All the Numbers - Numberphile

We're gonna do all the numbers. On Numberphile

we've done a lot of the numbers, some would say, but we haven't done all the numbers. Originally

we did the whole numbers. And these are the classics, my goodness. We've done 11, that was an early one.

We've done 3435, seventeen - right, all the whole numbers sit in here.

But then there are other types of numbers. If we go one step out, the rational numbers the ones that are ratios,

these are - you know, you get a seventeenth, you get I don't know things over twelve, you get all sorts of...

So now you've got all the rational numbers. We've gone beyond that though. And the rationals technically include the whole numbers,

they're a subset, but I'm doing this as what people call Venn diagram, which is wrong,

it's an Euler diagram. Because I'm not showing every single possible combination. - (Brady: Are negative numbers whole numbers?)

You know, I

put in a twelfth because I thought I was being hilarious and then I immediately thought, urgh I wasn't gonna put negatives on this diagram.

So I regret that, for two reasons. Opening the negative can and the expression on your face.

So I'm gonna make that a plus, there we are. In fact, this whole sheet of paper is just gonna be the

reals.

Positive re- you know, it works for negative reals, what am I saying? Have the negatives, it's fine.

But the sheet is - are all the reals and

inside here I've put whole numbers and then I've put rational numbers. If you - you can obviously get complex numbers coming up,

we're not gonna do that,

we're gonna stay down here. And I'm gonna gradually work our way out until we get a greater distance out

than Numberphile has ever gone before, right? We're going for the new Numberphile record: how far out into the weird reals

can we go? But we've done rational. Next one up the constructables. And these often aren't mentioned,

you don't have to add this in as a category, but I quite like constructables.

More importantly what people tend to go for, the next one out are the algebraic. Okay

so Simon did a fantastic video about algebraic numbers, and when you go outside,

transcendental numbers.

"I mean this number is really, really important and no one knew." - And so a lot of the number categories refer to in or out

of these different sets. So rational numbers are everything inside the blue line,

irrational numbers are everything outside the blue line.

You've got constructable numbers are anything inside the purple line, unconstructable numbers are outside this. And this light blue line out here:

algebraic numbers are everything inside there, and

transcendental numbers are everything

outside of there. And so constructable numbers are things that you can construct with a pencil and a compass and a ruler.

So Phi, you can do that, the golden ratio because you can do root 5, so you can get Phi.

You can do root 2, that lives in here,

that's kind of fun.

Algebraic numbers are the solution to an algebraic equation. If it's a square root or lower you can put it in constructable,

you can draw it. If it's higher than that