What are irrational numbers. What does it mean when the decimal

>>9147237
>What are irrational numbers.
real numbers that aren't rational

>What does it mean when the decimal part of a number goes on forever (repeating or not)?
what exactly do you mean? all numbers can be written this way

Irrational numbers are numbers that can't be formed as a ratio of whole numbers. That's about the beginning and the end of it. Ancient Greeks formed a mathematics of geometry and ratios (not yet working with purely abstract numbers, i.e. they'd never just say "3", they'd say "a length three times of another length"). They quickly discovered that lengths existed that can't be interpreted as a ratio of another length. In particular a right triangle of two equal lengths will have a third side who cannot be interpreted as a ratio of any equal subdivision of the two equal-length sides. In other words sqrt(2) is irrational.

>>9147237
>What are irrational numbers.
Most likely, female.

>>9147237
is that elliott rodgers

>>9147244
>>9147243
But what does it mean?
How can I make sense of it in the abstract way.

>>9147517
for example,
[math]\sqrt{2} \neq \frac{a}{b} , \forall\ a,b \in \mathbb{N} [/math]

>>9147538
So, irrational numbers are just a hack?
But I guess it's a little useful.

>>9147517
You can think of it like this:
Suppose the number is a portion of a pie (1.2 representing a whole pie and a fifth of one). If you keep adding this amount to itself, you will eventually reach a whole number of pies, as this is a rational number (in this case, 5 lots of 1.2, giving 6). Every rational number has this property (although the number of times you need to add it may be extremely large).
An irrational number does not have this property; you can add an irrational number to itself an infinite number of times and still not receive a whole number.

>>9147543
It's a little bit complicated. We care about them because some important numbers that appear all the time cannot be expressed with a/b, with a and b naturals (like pi or e).

>But I guess it's a little useful.
This is relative, I do maths for the sake of it, and I don't really care if it's useful or not. Irrational numbers are a "fact" and we have to deal with them, and studying them will help us to understand mathematics even more, which is important because (...)

>>9147543
no, they just arise as values which can't be written as a fraction of any two integers. there's no "hack". it can be proven geometrically, and, in the case of sqrt of 2, originally was in ancient greece.

there are more irrational numbers than rational numbers.

>>9147553
Do you know COQ, can you write a proof for that statement?

Also, are imaginary numbers another hack that we don't really understand and just use for their usefulness in some arbitrary field?

>>9147557
>there are more irrationals than rationals
but I thought N was infinite.

>>9147543
Quite the contrary really. Irrational numbers were one of the first things that "don't make sense, but exist."

Of course modern mathematics can make ample sense of them, but for the Ancient Greeks, it was a conundrum. Two things that don't share an atomic length that they can be compared to? It was hard to even conceive of. It was the first expression of the consequences of infinity staring back at them.

>>9147568
>but I thought N was infinite.
it is, but some infinities are greater than others
https://en.wikipedia.org/wiki/Cardinality

>>9147568
N is infinite, but R is even more infinite. The concept of larger infinities (such as Aleph magnitudes) is quite substantial in the field of Analysis.

You know how infinity is, in its most simplest form, a set that is too large for any finite set to map into? Well the natural numbers are "countably infinite" and irrational numbers is a set that even countably infinite numbers can't map into.

>>9147572
I just need to make sense of it.
Ok they exist, but what does it mean (other than being a thing with the property of not being expressed as the ratio of two integers)

What about that guy that gets memed here sometimes, the one that didn't like infinity, what does he think about it.

>>9147237
>>9147538


irrational numbers are like prime numbers (and shadows), except the operation for which the number cannot be the result of is division instead of multiplication.

>>9147575
This is too much for me.
How do people arrive at these conclusions.

>>9147237
>What does it mean when the decimal part of a number goes on forever (repeating or not)?

Nothing. It's just a matter of what base you're using. In base pi, even pi is rational.

>>9147587
I'm embarrassed to say that I don't entirely remember from my Analysis course well enough to explain. It has to do with Cauchy sequences and completeness.

But at its core it's a concept of density. If you think of countably infinite numbers as rational numbers (i.e. any number a/b where a and b are real numbers), then it's not particularly difficult to demonstrate that there are WAY more numbers than the numbers that this describes. And even though you can subdivide the numbers more and more using countable patterns, in the end you can't sub-divide enough to reach all Real numbers (both rational and irrational) that can be described. At least not without already having a concept of Real numbers.

>>9147584
He probably thinks that you can't REALLY construct a right triangle with two equal sides, in fact the Platonic idea of a triangle probably doesn't exist in this world. And, honestly, that could potentially be right. But it doesn't stop the significance of the mathematical consequences that we have discovered using irrational numbers.

>>9147605
What's his name?

>>9147587
density of sets, countability/denumarability of sets. look up cantor's diagonal argument. the irrationals being a larger set is actually a very simple consequence of the reals being uncountably infinite, but if you really want to understand why.. look at cantor's argument.

set theory, i.e. math dealing with sets, sizes of infinite sets (cardinality) etc. wasn't really developed until the 1800s, and lots of people were skeptical and outright disregarded Cantor as a mathematician for his work on set theory, because, intuitively, how can one infinity be bigger than another? it was assumed that most of set theory was essentially trivial until Cantor provided it with rigorous grounding, and david hilbert, a guy with serious credibility, supported his ideas fully.

there are still skeptics but they are essentially a lunatic fringe in the math community

>>9147600
Isn't it possible that many irrational numbers do have some a/b = x in the infinite a,b set

>>9147646
If that were true they wouldn't be irrational. If they are irrational then that must mean it was proved that there are no such a,b in N.