The Continuity of Irrational Numbers

Are the irrational numbers continuous or discontinuous? They don't seem to take exactly fixed positions on the number line. For example, if you zoom in on pi the end list of decimal places continuous to grow to an infinite number of places - implying a nonexistent place on the number line. Alternatively, you can drive at 15 miles per hour but the car never goes through pi miles per hour.

intermediate value theorem assuming velocity is continuous implies that you do, just for a moment go at exactly pi miles per hour

How so? Wouldn't the car get stuck because of pi's endless decimal expansion? It would enter eternal slow-motion or undergo a similar phenomenon.

It's simply impossible for us to enumerate it with decimals, but the same applies to 2/3s. The fact that we can't use decimals to completely specify its position on the number line doesn't mean it doesn't exist.

You're confusing a mathematical abstraction with actual function. Cars don't have to "read out" all the decimal places.

bad b8

I know that it certainly exists. But what I'm trying to get a sense of its actual geometric meaning with respect to continuity and discontinuity. The car is just a rough example. My problem is that I think that if an irrational number is a distance too, then it is infinitely long but that contradicts any property of continuity because there aren't any numbers after it.

how could sqrt(2) be infinitely long when it is clearly less than 2

That is my problem. Irrationals don't make sense as quantities. There aren't any irrational quantities of synthesized chemicals because the atoms of the substances are discrete, whole, and finite in amount.

pi miles is a little more than 3 miles. Less than 3.2 miles. What more do you need to visualize?

By continuity I think you mean density, continuity is only defined for functions. A number X is dense in the space of real numbers if for any real number D, there exists a real number within the interval (X, X+D). You can prove that irrational numbers are always dense.

Just think of it as an approximation in the real world, what's the problem here?

It's in between 3 and 3.2 but it's not a cardinal number.

OP there are infinitely more irrational numbers than rational numbers. Just think about that for a moment.

*indescribable numbers than describable

kek whoops

>what's the problem here?

They aren't a good measure of distance. There is not a pi long piece of bread because the atoms are finite and discrete.

Any way I was thinking about irrational numbers because of the Hodge conjecture. It seems like an obvious problem because if the composition of the manifold in question is all rational then the manifold is of a continuous function. But any manifold is of a continuous function.

you definition of "continuous" is unclear. Mathematically the existence of reals poses no meaningful problem (in b4 wildberger). Physically, the fact that you would need infinite decimals to describe an irrational length (in terms of human units) does nothing to prevent an irrational length from existing in space, because again, it's relative to a human unit.

If you draw a right triangle with legs of equal length, the hypotenus of that triangle will be incomensurable to the legs, but clearly there is no "discontinuity". It just says something about our ability to discuss those lengths in terms of ratios, but there is nothing wrong with the actual length existing in space.

Nothing that it can't exist. Just that the label for such a length would be meaningless because it would just be an indefinitely long length. Unbounded positive infinity basically.

Only if you insisted on using decimal. There's no reason to do that, or believe all distances in space must be described in decimal.

Uh, am I missing something? The decimal expansion is the essence of the irrational number.

The real numbers exist independently of any decimal expansion.

If you'd like, you could look up the dedekind cut or cauchy sequence construction of the reals, but in either case, real numbers are not they're decimal representation, they are constructed prior to that.

You can also just say the real numbers are an ordered field with the least upper bound property. Then you can define which real number a decimal expansion is pointing to, but again, the real number existed before that.

>There is not a pi long piece of bread because the atoms are finite and discrete
the space between the atoms is finite but not discrete, then
and there is a piece of bread pi long

>The Continuity of Irrational Numbers
TRIGGERED

Continuous isn't really defined that way. The set of real numbers is connected, if that's what you mean. The rational numbers and irrational numbers (individually, obviously the union of the too is the set of real numbers) are disconnected, though.

>implying 15 miles per hour isn't 4pi yurrbovs per weishcabs

>4pi yurrbovs per weishcabs

Quantum Gravity may have some disagreement there desu.