Natural Numbers

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Math > Number theory > Natural Numbers

Natural Numbers

Properties of the set of positive integers.

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Definition of prime [tags: prime composite]
For `t in NN` where `t>1`, `t` is prime if the only divisors of `t` are 1 and `t` itself. More concisely, if `t` is prime, then:
`(AA x in NN)(x div t => [x=1 or x=t])`
If `t` is not prime (and `t>1`), then `t` is composite. That is:
`(EE u,v in NN)(u>1 and v>1 and t=uv)`
The number 1 is neither prime nor composite.
Theorem: Product of natural number factors is not less
For all `a,b in NN`, `ab>=a and ab>=b`. That is, the product of two natural numbers is at least as large as both of its factors.
Proof: We'll first prove that `ab>=a`. Because `b` is a natural number, we know that `b>=1`. This gives two cases:
  1. `b=1`. Since `n xx 1=n` for all `n`, we know that `ab=a`.
  2. `b>1`. Recall that if `x>y and c>0`, then `cx > cy`. If `x=b`, `y=1`, `c=a`, we have `b>1` and `a>0` (because `a` is a natural number), so `ab>a`.
Therefore, `ab>=a`. By a parallel argument, `ab>=b`.