By John Ewing

This can be the tale of yankee arithmetic up to now century. It includes articles and excerpts from a century of the yank Mathematical per month, giving the reader a chance to skim all 100 volumes of this well known arithmetic journal with out really commencing them. It samples arithmetic yr by way of yr and decade by means of decade. The reader can glimpse the mathematical neighborhood on the flip of the century, the talk approximately Einstein and relativity, the debates approximately formalism in common sense, the immigration of mathematicians from Europe, and the frantic attempt to prepare because the conflict begun. newer articles care for the appearance of desktops and the alterations they introduced, and with a few of the triumphs of recent examine.

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Any number is less than its following and greater than its preceding. All natural numbers being positive integers, there is no natural number less than one. Hence 1 is the smallest natural number and it has no predecessor but it has a successor ; every other natural number has a unique ( i e. one and only o n e ) predecessor and a unique successer. If q, r, s, t e N. then we also have ( i i ) i f r > 5 , r + t>s + t, ( i i i ) if J">s, rt > st, ( i v ) i f r > s and tyq, r-\-t>s + q, ( v ) if r > 5 and tyq, rt>sq.

Similarly if A ~ \ a, b, c, d. , 10 j, then the two sets can be 43 44 : COLLEGE ALGEBRA put in one-one correspondence and we may write this relation as n (A) = 10. , n, n + 1, .. . j. Observe that ( i ) N and A are infinite sets; ( ii ) A is a proper subset of N, since 1 e N and 1 g; A ; (iii ) there is one-one correspondence between the infinite set N and its proper subset A; and ( iv) the two sets N and A are equivalent. Thus, an infinite set can have one-one correspondence with its proper subset whereas a finite set cannot have one-one correspondence with any one of its proper subsets.

We can also see that the set N U ) 0 j is closed for the operations of addition and multiplication and that these operations are commutative and associative We can also see that the distributive property of multiplication over the sum is also true in this set. It may also be noted that the set N U ) 0 { has both the identities-the additive identity " z e r o " and the multiplicative identity " one ", ( also called unity ). REAL NUMBERS : 4 9 3. The Integers. The set N U ) 0 j is closed for the operations of addition and multiplication.