By Arthur Cayley

This quantity is made from electronic pictures from the Cornell collage Library old arithmetic Monographs assortment.

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More often we do not confine our attention to a single system, but consider a whole class of systems defined in the following way: we specify the number of operations and certain laws which the operations are to obey. ) and the laws which form part of the definition are often called axioms with the appropriate prefix. Thus laws (i)–(iv) below could be referred to collectively as the hemigroup axioms. To see whether a given system belongs to a given class of systems we have simply to check that it has the right number of operations and that the axioms are obeyed.

If this had not been so we should have wanted to make a similar extension to the new system, and so ad infinitum. We have an analogous situation here: if we try to extend the Cauchy field by applying the same process again, the new field is isomorphic to the old. In other words, the Cauchy field contains the limits of all its Cauchy sequences. A field with this property is called complete. 6. An important idea connected with limits is that of continuity. The intuitive idea of a continuous function f of one variable is simply that f( x) should not suddenly change by large amounts when x changes by small amounts.

2 = 1′ and so, by (ii), x + 2 is defined to be (x + 1)′, that is, x″. Then x + 3 is x‴, and so on. Similar arguments apply to multiplication. Multiplication is essentially repeated addition. “Three fours” is 4 + 4 + 4. Thus, having multiplied m by n, to multiply m by n + 1 we need only add an extra m. This gives us the following definition of multiplication: From these two definitions and our axioms we can now develop the complete theory of whole numbers. In particular we can prove the laws of arithmetic, one of which—x + y = y + x—has already been mentioned.