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I am just learning some higher level mathematics, and there is a continuous emphasis on definitions and theorems, which ultimately form the foundation of proofs.

I am trying to look at what sort of information should go into a formal definition.

For example, i came across the formal definition of a composite number:

'A natural number $n \in \mathbb{N} $ is called a composite number if $n \neq1$ and there exist $a, b\in \mathbb{N} $ with $1 \lt a, b\lt n$ such that $n=ab$.'

This captures the notion of a composite very precisely, but I'm not sure why you would want to explicitly state '$n\neq1$', as, if $n$ is composed of $a, b \gt 1$, then surely it is obvious that $n\neq 1$?

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    From which source / reference did you get this "definition" for a composite number? – 2017-01-18
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    It already follows from $1\lt a\lt n$ that $n\ne1.$ So you are right, the definition is redundant. Personally, I would have left out $n\ne1.$ On the other hand, redundancy is not all bad and to be avoided at all costs. – 2017-01-18
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    @JoseArnaldoBebitaDris lecture notes. – 2017-01-18
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    Perhaps the motivation for such a definition is to partition the natural numbers into the unit $1$, the primes and the composites? Nonetheless, as it stands, indeed the definition is redundant. – 2017-01-18
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    Maybe the issue is with the def of [Prime](https://en.wikipedia.org/wiki/Prime_number) : "A *prime* number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself." We want to exclude $1$ from primes, and we want that the def of *composite* is the negation of that of *prime*, but at the same time we want that $1$ is not a *composite*... Thus we have : "A natural number greater than $1$ that is not a prime number is called a *composite* number." – 2017-01-18

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