Rational numbers are impossible!


#1

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#2

Numbers may not be rational, but they are always sensible.
Because one plus one equals a bigger one and all equations equal infinity…


#3

I love math; conceptually. Applying artificial barriers around concepts (the natural world) to try and make them understandable or at the very least usable as a resource to humanity. Humans (and other living beings) seem to really like the idea of predicability and maths help us accomplish that (how many people in the village? how much land will we need to harvest/hunt upon? how much shall we harvest and store vs. use? how many deer will we need to feed the honoured guests?).

postscript, money is only useful if it is being used…


#4

I’m probably not the target audience for these videos, but I’m very glad they exist. It’s impressive how she manages to touch on multiple areas of more advanced mathematics, while sticking to a popular format.


#5

I can’t pretend I understand all of this, but the 0% thing doesn’t make sense to me. Just because the dart has a 0% chance of hitting a specific real number you choose in advance doesn’t mean it won’t hit a different real number on its own. Even if it’s impossible to identify which number it did hit, that doesn’t mean it will never hit a real number. In fact given infinite throws, doesn’t the chance of hitting a real number at least once become 100%? If so, how can it ever have been 0%?


#6

If I follow what’s going correctly, it’s because you’re dividing by infinity. These maths start to get pretty counterintuitive.

One of these days I’m going to get around to learning more math than the smattering of self-taught calculus I have now.


#7

It may help to think of it this way - I really hope this isn’t more confusing.

The number line you usually see contains only integers, whole numbers (and zero). It goes on forever to the positive and negative, and so the number line itself is infinite. That number line has no smaller division than the whole numbers. “…-3,-2,-1,0,1,2,3…”

What Vi is talking about is a number line containing rational numbers - fractions.

That number line also includes the space between the whole numbers, and that space is bigger that you might think it think is, because that space can be infinitely divided. It isn’t truly represented by the picture on the page. In fact, if you want to find every possible fraction between “1” and “2” you can’t — that number is infinite as well!

So, (even though a TARDIS isn’t infinitely big) it’s like someone has a TARDIS (that’s the number line), and then sitting inside that TARDIS is another series of TARDISes. (Because there’s a new infinity between each whole number set, and each one is infinitely big.)

Because there are so very many numbers between the whole numbers on the number line, the odds are basically infinitely stacked against you ever hitting a whole number.


#8

This is more-or-less because the concept of infinity is hard to grasp. There are “infinitely many” integers (i.e. whole numbers), “infinitely many” rationals, and “infinitely many” irrationals, however it is still possible to compare the “sizes” of these different collections of numbers. There is a certain sense in which there are “as many” integers as rationals, but there are far more irrationals (and hence more real numbers) than either. Infinitely more, in fact.

If you use the standard intuition for probability (number of favorable outcomes divided by number of total outcomes), taking into account that there are “infinitely more” reals than rationals, you would get that the probability of hitting a rational number is 0.

If you want a more rigorous explanation, then it would be a good idea to look up the concept of cardinality of a set. It gets pretty weird.


#9

Great explanations, thanks, and I realize I confused real numbers with
rational numbers in my original question. Even so, it seems like the
probability approaches zero or calculates as virtually zero. Is my limited
brain just being stubborn?


#10

I think she has it exactly backward. You can only hit rational numbers on the number line.
Consider 1/2, that is the ratio of 1 to 2, not one divided by two. We can hit that with a dart every time, if you mean by “hitting with a dart” “retrieving the number’s value”.
It’s the irrationals that you can’t represent exactly, except symbolically. You can find a rational number as close as you please to an irrational number, but it will be only be the sum of a finite part of the infinite sequence in every case.
Some rational numbers (e.g. 1/3) must be represented as an infinite sequence of repeating digits in base 10, but that is irrelevant. 1/3 is a finite concept. Any number of infinite sequences will sum to 1/3, but 1/3 remains simple and finite.


#11

The metaphor of a dart is about picking a number at random rather than writing out an exact value. It’s true there are more real numbers than there are ways to write a finite string of symbols.

But as far as that goes, why would you imagine 0.5 or 1/3 are any less symbolic than √2? All of those represent simple operations applied to integers, defining numbers exactly, even if one is a little more challenging to convert to a decimal representation.


#12

Hello Mike, welcome to the BoingBoing forums :smile:

I think by “hit with a dart” she means: randomly select one of the infinite number of points on the real number line.

Put another way, let’s make it the real number haystack. We’ll let every one of the irrational numbers be represented by a strand of hay. Now in that infinite pile of hay, we’re going to have an infinite number of needles, which will each stand for a rational number.

