readyeveryday508]

sqaure roots -1 =i , -4=2i , -9=3i , -16=4i , -25=5i

[@gnikola2013]

There is a slight, non important misconception used on this video. The square root of nine is three. Not three and/or negative three. It's a matter of definition. You want the operation "square root" to drop out a single number, because, well, it's an operation. A function if you might. And functions have only one output for each input. However, it is indeed true that x^2-9=0 holds for both x=3 and x=-3. But x is not just sqrt(3), because how could a single number, x, be two numbers. It's something not even written properly in the mathematical language. In that case, we say that the equation holds for x1=sqrt(3) and x2= -sqrt(3), which are two different numbers. You might write this as x= ±sqrt(3), as short notation, implying the existance of two solutions, x1, and x2. If this comment does not convince you, maybe you could check out Epic Math Time channel's video "The meaning of the square root (√) symbol". He is an actual Maths graduate school student (or undergrad, I'm not sure). He very clearly explains that the misconception is due to not paying attention to the actual definition of the square root, which is the only important thing, as always, in Maths.

[@KTFG]

Excellent videos. Added to playlist.

[@rbmarik6082]

Del Ferro's formulae at 1:34 is different from that of 2:04...coefficients c and d are interchanged.... the correct one is where coefficient of x i.e. on the numerator of the fraction c^2/27

[@younes01976]

Beautiful video, please another with example equation and explain.

[@camontes]

I've learned that SQRT(9)=3, and only 3, not -3. It's different when you have to find x in SQRT(x^2)=9, where x=3 or x=-3. The square root of a positive number is always a positive number, and it's called principal square root.

[@ufotofu9]

This series is incredibly well made and explained. Thanks.

[@WelchLabsVideo]

Thanks for watching!

[@brettnelson7518]

Damn, I think the most impressive thing was not only did Del Ferro solve the cubit problems that were thought impossible but he then went 20 years without even needing to use it in a math duel. Guy must have been a beast in duels!

[@hariramnarayanan5157]

your videos are of top-notch quality and content !

[@urmilb6812]

3:45 @Welch labs , you are wrong . the square root of a positive number can only give you a positive result. please bring it to notice and correct it as it confuses a lot of people like me and makes them question their own fundamentals. Great videos by the way !!!

[@portersedge]

While the square roots of 9 are -3 and 3, the principal square root is defined to be the positive square root and is denoted by the radical symbol. Note: the principal square root of x^2 is the absolute value of x. Unfortunately, this step is often skipped in solving x^2 + 1 = 0. So much detail is presented in these videos, but this small error that you caught can be a source of confusion for students.Hopefully, more awesome videos like this one will be made.

[@emilpysenisoncrack420]

Please note that the square root of any complex will give you positive or zeroed "real" part.

[@bwstucky]

Great explanation so far. However, it is a potential source of real confusion to say "the square root of 9 is -3." The radical square root symbol is DEFINED as the positive (principle) square root.

[@이범찬-z2p]

한글자막 고마워요!! 다른 시리즈도 있으면 좋겠습니다ㅎㅎ

[@ripsumrall8018]

Look there is enough confusion here with out bring that into it!

[@jessebosker122]

3:37 wrong, it only equals 3. If the equation x^2 = 9 is given, the solutions are obvious 3,-3. But if sqrt(9) is given, the only solution is 3. It has to do with what you’re asking.