Page:Cantortransfinite.djvu/140

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OF TRANSFINITE NUMBERS
121

The ordinal type of depends, as we easily see, only on the types and ; we define

(5)
.

[503] In this product is called the "multiplicand" and the "multiplier."

In any definite imaging of on let be the element of that corresponds to the element of ; we can then also write

(6)
.

Consider a third ordered aggregate with

.

But the two ordered aggregates and are similar, and are imaged on one another if we regard the elements and as corresponding.

Consequently, for three types , , and the associative law

(7)

subsists. From (1) and (5) follows easily the distributive law

(8)
;

but only in this form, where the factor with two terms is the multiplier.

On the contrary, in the multiplication of types as in their addition, the commutative law is not