Algebraically Representing Integers

Dec 2019
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I'm wondering about algebraically representing a number as belonging to a certain set. For example, you can algebraically represent a rational number as \(\displaystyle \frac{p}{q}\) (which is useful in proofs involving rational numbers, such as the proof that \(\displaystyle \sqrt{2}\) is irrational). I'm also including functions as an algebraic representation, so representing a positive number as \(\displaystyle x>0\) counts for my purposes (I just want something that I can manipulate and use in solutions).

Is there a way to similarly represent integers while using only the algebraic operators?
 

Jhevon

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Feb 2007
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Interesting question. As far as I know, the answer would be "no".

Here's as far as I've seen anyone go with it: Define the natural numbers, some include zero, some don't. However, the Naturals are defined axiomatically. Meaning, they are assumed to be things that exist that don't need to be justified. We would define them based on something like the Peano axioms. We would denote the set of natural numbers as \(\displaystyle \mathbb{N}\), and think of it as \(\displaystyle \mathbb{N} = \{ 0,1,2,3,\dots \}\).

We would then define the integers as \(\displaystyle \mathbb{N} \cup (- \mathbb{N})\), where I'm using \(\displaystyle -\mathbb{N}\) to denote the negatives of all the natural numbers. We then denote the set of integers by \(\displaystyle \mathbb{Z}\). Where the "Z" comes from the German word for number, "zahl".

But honestly, after you do that, we usually just invoke them by saying something like, "Let \(\displaystyle n\) be an integer." or "Let \(\displaystyle n \in \mathbb{Z}\)." And no one questions it. The integers are thought of as the building blocks, the atoms of number systems. We build the other number sets from them via operations, but the integers themselves just...are.
 
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Dec 2019
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I was asking because I am wondering how you would go about prove something to be true for natural numbers specifically and not just for all real numbers or rational numbers (which would not necessitate the former because the integers are a subset of the rationals). How do people ever prove anything about naturals and integers beyond their basic properties?
 

Jhevon

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Feb 2007
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I was asking because I am wondering how you would go about prove something to be true for natural numbers specifically and not just for all real numbers or rational numbers (which would not necessitate the former because the integers are a subset of the rationals). How do people ever prove anything about naturals and integers beyond their basic properties?
Well, that's a very broad question! It would depend on what it is that you want to prove. But the area of math that deals with what you're asking is called Number Theory, and it's pretty extensive.

However, the farther you go in math the more you will realize the following: it doesn't matter what you are, it matters what you do. And eventually, there comes a point where most definitions are formulated that way. We call an object X by This-Name if it performs This-Action. And so, what you will realize, it matters less what the integers are, but rather, how they behave. They will have certain properties and behaviors with regards to operations we'd want to do that would distinguish them, from say, the real numbers. For example, they are "closed under addition and subtraction" but they are "NOT closed under division". So we would know in a proof, for instance, that we'd have to be careful with how we deal with division, but sometimes less so with addition, etc. But this is all getting very vague. Pick up a Number Theory text and start the journey!
 
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That's a very interesting way of thinking of numbers, I'll definitely read up on that! thank you so much for pointing me in the right direction!
 

Plato

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I'm wondering about algebraically representing a number as belonging to a certain set. For example, you can algebraically represent a rational number as \(\displaystyle \frac{p}{q}\) (which is useful in proofs involving rational numbers, such as the proof that \(\displaystyle \sqrt{2}\) is irrational). I'm also including functions as an algebraic representation, so representing a positive number as \(\displaystyle x>0\) counts for my purposes (I just want something that I can manipulate and use in solutions). Is there a way to similarly represent integers while using only the algebraic operators?
To Certusic, I think that you are simply out of you depth with this question. That may seen harsh but it is true.
You should have a textbook such as Paul Helmos' Naive Set Theory or James Henle's An Outline of Set Theory.
The natural numbers are well defined using set theory: zero is \(\displaystyle \emptyset\)
One Is \(\displaystyle \{0\}\) , Two is \(\displaystyle \{0,1\}\). Three is \(\displaystyle \{0,1,2\}\), etc, etc.
 
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Jhevon

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Feb 2007
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That's a very interesting way of thinking of numbers, I'll definitely read up on that! thank you so much for pointing me in the right direction!
Well, it's a way of thinking in general. Often the properties of something, or how it behaves in relation to something else is what's important. But yes, I think you should just pick up a basic number theory text if you're interested in that sort of thing.

EDIT: Plato's suggestions are good too. A good Set Theory text would also be good to read. Set Theory texts can get quite hairy if you haven't done advanced math though, so it depends on what level you're at.

If you've never done proofs seriously in your life, and you've only done up to say, calculus 2, I'd probably say a Discrete Mathematics text would be nice, or something like this book.