Imaginary Numbers Are Real [Part 4: Bombelli’s Solution]

Last time, we decided to let the square
root of minus 1 be its own new type of number, hoping it would help us solve Cardan’s problem. Doing so is helpful, but finding a solution requires one more insight from our friend, Rafael Bombelli. Bombelli knew that because of the way cubics are shaped, our function had to have a solution that didn’t involve the square root of minus 1. It had to be a plain old regular positive or negative number as he had seen before. The second big insight here was that for this to be the case, the root of minus 1 parts of each half of the equation must cancel out when added together. Bombelli use this idea to equate the two parts of the equation to a+b√(-1) and a-b√(-1), where a and b are constants that we need to find. To solve, we can first eliminate that annoying cube root by cubing both sides. The result is a particularly tough system of equations. Bombelli was able to get around this through some clever guessing and checking. If we look at our original equation and test a few integers, we eventually see that 4 is a solution. If we substitute 4 into our new equations, we can solve for a and b and obtain a=2 and b=1. These values make the two parts of our expression equal to 2+√(-1) and 2-√(-1). We can cube these and show that they are in fact equivalent to what we started with. And more importantly, when we add the two parts as the formula tells us to do, we just get 4, which we know is a solution to our original equation. We have found the solution to Cardan’s problem. And what’s really interesting is that our problem had nothing to do with the square root of minus 1 and neither did our answer. However, along the way, we found that by extending our number system to include the square root of minus 1, we were able to find a solution. And it turns out that extending our number system in this way is helpful in lots and lots of other problems as well. So what did Bombelli do to celebrate after discovering a number so crucial to the future of science and mathematics? He actually did nothing. He just scattered
his discovery and basically said it was a hack. As ridiculous as that might seem now, Bombelli drew a pretty reasonable conclusion at the time. It just seemed a little too convenient, like a trick devised just to solve problems like this. Squaring numbers had up until that point largely been associated with what the operation is named for: squares. A square’s area is equal to the length of its side squared, so positive areas makes sense. But what could a negative area be? What even is the square root of minus 1? Questions like these slow down the development of imaginary numbers. It turns out there is a much deeper and richer meaning lurking below the surface, but it would take long after Bombelli’s death to be revealed.

Bernard Jenkins


  1. i started this series thinking i was just going to watch one video and learn a couple of things

    but now i'm actually addicted and have learnt so much about the history of complex numbers i can't stop help

  2. Hey ! First thanks for those videos! I learned a lot ! There is just something (actually a couple but I didn't searched enough to ask about the other ones yet) that I don't get. You say at the beginning of this video:

    "Bombelli knew that because of the way cubics are shaped, our function had to have a solution that didn't involved the sqrt(-1)…".

    After looking in the math history of that time, it looks like the origins of analytic geometry only started with René Descartes (1596 – 1650) born a bit more than 20 years after Bombelli's death. From further research I came across Pierre de Fermat and his famous "Ad Locos Planos et Solidos Isagoge". Here is a quote of what britannica.com says about this paper:

    "Fermat considered an equation among two variables. One of the variables represented a line measured horizontally from a given initial point, while the other represented a second line positioned at the end of the first line and inclined at a fixed angle to the horizontal. As the first variable varied in magnitude, the second took on a value determined by the equation, and the endpoint of the second line traced out a curve in space."

    This paper was written in 1636. So a solid half century after Bombelli's death.

    So now, my question is. If he didn't knew the shape of a cubic, what lead him to think that there was at least a solution in "the plain old Regular positive or negative numbers"?

    I really don't try to be picky here or whatever, I'm just so curious, I have time and I decided to start to relearn the maths following the history of it. I hated that at school because teachers are often so bad and skip all the bits that could keep you interested so you're missing the big picture and only get apparently non-sense symbols mixed together. It's so much easier to remember everything when you have the context around. It's actually an amazing story.

    Thanks again for this playlist. You really have a gift, you should say hello to Khan Academy you could do a great deal of helping them change the education game.

  3. So mathematician = make shit up when you're stuck, do a ton of unnecessary work and guessing

  4. Just make the damn videos longer gosh it sucks to click every two seconds on a new video when all parts could be in at least two longer videos

  5. Weird. I was able to solve x3=15x+4 by factoring the zero I found with the Rational Zero Test and then using quadratic formula. Why didn’t they just do that?

  6. Very very very very….. interesting… Sir why don't you write a book about the history as well as applications of complex numbers in detail… If you are able to do that please send me a copy…. 😊

  7. Great series. Disliked every video and I still can't see how you made up that pretty graph. U realise u leave a lot of hanging stuff between 1 video and another an don't resolve it in the next one right? Pretty face, enfuriating material

  8. Great series. Disliked every video and I still can't see how you made up that pretty graph. U realise u leave a lot of hanging stuff between 1 video and another an don't resolve it in the next one right? Pretty face, enfuriating material

  9. Why did Bombelli think the real part of both the cube roots should be same ? Why couldn't it be that one of the cube roots is in the form of "a+bi" and the other in the form "c-bi" ?

