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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)

${\displaystyle P(0)=\{\{\}\},\,p(0)=|P(0)|=1.\,}$

Without the concept of empty sum, we would have to make the convention that ${\displaystyle \scriptstyle p(0)\,}$ 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 ${\displaystyle \scriptstyle {\rm {DPF}}(1)\,}$ of distinct prime factors of 1 is the empty set

${\displaystyle \scriptstyle {\rm {DPF}}(1)\,=\,\{\}\,}$

and the number of distinct prime factors ${\displaystyle \scriptstyle \omega (n)\,}$ is the cardinality of the set of distinct prime factors of ${\displaystyle \scriptstyle n\,}$

${\displaystyle \scriptstyle \omega (n)\,=\,|{\rm {DPF}}(n)|\,}$

we get the cardinality of the empty set, i.e. 0, for ${\displaystyle \scriptstyle n\,}$ = 1.

#### Number of distinct prime factors

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

${\displaystyle \omega (n)=\sum _{i=1}^{\omega (n)}{\alpha _{i}}^{0},\,}$

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 ${\displaystyle \scriptstyle n\,}$ = 1.

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

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

${\displaystyle \Omega (n)=\sum _{i=1}^{\omega (n)}{\alpha _{i}}^{1},\,}$

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 ${\displaystyle \scriptstyle n\,}$ = 1.