The true history of complex numbers.

I have adopted this story from Tristan Needham' s book "Visual Complex Analysis". This is a true origin of complex numbers !

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[MetaMaths]

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[flavrt]

In school, I saw history folded into science and math classes as a waste of time. Now as a teacher, I appreciate how historic personalities can be used to connect new abstract knowledge to the students’ subject base knowledge. Thank you for the presentation.

[manamritsingh]

Damn. This makes you see how math was literally telling you that i was a thing but we humans refused to believe it back then. It's kinda like Math has a life of its own. Math can never be wrong but our understanding of it might be flawed. Math is a beautiful subject that speaks to you. I loved it. Thanks

[francisarmitage]

Fascinating. I studied applied math and theoretical physics at the university of St. Andrews, Scotland (1976-80) and I taught math in a South African high school (1999-2019). I used to 'entertain' learners with stories about complex numbers and their use in fluid mechanics, but I had never heard of this origin story. In fact, I cannot remember how complex numbers were introduced to us all those years ago - I think we were just shown their use rather than their origin...

[EebstertheGreat]

In Europe, it was the rigorous treatment of negative numbers that enabled (and forced) the acceptance of complex numbers. Complex numbers are not so great a conceptual leap beyond negative numbers, in spite of the way they are obscured in contemporary education. Once the representation of the real numbers (or some dense subset) and a line was understood and internalized--that is to say, once real numbers were understood at all--complex numbers followed very quickly.

For centuries, Europeans only considered polynomials over the positive numbers, and addition and subtraction were treated separately. For instance, there was not one general quadratic univariate polynomial equation ax²+bx+c=0, but four: ax²+bx+c=0, ax²+bx=c, ax²+c=bx, and ax²=bx+c. Even then, when Scipione del Ferro
solved every special case of the general cubic equation, he did not admit negative solutions, yet he encountered negative discriminants. This was immensely frustrating and required a number of lengthy workarounds to avoid using complex numbers. It was just a few decades between this algebraic masterpiece, working effectively in the positive algebraic numbers of degree 4 or less, and Rafael Bombelli's non-rigorous but fairly modern treatment of the complex numbers. Of course, a proper formalism even of the real numbers remained centuries away, but that is related to a conceptual difficulty of the continuum, not of the imaginary axis.

It is important to remember that the geometric intuitions we take for granted did not yet exist for real numbers. There was no concept of the real line or the Cartesian plane. Descartes's intuition caught fire for a reason: algebra was desperately in need of visualization. So this idea of plotting complex numbers on a plane was inconceivable to an establishment that had not yet thought of plotting real numbers on a line. In truth, the algebraic understanding came first, in terms of the rules that must necessarily apply to abstract quantities, and the geometric understanding came later, in terms of the rectangular and polar representations of complex numbers.

[jonahansen]

This is a very interesting video. I think, however, that you could improve it substantially and really drive home the point of the need for complex numbers in the last segment where x is the sum of the cube root of 2+11i and the cube root of 2-11i. What is happening is that the formula requires complex numbers, but they conspire to produce a real value because it is a sum of a complex number and its conjugate.

This idea is profound and important, and comes up in a number of physics problems where the answer must be real, and what happens is a sum of a complex value and its conjugate "conspire" when added to produce such.

It is similar to how the fibonacci sequence can be represented as a set of discrete difference equations whose solutions are sums of powers of weird values containing irrational roots but which, when summed, conspire to produce integer values.

[levprotter]

"Judging Mathematics by its pragmatic value is like judging symphony by the weight of its score."
-Alexander Bogomolny

[srenladegaardkristensen]

1:28. François Viète called. He wants his portrait back.

[paulbetts]

Amazingly concise, yet beautifully conveyed video. Thank you for summarising this fascinating intersection of mathematics, history, and human discovery.

[MaximQuantum]

WOW I thought I was watching a youtube video from someone with 500k+ subs, you produce the same or even better content than them! :D
Checkpoint: 2.1k subs - 35, 531 views

[SimchaWaldman]

01:12 The right cubic root has a small error to correct.

[crustyoldfart]

" possible and necessary for mathematics ". More precisely stated : use of ' I = sqrt(-1) ' can produce results which are demonstrably correct [ as in the example of the intersection of a line and the simple cubic curve ], and it also points a way to derive two solutions to every quadratic. In mathematics we can always find examples of pragmatic use of a concept on the one hand, and rigorous reasoning on the other. One example is G.H.Hardy's work in putting Newton's calculus on a firm basis based on clear axioms.
The invention of the Argand method is certainly pragmatic and useful in arriving at conclusions which are demonstrably correct.
This should be good enough for most purposes, but will be offensive to those who believe that a priori reasoning based on rigorously defined axioms, which in turn are akin to laws of the universe - handed down from on high, pristine and immutable. Kant asks the question did mankind bring mathematics to the world, or does it exist in some absolute sense. In posing his famous set of problems David Hilbert states " wir muessen wissen, wir werden wissen " - a statement which was refuted by Kurt Goedel, so that we cannot always " know " even though there is no evidence to the contrary.
A further example of the utility of complex numbers, as if it were needed, is the its use in taking higher powers of (a+I*b) to evaluate tan(2*x), tan(3*x), .... where tan(x)=b/a. This is much simpler that embarking on a proof based on Euclidean geometry. The proof of the pudding is in the eating.

[rtravkin]

A mistake @ 1:14 — the 2nd term in the Cardano–Tartaglia formula has the operation -q/2 - sqrt(…) applied twice instead of once. The two cube roots should look identical except for the sign of the square root inside. Thank you for your work!

[ne]

I have been looking for this for so long. Great work.

[spacelogisticsinc]

Phenomenal video. Complex numbers always looked arbitrary to to me. Was also told the old false origin story about them. This video solves both issues, satisfyingly too.

[dcterr]

A fascinating thing about the history of math and science is that it shows how many dead ends were taken first to get where we are today, so that the development of these subjects isn't anywhere nearly as clean-cut as you might think, and for all we know, we're still heading towards lots of dead ends! I for one suspect string theory is one of them!

[ytashu]

Very interesting and compelling. I don't think i have ever seen complex numbers motivated in this way. Thanks!

[ToBeFree]

Great👍! This has been the best explanation about how the complex numbers have come into being.

[channelsixtyseven]

I saw complex numbers for the first time and just accepted it, because it worked. It made all the more sense when capacitive and inductive reactance are complex values.

[mostexcellentlordship]

You have a number line. You decide to place an "origin" somewhere - where this line "ends" - and on the "other side" the same number line is reflected and you call it "negative numbers". This is all so arbitrary. Reflected around what axis precisely? Introducing this arbitrary "origin" is completely equivalent to introducing another orthogonal axis without which this weird "reflection" argument makes no sense. You create a 2D space with this move. In fact, we should be weirded out by negative numbers alone. How can you just make this move? Reflect a line in "another" direction? Which direction? How can you just do this? And what do you do with this remaining dimension you added? Complex number on the other hand just make perfect sense.

I now challenge you to apply the "complex plane" to conceptual space: if at 1 is "existing" and at -1 is "non-existing", what does a 90degree CCW rotation away from 1 represent?

If you'll applaud in silence I will go gently on you during my Nobel acceptance speech.