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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 $\scriptstyle p(0) \,$ 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 $\scriptstyle {\rm DPF}(1) \,$ of distinct prime factors of 1 is the empty set

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

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

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

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

Number of distinct prime factors

The number of distinct prime factors of $\scriptstyle n\,$ 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 $\scriptstyle n \,$ = 1.

Number of prime factors (with repetition)

The number of prime factors (with repetition) of $\scriptstyle n\,$ 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 $\scriptstyle n \,$ = 1.