Rational vs. Irrational (and Algebraic vs. Transcendental)

There are other families of real numbers that are named/defined, such as "computable numbers", "constructible numbers", etc., as well as types of number that aren't "real", but every "real number" will either be an integer, a non-integer rational number, an algebraic irrational number, or a transcendental number. In this video I explain the details about what those categories mean. Sometime In the future, I'll explain more about how these terms would apply if we were looking at imaginary and/or complex numbers.

To clarify about the "degree 5 polynomials are weird" thing I mentioned, their roots are still considered algebraic numbers, but some cannot be expressed using radicals in the same simple way a number like the golden ratio can. I'll make a future video about that topic and about the mathematician I mentioned (Galois) someday.... Also, one more clarification about the types of polynomials allowed for "algebraic numbers" (like how I mentioned they cannot be infinite length or have irrational coefficients) is that they also must be in a single variable (typically "x" is chosen).

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Domotro
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Combo Class, taught by Domotro, is a crazy educational show where you can have fun learning rare things about math, science, language, and more! This is the channel for Domotro's Combo Class bonus videos, shorts, and livestreams - make sure you're also subscribed to the main @ComboClass channel where the main episodes go!

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[Bibibosh]

You're a really good presenter. You very rarely mess words up and you're always clear with your pronunciation. I enjoy watching you and the way you explain certain mathematical phenomena. Maths is truely mind boggeling ! Great video!
Keep up the amazing content. Every video has been ideas and new ways of thinking! LOVE IT

[JM-elvk]

Fantastic show, the whole family really enjoy watching your content 🤝

[Ezzell_]

Nice to see you cleaned up. You are looking professional.

[tavishbryan]

Dude I love your content. It's a real gift to math youtube ❤

[ShandoGuardian]

I love the mad scientist vibe this guy gives off

[illogicmath]

Suggestion for a video: explain the cardinality of ℕ, ℤ, ℚ, ℝ, ℝ\ ℚ, 𝔸, 𝕋.

[produck]

Maths get me so hard. Keep up the good work

[SunroseStudios]

integers aren't closed under division, because you can make non-integer rationals that way; irrational numbers aren't closed under multiplication because some of them can be multiplied to get rational numbers (see: multiplying the square root of 2 by itself!)

not sure about the others, those are less obvious

[grodifer]

I haven't messed around that much with "imaginary" numbers (i've always kind of hated that name because ive had math teachers say they arent real, even though they are) but im wondering if there are similar kinds of distinctions that can be made among the "imaginaries"

ive heard them called perpendicular numbers
maybe rotational numbers is a decent name since they are a "rotation" from the classic number line
i think that all directions are in reference to the positive integers
meaning that negative is a direction, and its the opposite one from the positive numbers
likewise imaginary is a direction, and using trig we can get any distance from the origin along an angle that isnt a 90 degree (pi/2) rotation from the positive or negative numbers
maybe we should call the number line the linear numbers
or base numbers, since they are the base from which you rotate for i
i feel like positive and negative need a distinction, but we already have to names positive and negative
but maybe call them right numbers, cuz positive numbers go to the right on the number line
and then left numbers for negatives

idk, just spitballing
a lot of those names are probably bad
i need to spend more time on it, that's for sure

[stickfiftyfive]

Any ideas as to alternative titles for "real" and "imaginary" numbers?

[themightyripples]

I can remember being a little kid and dismissing the idea of irrationality and also the idea of repeating decimals entirely. I remember saying to my mom "I'm going to become famous for figuring out how to divide 100 by 3 without a repeating decimal", lmao. Of course thats absurd, but my little brain did not like that something so organized seeming such as maths could have these 'glitches' where no matter what you do you reach some sort of infinite string

[evandrofilipe]

I don't know man the reason why we have "i" the number falls out really good in geometric algebra, I think you should check it out

[SteinGauslaaStrindhaug]

Love that the camera operator was just as distracted by the squirrel as I was...

[monoman]

ratio of hair to face has changed. right....

[whfsyv]

That actually very comlicated even you understand that