  10. It turns out, that I'm 45 and a much bigger nerd than I thought… Why on Odins bloody earth would I spend my saturday evening watching the origin of imaginary numbers, I ask my self. I do NOT have a solution. Oooo, there is a part five. Awesome. 😆

  11. There's another intuitive way of looking at why some numbers need to be represented with i. Take the 2D coordinate plane. Take coordinates represented by a symmetric point. For example, (x,y) = (3,3), (-6,-6). Now consider the total number of countable integral points represented within that square region closed by the points. (3,3) = 9, (-6,-6) = 36 points. In both these cases, one can tell that square root of either of that integral number by simply looking at the coordinate representation. Either of the axes can be selected to represent the root. Then, how about (-5,5) or (7,-7)? What would the root be? The countable region tells us it's 5 and 7, but looking at the coordinate pair, which one would you pick? We have an ambiguity of choice w.r.t sign of the number due to asymmetry of sign and since we want to represent it on the Cartesian plane. So, i, the root of -1 is a convenient way to resolve this ambiguity of choice. With 5i or 7i, I'm simply allowing the choice or rather deferring the choice until the square is actually calculated.

  12. but isn't this bombelli's solution still just a guesswork? I mean, he didn't solve the root of negative problem, he found the solution by guesswork (analytically) and then arranged his formula to work. If I change one number in his given equation he is still stuck with new guesswork

  13. So incorporating sqrt(-1) into the equation helped lead to the obvious answer of 4 in that example, but does that work with the earlier examples, where it was solvable without using sqrt(-1)?

  14. Goes far too quickly for any non mathematician to follow. Pity, cos it's well done.

  15. I've tried to understand lateral or "imaginary" numbers by myself before but did not understand any explanations online :(. This series however is really helpful and i understand a lot more than i did before, so thank you very much for it!

  16. Why the hell I'm still watching this series if I'm a telecommunication engineer that deals with complex numbers every fucking day?

  17. 1:26 It should be 4³ instead of 4² (4³ = 15 * 4 + 4)
    Amazing video btw. One more subscriber

  18. At 0:56 seconds on the proof it says that 15 squared over 27 equals 125 but I'm sure you mean cubed

  19. That comment about the negative area got me thinking, what if the positive area is normal matter we all know and the negative mass was antimatter?

  20. Well done but not complete….Matrix and conforming maps are funny and usefull to see suggesting simmetries of coefficients

  21. OMG, I started somehow finding the first video. The topic and the quality were so good that I've watched up to here. Well done Welch Labs!! =]

  22. If you pause at 1:09, the last line of the paper on the right is wrong lol. B squared = -1 should be b squared = 1

  23. Ааа а где русские субтитры в следующих видео???

    Where is subtitles in the next parts?!!!?? 🙁

  24. 1:00 what if this scientist just puted numbers and solved and didn't invented any imaginary?

  25. there is no end to this . . . jyst tell us how Steinmetz used i solve power transmission in a/c systems.

  26. Increíblemente explicado, …….Muchísimas gracias por los subtítulos amigos!!

  27. Great video, but you need to stop splitting these up so often and make longer videos

  28. this 3d-graph is wrong. there should be 2 surfaces: 1 for real component and 1 for imaginary.
    1 surface intersecting 'x' plane generates infinite set of roots for (re(x^2+1)=0).

  29. Easy. Is this what you do in the US?
    This is 8 grade stuff in India.

  30. I just wanted to see the the graph in part 1 video. I hope this is not an infinite part playlist.

  31. I was following the earlier videos but iknow my limits… I show my palms to the eye in the sky and I'm out. 😂

  32. 0:39 I get why the imaginary coefficient has to be the same of both terms, but what about the real coefficient (a)? Why does that have to be equal

  33. Lol yyeeaaa there’s over 13 episodes I’m done right here

  34. Woooowww…. Ok, that is amazing!!
    I never saw it that way.
    This is amazing work!!!

  35. I really liked the historical approach in parallel with the math class, after all some of us are still humans

  36. Minute 1:07 .the correct is b^2 = 1 the minus was a mistake. I was wondering if someone noticed

  37. where is the subtitles :((((((, I don't understand english very well, so I need the subtitles, automatic subtitles is don't working too, af

  38. Minimal research went into this evidently. Waving around a picture of Viete and passing it off as Bombelli without so much as an editor's correction is frankly astonishing. Pop math was a mistake.

  39. thanks for your marvellous work….at 00:44 cubic root of (2 +square root of -121) should equal cubic root of (a+b square root of -1) not plain (a+b square root of -1) or am i not seeing the insight?????

  40. math is the language of the universe. if you can understand math, you will be able to understand physics.

  41. is cubic root of (2-11i) has one value as i think that the answer could be (2-i) and could be -0.1339745963+ 2.232050808 i in this case x will not equal to 4 !!

  42. I don't agree with this method at all, what is the meaning of guessing? I don't have time to guess all the day! I need some serious math

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