Problems from Astronomy and Geometry

Problems from astronomy and geometry

The last problems use simultaneous equations and quadratic equations. In a geometric problem draw a figure and mark its dimensions in terms of the unknown; in a quadratic problem check which roots fit the conditions of the problem, and reject the others.

Note on negative numbers

One who has a limited acquaintance with algebra is inclined to wonder over the reluctance with which the early masters of mathematics accepted the concept of negative numbers. If one has not really mastered the concept, however, he will have considerable difficulty in giving consistent answers to the following questions:

1. Is -20 greater than +5?
2. May a distance east of a certain point be considered positive and a distance west of that point negative?
3. Would a lazy man prefer to walk -20 miles from such a point rather than +5 miles from it, if he were convinced that -20 is less than +5?

Here the difficulty is due largely to the use of the terms greater than and less than. That 7 is greater than 3 is obvious from common-sense reasoning like this: The number 7 is greater than 3, because from a group of 7 objects 3 objects may be taken away and some objects will remain. Now algebra parts with this clear arithmetical notion of greater and less when it extends subtraction so as to include the taking away (subtraction) of a greater number from a less. Algebra states a definite rule for subtraction, a rule which in its application contradicts no result obtained in arithmetic. The rule, however, does give new results. For instance, it determines the result when 7 is subtracted from 3, an operation which arithmetic does not attempt. Upon this rule of subtraction, which in a certain sense is a definition, algebra bases its definition of the expressions greater than and less than, which follows:

The number a >, =, or < the number b according as a - b is positive, zero, or negative.

Therefore -20 is less than +5 not in the concrete or arithmetical sense, but in the algebraic sense; that is, in the sense that it accords with the preceding definition.

A little closer inspection of the nature of a negative number may help to clear matters farther. The result of subtracting 7 from 3 may be viewed in two ways. The more abstract view is to regard -4 as a mark of order in the scale of numbers:

\ldots,\ -5,\ -4,\ -3,\ -2,\ -1,\ 0,\ 1,\ 2,\ 3,\ 4,\ \ldots

Then -20 would be less than 5 in the sense that -20 comes before 5 in the scale. From this point of view, then, negative numbers have ordinal but not cardinal values. This is only another way of saying that no group of objects exists whose number is negative.

A different view of a negative number, -4 for example, is to regard it as an operand (a symbol denoting both a number and an operation); that is, as a subtraction of 4 delayed only until there arises in the course of further operations a number from which to subtract it.

Those who desire a fuller discussion than the one which precedes are referred to Fine's "College Algebra," or to Chrystal's "Text-Book of Algebra."