# Real-numbers sentence example

But among the

**real numbers**all classes have limits.The arithmetic of

**real numbers**follows from appropriate definitions of the operations of addition and multiplication.The addition and multiplication of these "signed"

**real numbers**is suitably defined, and it is proved that the usual arithmetic of such numbers follows.But now the irrational

**real numbers**have all made their appearance.Such a number is a "one-many" relation which relates n signed

**real numbers**(or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers I, 2..AdvertisementThis is exactly the same reason as that which has led mathematicians to work with signed

**real numbers**in preference to**real numbers**, and with**real numbers**in preference to rational numbers.The application of the theory of

**real numbers**to physical quantities involves analogous considerations.Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of

**real numbers**.Complex numbers are conveniently treated in connexion not only with the theory of equations but also with analytical trigonometry, which suggests the graphic representation of a+b,l - by a line of length (a 2 +b 2)i drawn in a direction different from that of the line along which

**real numbers**are represented.Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and

**real numbers**generally, which include incommensurable numbers.AdvertisementThe importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of

**real numbers**and of signed**real numbers**.Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of

**real numbers**.The

**real numbers**will probably be similar to what the calculator determines, but the exact dollar amount may vary slightly.