The reason the chances of drawing one of the needles from the haystack is 0% is because of the different size (cardinality) of the infinities involved. There are a bunch of needles, make no mistake, but for every needle, there is an infinite number of pieces of hay. So, the chance that the thing you draw from the haystack is a needle is 1/infinity.

We don’t have a good way to understand what that means until we approach it with the concept of limits. We say “limit of f(x)= 1/x as x approaches positive infinity” and it turns out, that’s 0.

Actually, it’s that every rational number, when expressed as a “decimal” is an infinite sequence of repeating digits, no matter what the base. It so happens that some rational numbers tail out with an infinite sequence of repeating zeros, which we customarily don’t bother to write.

It’s about a wash as to whether it’s tidier to write 1/3 or 0.3… But it’s definitely easier to write 1/7 than it is to write 0.142857… In most cases the representation of a rational number as a ratio is more compact.


#13

Keep in mind that my explanation is very informal, since it basically amounts to saying “even though they’re both infinite, there are infinitely many more real numbers than rational numbers”.

Part of the problem (essentially, what makes it counterintuitive) is that there’s a lot of machinery needed to even correctly state the question. Standard probability computations work fine when you’re dealing with finite situations, but they run into trouble when you’re looking at something “sufficiently infinite”.

That is, assuming everything is random, the odds of getting a coin to come up heads twice in a row is 1/4. There are 4 possible outcomes (HH, HT, TH, TT) and 1 favorable outcome (HH). So, you can just divide 1 by 4 to get the answer.

In the real numbers case, however, you can’t just divide by infinity, so you need to correctly define what it means to determine the probability of hitting a number before you can even ask what is the probability of hitting a real number.

The correct approach is to define a specific way of measuring probability for continuous spaces (i.e. ones that don’t have any gaps; consider real numbers vs. rational numbers or finite sets). This is again somewhat complicated, and it’s where you might encounter probability distributions (e.g. the normal distribution) or even probability measures (special functions that mimic the standard probability computation outside of finite/discrete cases).

Using these notions it is possible to rigorously state what we mean by “the probability of an arbitrary number on the real number line being rational is 0” (the actual statement would look more like: “the measure of the set of rational numbers on the real number line is 0”).


#14

Hi Mike, I hope you don’t feel piled up on.

The irrational numbers just sit between the rational numbers. You can give them any useful name, and you can estimate their location on a number line in relationship to another number. After all, we know that “pi” sits between “3” and “4” on an integer line, and is between “3.125” and “3.25” on a rational number line.

Because you can’t express irrational numbers algebraically, you have to arrange them in relationship to real numbers and other irrational numbers, but they’re still countable/estimate-able.

I agree with chenille that metaphor of the dart is about chance - not physicality. Vi is trying to express how odds/probability are effected by the multiple types of infinities contained on a single number line. Here’s a simplified example:

In this integer set, I’m counting by tens: -10,0,10

There are three items, and my chance of hitting my choice is 1/3. (33%)
My odds of hitting any of the three is 3/3. (100%)

In this integer set, I’ll count by ones:
-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10

There are 21 items and my odds have changed.
Now my odds of hitting any one I choose is 1/21. (5%)
My odds of hitting any of the original 3 is 3/21. (14%)
My odds of hitting any of the new numbers is 18/21. (86%)

So that’s just from changing a short number line with easily counted numbers, and Vi is talking about how you fill the gaps with infinities.


#15

I don’t think that’s what she meant, but I like what you said!


#16

No, that’s not true

Examples of countable sets include the integers, algebraic numbers, and rational numbers. Georg Cantor showed that the number of real numbers is rigorously larger than a countably infinite set, and the postulate that this number, the so-called “continuum,” is equal to aleph-1 is called the continuum hypothesis. Examples of nondenumerable sets include the real, complex, irrational, and transcendental numbers.

Algebraic numbers are roots, and ratios of roots. Things such as pi and e and so forth are transcendental. Only a few of the transcendental numbers have names, but there are infinitely more of them than there are of algebraic numbers, or integers, or rational numbers)


#17

Edited for clarity: I wasn’t referring to the set of all irrationals - I was referring to their individual placement on a number line. Sorry if that wasn’t clear somehow.

To better explain - pi, which we often estimate at 3.14159, rests between 3.14 and 3.15 on a number line. I was just explaining to Mike how you could fill a number line with irrational numbers - by knowing the location of the rational numbers surrounding them, you can name the placement of the irrational numbers between.


#18

Watch it all the way through. Wait for it… wait for it… the last 3 words are gold.


#19

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