numbers has $\frac{\G1}{2}$ elements and the set of integers has $2\mbox{\G1\small{+}}1$ elements. Thus, the new numeral system allows one to distinguish within countable sets many different sets having the different infinite number of elements (remember again the analogy with the microscope). Analogously, within uncountable sets it is possible to distinguish sets having, for instance, $2^{\mbox{\tiny{\G1}}}$ elements, $10^{\mbox{\tiny{\G1}}}$ elements, and even $\G1^{\mbox{\tiny{\G1}}}-1$, $\G1^{\mbox{\tiny{\G1}}}$, and $\G1^{\mbox{\tiny{\G1}}}+1$ elements and to show (see [@first; @Lagrange]) that $$\frac{\G1}{2} < \G1 < 2\mbox{\G1\small{+}}1 <
2^{\mbox{\tiny{\G1}}} < 10^{\mbox{\tiny{\G1}}} <
\G1^{\mbox{\tiny{\G1}}}-1 < \G1^{\mbox{\tiny{\G1}}}<
\G1^{\mbox{\tiny{\G1}}}+1.$$

Formally, grossone is introduced as a new number by describing its properties postulated by the *Infinite Unit Axiom* (see [@informatica; @Lagrange]). This axiom is added to axioms for real numbers similarly to addition of the axiom determining zero to axioms of natural numbers when integers are introduced. Inasmuch as it has been postulated that grossone is a number, all other axioms for numbers hold for it, too. Particularly, associative and commutative properties of multiplication and addition, distributive property of multiplication over addition, existence of inverse elements with respect to addition and multiplication hold for grossone as for finite numbers. This means that the following relations hold for grossone, as for any other number 0 