Prove that there are no integers between 0 and 1?

Question

Please don't tell that it is how integers are distributed, or that is definition of integers and so on. I found this problem in a book, and so please try to provide some proof based on number theory. I have some basic number theory background.

Answer 1

I got so tied up in complicated possibilities, I overlooked the simple. Here's your proof using number theory.

Use the well-ordering principle of the positive integers (sometimes called the least-natural-number principle, or more informally, the "minimal criminal"). Every nonempty set of positive integers contains a smallest element.

Suppose that there exists an integer between 0 and 1. Then the set of positive integers between 0 and 1 is nonempty, so by the well-ordering principle, let n be the smallest such integer. Now consider n^2, which is a positive integer since n is a positive integer. Since n<1, we know that n^2
And to those who think the question is pointless, I couldn't disagree more. The point of the exercise is to examine what makes 1 ("unity") unique in the scheme of the natural numbers. In my 30 minutes of tinkering with ideas, I came across some interesting things. I was almost able to construct infinitely many integers between 0 and 1. The only reason I couldn't was... the well-ordering principle! To me, that's interesting. I learned something new, and I've already taken number theory.

Answer 2

Answers to a question this basic only make sense within a given axiomatic system. In some systems, it might be an axiom that there is no integer between 0 and 1; in others, it must be proven from more primitive notions. So before you try to answer this, you should look at your book and answer two other questions: (1) What is an integer? And (2) What does it mean to say a is between b and c?

Unless you can answer these questions, you can't give a proof.

Answer 3

there are a few integers between o and 1 i'll prove it and i'm only ten. Integers are just positive numbers and negetive numbers so an integer between 0 and 1 is 0 1/7 and 0 2/4 see really easy you should have learned it in 5th grade.

Answer 4

What book did you find this weird question in?!? BY DEFINITION, and it's nonsense to say NOT to use this since we MUST rely on definitions, at least, of stuff in order to prove things, every integer positive number k can be expressed as 1 + 1 +...+ 1 (k times), and this is so whether you rely on Peano's Axioms or whether you use the definition of the integers Z as (an abelian) group under addition.
Then, as a direct consequence of either basic assumption as above, for ANY positive integer k DIFFERENT from 1 we get that 1 < k. Q.E.D.
Of course, non-positive integers are out of the question since we require between 0 and 1 and thus positive ones.
That's all, whether you like it or not.
Now, if you have ANOTHER set of basic assumptions, axioms or whatever according to which you wanna prove this, then you better tell about it...
Regards
Tonio

Answer 5

A basic outline of a proof would go like this.

You can construct the the set of all integers by adding or subtracting 1 from zero any number of times.

So if there was a integer greater than zero and less than one, theyou n should be able to find it by adding 1 to zero a certain number of times.

but 0=0
1=0+1
everything else is greater than 1 (0+1+,..,+1)
therefore there are no integers between 0 and 1.

Answer 6

An integer is a whole number (not a fraction) that can be positive, negative, or zero. Therefore, the numbers 10, 0, -25, and 5,148 are all integers. Integers cannot have decimal places.

Between 0 and 1 on the number line there are only fractions, decimals, and irrational numbers such as 1/sqrt(3

Although you don't want to hear it there are no integers between 0 and 1 because of the very definition of integers.

Answer 7

im not sure bout this but i think i have something

so lets say we know that integers are 0,1,2,3,4,5,6.....
lets look at the positive one only including zero.

okay lets construct an equation from the stuff we know

lets say we a re suppose to find whether there are integers between 4 and 1
we know that there are 2 integers between 4 and 1

now are there integers between 10 and 6
there are 3 integers between 10 and 6

i figure u can see the pattern now

4 - 1 = 3 (there are 2 integers)
10 - 6 = 4 (there are 3 integers)

so we can say hte number of integer between two integers are given by

number of integer = m - n - 1; where m > n;

to see wheter there is an integer between 0 and 1.

(1 - 0 - 1) = 0 (this means no integer between 0 and 1)

hope i answered your question

Answer 8

integers are whole (fractionless) numbers. therefore no integer between 0 and 1 exist because ALL numbers BETWEEN 0 and 1 are fractions.

Answer 9

integers are whole numbers like 1,2,3 and not 0.1,0.2,0.3, and these decimals are not in the whole numbers range and these decimals above from 0.1 - 0.9 are the only numbers btn 0 and 1 hence they are not integers

Answer 10

i can prove that there are numbers between them like 0.1 0.2 0.3 bla bla bla