By George Chrystal
This Elibron Classics ebook is a facsimile reprint of a 1904 variation by way of Adam and Charles Black, London.
Read or Download Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 PDF
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Additional resources for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1
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 .