### Author Topic: Maths  (Read 5685 times)

#### spike43884 ##### Maths
« on: April 21, 2016, 12:21:44 PM »
Q: Prove the ratio of integers to rationals is/isn't a 1:1 relationship?

Answer 1: Yes as integers and rationals go on infinitely so the ratio is infinity:infinity which simplifies to 1:1

I came out with an alternative solution.
Take the range between 0 and 1, inclusively.
All the rationals can be expressed as y/x (y over x, aka fractions) where x is any number between 1 and infinity, as y is any number between 1 and x (so in turn, can be 1 to infinity).

So to work out all the fractions we could have, we should do x*y, or infinity * infinity, which is infinity squared.
So the ratio is 1:infinity for integers to rationals.

This holds up when extrapolated to infinite integers, as you multiply both sides by infinity, getting a ratio of infinity:infinity cubed
as we found the rationals for any increment of integers by 1, we can do this double-sided multiplication.

Now I have a feeling this is wrong, as these are mathsy people who did answer 1, and I'm a secondary school'er.
Please find me error for me!
Autism can allow so much joy, and at the same time sadness to be seen. Our world is weird, and full of contradiction everywhere, yet somehow at moments seems to come together, and make near perfect sense.

#### Numsgil ##### Re: Maths
« Reply #1 on: April 22, 2016, 08:09:32 PM »
Q: Prove the ratio of integers to rationals is/isn't a 1:1 relationship?

Answer 1: Yes as integers and rationals go on infinitely so the ratio is infinity:infinity which simplifies to 1:1

[socratic question]Can this proof be used to show that the real numbers have a 1:1 relationship with the integers?  If so, consider reading this[/socratic question]

Quote
Now I have a feeling this is wrong, as these are mathsy people who did answer 1, and I'm a secondary school'er.

Your #1 is wrong, so whoever gave it to you probably shouldn't be trusted on this stuff.  See, problem is, not all infinities are equal.  See, there are infinities and then there are infinitiesThe hotel paradox is a good introduction to the idea that not all infinities are equal.  Infinities don't cancel.  You have to find other ways to get rid of them, usually involving some form of induction (prove it's true for n, then prove it's true for n+1 assuming it's true for n).

You're on the right track of breaking down the rationals to y/x, but you need to find an explicit formula that maps a given rational number to a unique integer (or natural number if you want to feel smart.)  That is, find a way to give each rational number y/x a single integer n that uniquely defines it.  Not every integer n needs a rational to go with it, though.  It's okay to have holes.  Also don't use the term "number" in your proofs, because it's necessarily ambiguous when you're talking about different kinds of numbers.  Better just to say something's a rational or an integer.

It's not an easy proof.  If you're interested in this stuff, you might pick up a book on Real Analysis.  It's a grad level course, and the later stuff gets in to calculus, but the early stuff is all about building the notion of numbers from first principles, assuming as little as possible except a basic knowledge of proofs (proof by induction especially).  The one I used for my class had sections about the history of civilizations and their concepts of what was or wasn't a number, which is also quite interesting.

#### Botsareus ##### Re: Maths
« Reply #2 on: April 23, 2016, 10:04:02 AM »
Rational numbers can be expressed as a fraction (hence the name from 'ratio')

#### spike43884 ##### Re: Maths
« Reply #3 on: April 23, 2016, 12:49:11 PM »
Q: Prove the ratio of integers to rationals is/isn't a 1:1 relationship?

Answer 1: Yes as integers and rationals go on infinitely so the ratio is infinity:infinity which simplifies to 1:1

[socratic question]Can this proof be used to show that the real numbers have a 1:1 relationship with the integers?  If so, consider reading this[/socratic question]

Quote
Now I have a feeling this is wrong, as these are mathsy people who did answer 1, and I'm a secondary school'er.

Your #1 is wrong, so whoever gave it to you probably shouldn't be trusted on this stuff.  See, problem is, not all infinities are equal.  See, there are infinities and then there are infinitiesThe hotel paradox is a good introduction to the idea that not all infinities are equal.  Infinities don't cancel.  You have to find other ways to get rid of them, usually involving some form of induction (prove it's true for n, then prove it's true for n+1 assuming it's true for n).

You're on the right track of breaking down the rationals to y/x, but you need to find an explicit formula that maps a given rational number to a unique integer (or natural number if you want to feel smart.)  That is, find a way to give each rational number y/x a single integer n that uniquely defines it.  Not every integer n needs a rational to go with it, though.  It's okay to have holes.  Also don't use the term "number" in your proofs, because it's necessarily ambiguous when you're talking about different kinds of numbers.  Better just to say something's a rational or an integer.

It's not an easy proof.  If you're interested in this stuff, you might pick up a book on Real Analysis.  It's a grad level course, and the later stuff gets in to calculus, but the early stuff is all about building the notion of numbers from first principles, assuming as little as possible except a basic knowledge of proofs (proof by induction especially).  The one I used for my class had sections about the history of civilizations and their concepts of what was or wasn't a number, which is also quite interesting.

I wish we had the ability to display fractions on this forum clearly (rather than having to go /).
I might be able to display the y/x utilizing inequalities. (Considering they can have numerous values, its probably the clearest display for them).
Autism can allow so much joy, and at the same time sadness to be seen. Our world is weird, and full of contradiction everywhere, yet somehow at moments seems to come together, and make near perfect sense.

#### Peter ##### Re: Maths
« Reply #4 on: April 23, 2016, 01:08:08 PM »
I wish we had the ability to display fractions on this forum clearly (rather than having to go /).
I might be able to display the y/x utilizing inequalities. (Considering they can have numerous values, its probably the clearest display for them).
You can display fractions clearly!

Code: [Select]
$$\frac{n}{1}= n$$
« Last Edit: April 23, 2016, 01:11:10 PM by Peter »
Oh my god, who the hell cares.