How can irrational numbers exist on a number line?

>>8449985

>How can pi exist on a continuous number line?

How can't it?

You just draw stuff. Of course, you can bound the expansion of pi after the 10^x decimal and get down to distances small as atoms, something you can't resolve in reality.
The reals are merely a model for space. The assumption that our world is a manifold with a commutative coordinate function algebra is also just a physical assumption, there may not even by "points".

>>8449991

Because if we got infinitesimally close to 3.141592... we could always add decimals but we would never reach the value of pi - only approximations of it. I'm quite new to proper mathematics to be honest so I'm not explaining myself very clearly. Basically I just don't understand how irrationals - which as I understand are not just decimal approximations of numbers - can be represented on a continuous number line built around representing decimals

>>8449985
You claim to have a continuous line. We want that. Suppose we have a line made of only rational numbers. Lets call that set Q

Now consider the function f(x)=x^2 - 2

Now consider the graph of this function. Graph it and look a it. The curve passes through the x axis twice but this it actually touch it? Well...
x^2 - 2 = 0 iff
x^2 = 2

But there is no rational number that satisfies this. Therefore there is a gap in that graph. A gap between the upper graph and the lower graph. A discontinuity.

But the graph is f(Q) which is equivalent to Q. If Q is a line then f(Q) is that line curves a little.

So Q must have discontinuities so it is not the line we want.

That said, this only proves the existence of algebraic numbers. PI acually has many logical numbers that Wildberger explains thoroughly. Most likely, PI doesn't really exist.

>>8450002

>we could always add decimals but we would never reach the value of pi - only approximations of it.

Everything is an approximation. Numbers themselves are just symbols that represent values. The scale at which the dot you're placing at the point of pi is written at is way too large for anyone anywhere near the sort of precision you're talking about. If you had a magical perfect pi marker its place on the line would look indistinguishable to the place on the line a normal person would stick it at.

irrationals are even more "common" than rationals.

File: sqrt2.png (17KB, 350x500px) Image search: [Google]

17KB, 350x500px

>>8449985
Why would rationality have anything to do with it?

>>8450002
> continuous number line built around representing decimals
Uh, what?

What's the connection between the number line and decimals?

>>8449985
If infinite decimal expansions distub you, then you should also have a problem with rationals
0.333333... is just an aproximation of 1/3, which is a rational
However, 1/3 in base 6 is exactly 0.2

>>8450002
I don't really understand what you are trying to say, but it's possible for pi to exist on a continuous number line, same as decimals, it's not possible to exactly put a point where pi is located because its a infinite number and therefore a representation of it in the x axis sometimes is just the closer decimal exa 3,14 or 3,141592

>>8450112
wrong
thanks for the bait tho

>>8449985
They don't anon.

Irrationals are limit points of Cauchy sequences of rationals that we "fill the gaps" with. They are uncountable, so trying to pin point exactly where it is on a number line is hopeless.

I think you're confusing things, OP. Just because we usually represent numbers on a line with a rational scale doesn't necessarily mean it's the only way we have to represent numbers. Consider, for example, a number line that has a scale of pi. In that case, locating pi would be quite simple, even though it has infinite decimals. What I mean is that just because you can't express pi as a number with finite decimals doesn't mean that you can't represent it in a continuous number line.