Exploring the Equivalence of NxN and N: A Proof Using Gaussian Integers

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In summary, the conversation discusses the use of Gaussian integers to show that NxN is equivalent to N. However, there is a problem with the mapping as (x,y) and (y,x) get mapped to the same integer. It is pointed out that N x N is not equivalent to N, but rather has the same cardinality as the set of rational numbers. The conversation ends with a disagreement on whether or not the rationals and the naturals have the same cardinality.
  • #1
cragar
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I thought of a way to use Gaussian integers to show that NxN~N
We look at (1+i)(1-i) and this corresponds to the coordinate (1,1)
then (1+2i)(1-2i)-->(1,2) then (1+3i)(1-3i)-->(1,3)... and you keep doing this, so we have injected NxN into N.
 
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  • #2
actually there is a problem with this (x,y) and (y,x) get mapped to the same integer
 
  • #3
It looks to me like your mapping goes from N to N x N. Is that what you intended? (1 + i)(1 - i) = 1 - i2 = 1 + 1 = 2. So here the integer 2 is mapped to (1, 1). Did you mean for it to go the other way?
 
  • #4
The fundamental problem is that N x N is NOT equivalent to N, it has the same cardinality as the set of rational numbers. It appears that your assignment is "one-to-one" but not "onto".
 
  • #5
HallsofIvy said:
The fundamental problem is that N x N is NOT equivalent to N, it has the same cardinality as the set of rational numbers. It appears that your assignment is "one-to-one" but not "onto".

But ##\mathbb{N}## is equivalent to ##\mathbb{N}\times\mathbb{N}##...
 
  • #6
HallsofIvy said:
The fundamental problem is that N x N is NOT equivalent to N, it has the same cardinality as the set of rational numbers. It appears that your assignment is "one-to-one" but not "onto".
The rationals and the naturals do have the same cardinality.
 

1. What are Gaussian integers?

Gaussian integers are complex numbers in the form of a+bi, where a and b are both integers and i is the imaginary unit. They are named after mathematician Carl Friedrich Gauss and are used in number theory and applications involving complex numbers.

2. What does the notation "NxN~N" mean in relation to Gaussian integers?

The notation "NxN~N" means that the set of Gaussian integers of the form a+bi, where a and b are both integers, is isomorphic to the set of integers. In other words, there is a one-to-one correspondence between the two sets, meaning they have the same cardinality or number of elements.

3. How is the proof for NxN~N derived?

The proof for NxN~N can be derived using the concept of Gaussian integer factorization. By showing that every Gaussian integer can be uniquely factored into a product of prime elements, it can be proven that the set of Gaussian integers is isomorphic to the set of integers.

4. What is the significance of the NxN~N proof?

The NxN~N proof is significant because it provides a mathematical framework for understanding and working with Gaussian integers. By establishing the isomorphism between Gaussian integers and integers, it allows for the use of familiar algebraic and number theoretic concepts in solving problems involving Gaussian integers.

5. What are some applications of Gaussian integers?

Gaussian integers have applications in many areas of mathematics, including algebraic number theory, cryptography, and signal processing. They are also used in physics and engineering to model and analyze systems with complex numbers.

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