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The empty product is defined as the multiplicative identity, i.e. 1.
Applications of the basic concept of empty product
With the basic concept of empty product we don’t need to make a multitude of secondary conventions, since we can just fall back on this one basic convention (in the spirit of Ockham’s razor).
Zeroth power
If we define the
zeroth power of
as the
empty product, then we don’t need to explicitly define
0 0 := 1, which is what we need for
in the
binomial expansion (see
Pascal triangle). The concept of empty product means that we ignore the base in
0 0, just what we need to get
1 as result (since otherwise
tells us that it should be
0, while
tells us that it should be
1, leaving us with an unsolvable conundrum...).

i.e. for the constant term, we need
to be
1 for any value of
, including
.
Prime factorization of 1
Considering only the primes with positive exponents, a
positive integer has a unique (up to ordering)
prime factorization
 ${\begin{array}{l}\displaystyle {n=\prod _{i=1 \atop {{p_{i}}^{\alpha _{i}}\parallel n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},}\end{array}}$
where
is the
number of distinct prime factors of
and
are the
distinct prime factors of
and
means the highest power
of
that divides
.
For prime numbers, exactly one prime exponent is positive. 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.
Without the concept of empty product, we would have to make the convention that the prime factorization of 1 is undefined (or consider 1 to be prime, like it used to be in the past!).
Factorial of 0
The
factorial of a
nonnegative integer is defined as the product of all
positive integers up to
, the factorial of zero being the
empty product, defined as the
multiplicative identity, i.e.
1.
Without the concept of empty product, we would have to make the convention that 0! := 1.
Zeroth primorial number and primorial of 0
Definition for prime numbers (primorials)
The
th primorial number, denoted
, is defined as the product of the first
primes, the
0th primorial number being the
empty product, defined as the
multiplicative identity, i.e.
1.
Without the concept of
empty product, we would have to make the convention that
.
Definition for natural numbers
The
primorial of a
natural number , denoted
, is the product of all positive
prime integers up to
, the primorial of
0 and
1 being the
empty product, defined as the
multiplicative identity, i.e.
1.
Without the concept of empty product, we would have to make the convention that 0 # = 1# := 1.
Zeroth compositorial number and compositorial of 0
Definition for composite numbers (compositorials)
The
th compositorial number, denoted
, is defined as the product of the first
composites, the
0th compositorial number being the
empty product, defined as the
multiplicative identity, i.e.
1.
Without the concept of
empty product, we would have to make the convention that
is set to
1 (unless we use the conventions for
0! and
).
Definition for natural numbers
The
compositorial of a
natural number , denoted
, is the product of all positive
composite integers up to
, the compositorial of
0, 1, 2 and
3 being the
empty product, defined as the
multiplicative identity, i.e.
1.
Without the concept of
empty product, we would have to make the convention that
(unless we use the conventions for
0!,
0 # and
1#).
Empty power tower
Since a power tower is repeated exponentiation (which is repeated repeated multiplication), the “empty power tower” gives the empty product, i.e. 1.
See also