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Empty product

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The empty product is defined as the multiplicative identity, i.e. 1, as numbers are concerned.

Contents

Applications of the basic concept of empty product

With the basic concept of empty product we don't need to make a multitude of conventions!

Zeroth power of 0

If we define the zeroth power of \scriptstyle x \,\in\, \mathbb{C} \, as the empty product, then we don't need to explicitly define \scriptstyle 0^0 \,:=\, 1 \, (see 0^0), which is what we need for \scriptstyle x \,=\, 0 \, in the binomial expansion (see Pascal triangle). The concept of empty product means that we ignore the base in \scriptstyle 0^0, just what we need to get 1 as result (since otherwise \lim_{x \to 0^+} 0^x tells us that it should be 0, while \lim_{x \to 0^+} x^0 tells us that it should be 1, leaving us with an unsolvable conundrum...).

(1+x)^n = \sum_{d=0}^n \binom{n}{d} \, x^d := \sum_{d=0}^n \frac{n!}{d! \,(n-d)!} \, x^d, \,

i.e. for the constant term, we need \scriptstyle x^0 \, to be 1 for any value of \scriptstyle x \,, including \scriptstyle x \,=\, 0 \,.

Prime factorization of 1

Considering only the primes with positive exponents, a positive integer \scriptstyle n \, has a unique (up to ordering) prime factorization

n = \prod_{i=1 \atop {{p_i}^{\alpha_i} \parallel n,\, \alpha_i \,\ge\, 1}}^{\omega(n)} {p_i}^{\alpha_i}, \,

where \scriptstyle \omega(n) \, is the number of distinct prime factors of \scriptstyle n \, and \scriptstyle p_i,\, p_i \,>\, p_{i-1} \, are the distinct prime factors of \scriptstyle n \, and \scriptstyle {p_i}^{\alpha_i} \parallel n \, means the highest power \scriptstyle \alpha_i \, of \scriptstyle p_i \, that divides \scriptstyle n \,.

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 1

The factorial of a nonnegative integer \scriptstyle n \, is defined as the product of all positive integers up to \scriptstyle n \,, 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 \scriptstyle 0! \,:=\, 1 \,.

Zeroth primorial number and primorial of 0

Definition for prime numbers (primorials)

The \scriptstyle n \,th primorial number, denoted \scriptstyle p_n \# \,, is defined as the product of the first \scriptstyle n \, 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 \scriptstyle p_0 \# \,:=\, 1 \,.

Definition for natural numbers

The primorial of a natural number \scriptstyle n \,, denoted \scriptstyle n\# \,, is the product of all positive prime integers up to \scriptstyle n \,, 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 \scriptstyle 0 \# \,=\, 1 \# \,:=\, 1 \,.

Zeroth compositorial number and compositorial of 0

Definition for composite numbers

The \scriptstyle n \,th compositorial number, denoted \scriptstyle \frac{c_n!}{c_n \#} \,, is defined as the product of the first \scriptstyle n \, 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 \scriptstyle \frac{0!}{p_0 \#} \, is set to 1 (unless we use the conventions for \scriptstyle 0! \, and \scriptstyle p_0 \# \,.)

Definition for natural numbers

The compositorial of a natural number \scriptstyle n \,, denoted \scriptstyle \frac{n!}{n \#} \,, is the product of all positive composite integers up to \scriptstyle n \,, 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 \scriptstyle \frac{0!}{0 \#} \,=\, \frac{1!}{1 \#} \,=\, \frac{2!}{2 \#} \,=\, \frac{3!}{3 \#} \,:=\, 1 \, (unless we use the conventions for \scriptstyle 0! \,, \scriptstyle 0 \# \, and \scriptstyle 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 tetration.)

See also

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