Download The Number System by H. A. Thurston PDF
By H. A. Thurston
The instructing of arithmetic has gone through huge alterations in strategy, with a shift in emphasis from rote memorization to buying an realizing of the logical foundations and method of challenge fixing. This ebook deals counsel in that path, exploring arithmetic's underlying options and their logical development.
This volume's nice benefit lies in its wealth of explanatory fabric, designed to advertise a casual and intuitive realizing of the rigorous logical method of the quantity procedure. the 1st half explains and reviews on axioms and definitions, making their next remedy extra coherent. the second one half provides an in depth, systematic development of the quantity structures of rational, genuine, and intricate numbers. It covers entire numbers, hemigroups and teams, integers, ordered fields, the order relation for rationals, exponentiation, and genuine and intricate numbers. each step is justified through a connection with the best theorem or lemma. routines following every one bankruptcy partially II support readers try out their development and supply perform in utilizing the correct concepts.
Reprint of the 1967 Dover variation.
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Extra info for The Number System
Example text
Suppose that a·x = b and c·x = d have the same solution, x = k. Then a·k = b and c·k = d. Therefore (a·k)·d = b·(c·k). Therefore, if the commutative and associative laws are to hold, we must have (a·d)·k = (b·c)·k. Then the cancellation law tells us that a·d = b·c. Suppose, on the other hand, that a, b, c, and d are numbers such that a·d = b·c. Let k be a solution of a·x = b. Then a·k = b. Therefore (a·k)·c = b·c = a·d. Therefore by the associative law, a·(k·c) = a·d. Then, by the cancellation and commutative laws, c·k = d, and so k is a solution of c·x = d.
4. Now let us return to the definition of “fraction”. Assuming the existence of whole numbers, there is no doubt of the existence of symbols like m/n, each obtained by writing a whole number, a stroke, and then another whole number. They will not, however, do for fractions because the symbol 1/2, for instance, is different from the symbol 2/4, whereas we want these two fractions to be equal. Let us consider all those ordered pairs (m, n) of whole numbers which have the property that n = 2·m. We take this set of ordered pairs as one of our objects, and call it the fraction 1/2.
The condition for (m, n) to be in the fraction 1/2 is n = 2·m. The condition for (m, n) to be in the fraction 2/4 is 2·n = 4·m. It is one of the properties of the arithmetic of whole numbers that n = 2·m if and only if 2·n = 4·m. Therefore an ordered pair is in one fraction if and only if it is in the other, and so the two fractions are equal. In this way we can show, in general, that a/b and c/d are equal if and only if a·d = b·c. This is precisely what we wanted. 5. In everyday language we use fractions and whole numbers together: when we ask for a three-halfpenny stamp we are using the fraction 3/2; when we ask for a penny-halfpenny stamp we are using the mixed number .