Algebraic vs. Transcendental Numbers

[johnmellor]

im 68 yrs of age and tried to learn this at college many years ago and failed but ive found your way of explaining and enthusiasm really great and im finally able to understand more , your students come across as really engaged in the course well done

[mapleandsteel]

You have such a talkative, yet involved class !
Awesome !!!!

[desmondbrown]

You really see how interesting those transcendental numbers are when you start dealing with limits in calculus.

[skylardeslypere]

It hurt my soul when you wrote rationals instead of Q (0:36)

[morezco]

Reviewing this as I write a paper for uni... I take it that this is a class for children, but it did get me excited and interested in watching it all the way through. What an awesome teacher.

[quantumking]

This is awesome. I wish my high school had been this in depth. I'm doing all the catch up I can as I work on my associates in engineering, and ultimately PhD in Physics. Thank you so much.

[markkaravan]

Love your enthusiasm. Great work Eddie :)

[jannickharambe]

I didnt really understand algebraic numbers but I did understand how to calculate e now. Thanks :D

[wiz]

OMG!.... I wish if everyone has a teacher like you with your full enthusiasm!
RESPECT :)

[tompurcell]

I think something was omitted here. Certainly I can write the algebraic equation x - e = 0, from which x = e is a solution. What prevents e (and other transcendental numbers from being algebraic is that they cannot be solutions to any polynomial equation which has integer coefficients and some integer constant.

I'm not being petty; this is important.

[lemoncryptonfa]

There are infinite more transcendental number than algebraic number. Wow

[Questiala]

I’m not even needing to learn this but I’m doing it for fun.

[alejrandom]

This is so beautiful, I wish I was introduced to math something like this but I'm glad we have you now ;) nice work!

[psyche]

Can you please start a series on all the disagreements there were in maths 3:23 By the way really love the way you teach maths ❤❤

[davidwright]

Eddie - it would be helpful if you could number the videos so dependencies would be clear. As in, 'Video y depends on the information in videos a, b, ... . You did this in the set introducing complex numbers.

[nicolasargon]

In the question as to whether math is discovered or created I think the transcendental numbers gives us great insight. I would say that math is a representative paradigm (yup, created). The symbols (1, 2, +, =, ..., etc.) are obviously glyphs which label something about phenomena. That pi = 3.1415... is not an absolute value but the symbolic representation for what pi is. Its like height or length. Identities existing in phenomena (the 'actual') can be characterized by height or length, but the measurement units (inches, centimetres, miles, ..., etc.) for such phenomena are created. And math is the same thing. Exponentials and the relationship between circumference and diameter are real phenomena, but the intellectual structure to measure or 'make sense' of such phenomena is created.

By 'make sense' I mean that the phenomena which are real (length, exponentials, etc.) are intelligible to us as human beings under certain circumstances. This is why 'created' does not imply 'arbitrary'. We use symbols like "1, 2, 3..." because we can see them. If we were all blind written language (if written at all) would likely take alternative symbolic representations for such values. 'We can see them' is simply one of the constraints for the intelligibility of our representations. This 'intelligibility' constraint is by origin of what we are as humans. I can imagine a 'math' from aliens which borrows none of the symbolic representations of our own mathematics. Depending on what senses they have - how they apprehend the universe and its phenomena, their expressions of mathematical realities may likely be incomprehensible (ie, un-decodable) to us as humans. So that's one reason why math is created, because it has an intelligibility funnel making math take on a certain symbolic representation comprehensible to us humans.

But Intelligibility is only one half of it. The expressive quality of the representation is the other half. Centimeters and inches work well not by virtue of their units, but by the relations between their units. That each unit of length is standardized and equal, permits quantification and comparison in height between disparate phenomena characterized by height. This 'standardization' is a property of the system of measurement, not the particular measurement itself. We can think of the "properties of the system of measurement" as the expressive constraint of any representative paradigm, and the "particular measurement" as the intelligibility constraint of any representative paradigm.

When thinking of the expressive potential of mathematical equations, it follows the same principles as height and length. There are mathematical representations of phenomena that approximate actual phenomena to a higher or lower degree of fidelity. That degree of fidelity is the 'expressive quality' or 'potential' of the representative paradigm. Just like a drawing of a bird depicts many aspects of the actual bird while losing others aspects, mathematics excels are depicting many aspect of the actual world while losing others. The numbers (1, 2, 3, ...) are the particular measurement of the representative paradigm, and the operators (+, -, =, ... ) reveal the properties (patterns, symmetries, etc.) of the system. So that's the other reason why math is created, because there's an expressive constraint for what it can describe.

So ultimately math is still created, because of both an intelligibility constraint and an expressive constraint. That doesn't mean math isn't 'true' but that math is a representation of the actual, not part and parcel of the actual. It's a way to express aspects of the actual world in a manner intelligible to us humans.

I guess you could suppose that there's a representative paradigm (math) which describes the actual with perfect fidelity. Which even if it was the case, it's certainly not the case at this time. But I doubt it's the case. The 'expressive' quality or potential of math - or any representative paradigm, is a matter of epistemic scrutiny. That math is useful, meaning that it can allow us to predict things about the world is part of the 'expressiveness' of math, and also a epistemic test from which we derive certainty or confidence in the mathematical paradigm. That math is internally valid, doesn't have internal contradictions is also an epistemic test. And there are many others.

Why I would argue that math's expressiveness is characterized foremost as a constraint and not as a description of perfect fidelity is because what qualifies as an 'epistemic test' is itself constrained by what we are as humans. Our scope of experience or apprehension are the limits for that which we can derive certainty of. And since or scope of experience, and therefore our scope of epistemic certainty, is limited, than any representation which communicates (communication = expression + intelligibility) something within such scope is itself limited. And thus, I would argue that math is created.

[trendytrenessh]

Wow Eddie you make maths so fun, your class is really lucky! Thanks for the video :)

[feynstein]

This seems to be a technicality. It seems to me that transcendental numbers cannot be produced by *finite* polynomials. Once you remove that restriction and move on to *infinite* polynomials, everything's possible. There's certainly a distinction. However, saying that these numbers transcend algebra itself gives the wrong impression. Nothing transcends algebra.

[EvilErwin]

Grait video.
But i have one question:
WHO mounts a whiteboard on top of a whiteboard?

[Hishamhh]

when you compare the number of algebraic numbers in existence with the number of transcendental numbers.. that would be like comparing infinity vs infinity on steroids..😛