Algebraic numbers are countable

Transcendental numbers are uncountable, algebraic numbers are countable.

There are two kinds of real numbers: The algebraic numbers (like 1, 3/4, sqrt(2)) and the transcendental numbers (like pi or e). In this video, I show that the algebraic numbers are countable, which means that there are so many transcendental numbers that if you choose a real number at random, the probability that it's transcendental is 1!!! This is so suprising, because we're much more used to dealing with algebraic numbers!

Note: My apologies for not making my usual outro at the end, my phone ran out of battery, but luckily it caught most of the video!

[ffggddss]

Nicely done!
It gets only slightly easier if you explain at the top, that "zero of a rational polynomial" and "zero of an integer polynomial" are exactly the same; then you needn't rely on a prior proof that ℚ is countable. (And it's interesting to note that ℚ is just the degree=1 subset of algebraic numbers.)

BTW,  I was puzzled by the idea that you wind up with a countable infinity of algebraic numbers, by taking all integer polynomials of each (finite) degree, each of which is countable, and then, putting them all (countably infinitely many of them) together.

It isn't obvious to me, offhand, that a countably infinite union of countably infinite sets, is necessarily a countable set.
For instance, the reals aren't countable; yet one definition of ℝ is the set of all possible convergent sequences of ℚ. (Each such sequence being countable.) That's kind of a power set (2^ℵ₀ = ℵ₁), although not really, because it's constrained by having to be convergent; so I'm not sure how that plays out.
And especially, how it plays out differently here, than for the algebraic numbers.

Is there just something that should have been obvious here?

And why can't the geniuses of Silicon Valley come up with longer lasting batteries? Your ending was fine, BTW, even without your usual outro.

Fred

[bouteilledargile]

im so glad i found this channel. i never really liked proofs before but this channel has made me appreciate them so much

[marcosvb]

Very clear and elegant demo! :) Thank you so much!

[neilgerace]

"Almost all" real numbers are transcendental

[Nightlessness]

This is some nice analysis/algebra material :)

[robertschlesinger]

Excellent video. Many thanks for taking the time to prepare, produce, and post the video.

[anowarali]

in step 1; you have said there is atomst N roots in N degree polynomial but in the very next line you assume the N+2 roots which are Is it not contradict each other???😇

[Shaymin]

Nice video :). I was thinking at first on counting the number of equations with rational coefficients, doing pretty much the same methodology. But then the final set would have different equations with same roots and it's not as elegant.

[sugarfrosted]

Transcendence bases might be an interesting topic.

[aanchal]

Ur explanation is just wow 🤩🤩
Sir U r amazing ☺🙏🙏

[Debg]

It's interesting and maybe a little sad the fact that the number of trascendental numbers we usually work with and can compute is pretty much countable.

Happy (truncated) π day!

[ashutoshpandey]

You are a wonderful teacher. I am glad I found your channel.

[harrywotton]

Let me give you a unique proof though it is less strict.

If we make formulas or sentences to state something in mathematics, we arrange several figures, alphabets and mathematical symbols like these.

"2+2=4"
"sinπ=1 is not true."
"√2 is irrational"

How many kinds of symbols are used?

(Figures) 0123456789inf... probably less than a hundred
(Alphabet) abc..., αβγ..., probably some hundreds
(Mathematical symbols) =+-×÷ lim dx..., probably some hundreds or maybe thousands

There can't be infinitely many kinds of symbols. Also, formulas or sentences can't be infinitely long. Therefore, the patterns of mathematical statements are countable. It follows that all the numbers we can "define" by formulas are countable. And all algebraic numbers can be "defined" by a formula, so they're countable.

People say that there're more real numbers than natural numbers, but most real numbers can't even be defined. Random figures without any "rule" in a perfect sense of the word can't be defined. Such a number may be like

[AndDiracisHisProphet]

To clarify, Peyam sometimes says "algebraic numbers are zeros of polynomials with integer coefficients" and sometimes with "rational coefficients".
The last one is correct.

Also, happy pi day

[madhavpr]

Love the video !! However, a question has popped into my mind. How are we so sure that the real numbers are either algebraic or transcendental i.e the set of real numbers is the disjoint union of algebraic and transcendental numbers? Is there a chance that we may have missed some other kind of numbers, that are quite different from those two? Why is it "wrong" to imagine the existence of such numbers? In terms of probability, if I pick a number, what are the chances that it might neither be algebraic nor transcendental but something else?

[kapildeo]

Integration of ln(2cosx) from 0 to sir...

[FishSticker]

I don’t get the last step, I can’t see how you would be able to order Q x Q x … infinitely many times… x Q, I can understand ordering for arbitrarily
large ones, but not unlimited length ones

[shandyverdyo]

I like it. Eventhough i don't understand.

[funkysagancat]

Excelent video! Really easy to understand

[husklyman]

I want to challenge you to solve for the square root of while π=3.1415...