Empty product

The empty product is in mathematics, the special case of a product with zero factors. He is assigned the value one.

In combinatorial, enumerative considerations the empty product is normally involve, since there is exactly one way to multiply Nothing, which is why it is justified to speak of the empty product. It is to be distinguished from the product or a product with only a single factor (which will be equal to this factor).

In other areas such as groups, ring or field theory, in which the multiplication is considered as fundamental, inner join, each definition with less than two factors is not initially make sense. Nevertheless, the empty product emerges implicitly in several contexts, eg when powers and the faculty, and there is occasionally the reason for understanding problems. The common value assignment to one is not always intuitively obvious.

Relating to powers and the empty sum

Analog is the addition of 0 summands as the empty sum and gives it the value zero. This is clearly justified: When adding nothing to get nothing (nothing = zero is the neutral element of addition).

Now apply to any finite product factors and the logarithm to any base:

This is set to the empty product and gets right to the left in the exponent of the empty sum:

Since the assignment of the empty sum to 0 is very plausible, the empty product in the sense of consistency shall be given the value of which at least must also be constant for all.

Problems of value assignment

It is common to define for real. Thus, the real-valued exponential functions are continued steadily and analytically in point. In the complex numbers, it is a bit more complicated, since there is a branch point, for real, it stays there properly. Thus there is nothing against

A blemish is apparent when one tries to generalize on. To put the power is still compatible with the most common definitions, since it is valid for all: this provides for in the function with a discontinuity. See also " nullhochnull ".

Empty Cartesian product

The Cartesian product of two sets is defined as the set of all ordered pairs. Normally, you can this for any index set defined as follows:

Applies now

Then the -th power of any amount (even ) is given by

Thus results for the empty Cartesian product:

Because as a special relation

Since the numbers of set theory can be defined as and, further follows:

With the help of the Cartesian product is obtained as a satisfactory interpretation in the assignment of

Other correlations

• Considering the one that has no prime factors there is consistent, assigning it the empty factorization, that the empty product.
• Just as the empty sum is equal to the neutral element of addition, the empty product is equal to the neutral element of multiplication.
• From the definitions of empty product and faculty follows:
• There is exactly one way to select anything from pieces - shall apply mutatis mutandis for the binomial coefficients, in particular. They can be directly attributed to the Faculty of zero.