Fundamentals of Freshman Mathematics by Carl Allendoerfer

By Carl Allendoerfer

Moment Library reproduction. San Diego Air and area Museum.

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In the above example, x = Illustration 2. " All that we have to do is to find a single odd number whose square is odd. Since 3'^ = 9, we have established the disproof. We close with this warning: Although disproof by counterexample a valid procedure, theorems are not to be proved by verifying them in a number of special cases. Be sure that you do not confuse these is two ideas. 8 mathematical statements, some of Prove those which are true, and dis- In Probs. 1 to 18 you are given a series which are tru« and some of which are false.

The collection of all these, the rationals do not have is called the system of real numbers. It is quite a completely satisfactory definition of a real number, but for our present purposes the following will suffice: plus the irrationals, difficult to give Definition: A real number is a number which can be represented by an infinite decimal expansion. " Chap. 2. 2. Addition of Real Numbers + Addition is defined for pairs of real numbers such as 2 3 = 5, 21 = —1, etc. Indeed the sum of every pair of real numbers —3 + Fundamentals of Freshman Mathematics 22 [Chap.

It is slightly more difhcult to Associative Law of Addition, Actually the sum, a -{- b -{- c, defined; for originall}' Therefore we make (a -\- -{- h) c = a -\- (b -{- c). numbers needs to be add two numbers, a -{- b. of three real we knew only how to the following definition: Definition a + 6 + c is defined to be the sum (a + 6) + c. We now prove a theorem which illustrates the fact that the sum of three real numbers is the same regardless of the order in which the : addition is Theorem performed.

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