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# Empty sum

The empty sum is defined to be equal to the additive identity, 0, as numbers are concerned.

## Applications of the concept of empty sum

With the concept of empty sum we don't need to make many other conventions, e.g.

### Partitions of 0

The set of partitions of 0 is a set containing the empty set (the sum of elements of the empty set being the empty sum, defined as the additive identity, i.e. 0)

$P(0)=\{\{\}\},\,p(0)=|P(0)|=1.\,$ Without the concept of empty sum, we would have to make the convention that $p(0)\,$ is set to 1.

### Prime factorization of 1

For the unit, 1, there are no primes with nonzero exponents (the set of prime factors of 1 is the empty set) and we get the empty product, defined as the multiplicative identity, i.e. 1. (Cf. Empty product#Prime factorization of 1.)

Since the set ${\rm {DPF}}(1)\,$ of distinct prime factors of 1 is the empty set

${\rm {DPF}}(1)\,=\,\{\}\,$ and the number of distinct prime factors $\omega (n)\,$ is the cardinality of the set of distinct prime factors of $n\,$ $\omega (n)\,=\,|{\rm {DPF}}(n)|\,$ we get the cardinality of the empty set, i.e. 0, for $n\,$ = 1.

#### Number of distinct prime factors

The number of distinct prime factors of $n\,$ is (tautologically) given by

$\omega (n)=\sum _{i=1}^{\omega (n)}{\alpha _{i}}^{0},\,$ where we get the empty sum, defined as the additive identity, i.e. 0 (the final value of the index being lower than the initial value) for $n\,$ = 1.

#### Number of prime factors (with repetition)

The number of prime factors (with repetition) of $n\,$ is given by

$\Omega (n)=\sum _{i=1}^{\omega (n)}{\alpha _{i}}^{1},\,$ where we get the empty sum, defined as the additive identity, i.e. 0 (the final value of the index being lower than the initial value) for $n\,$ = 1.