EXERCISE I. 3

Find, by applying the axioms, the values which must be assigned to the letters in order that the following conditional equations may be true.

In each case the axiom applied should be stated, and the result obtained should be verified, by substituting for the letter in the given equation the value found.

CHAPTER II

AN EXTENSION OF THE IDEA OF NUMBER

1. IN arithmetic we have found it possible to subtract one number from another only when the number subtracted was not greater than the number from which it was taken.

Such combinations of numbers as 6-9, 10-11, etc., are from the point of view of arithmetic wholly destitute of meaning, since there exists no number, that is, no result of counting, which when added to 9 gives 6, or when added to 11 gives the sum 10.

Since such combinations of numbers occur frequently in mathematical work, it becomes necessary for us to give them a meaning if we are to allow them to remain in our calculations. To do this we find it necessary to extend our notion of number. The combination of numbers 6-9, as written, suggests to us a difference, and it will be convenient for us to reckon with it as with every other "real" or "actual" difference, such as 9 — 6, or 83, etc., that is, a difference in which the subtrahend is less than the minuend.

PRINCIPLE OF NO EXCEPTION

2. Mathematicians are accustomed to apply the names of familiar combinations of numbers and symbols which have recognized meanings to all similar combinations, even when these do not appear at first to admit of meaning, or even to make sense. This principle, that the old laws of reckoning and the old meanings must be carried over to include all special cases of a given general type, even those which may appear at first to be exceptions, will appear under many different forms throughout the whole science of mathematics, and will be referred to as the Principle of No Exception.

Instead of being an unwarranted stretching of language, as it may appear at first, the Principle of No Exception insists rather on a stretching or broadening of ideas to fit the language used, in order to avoid contradictions which might otherwise arise,

In Arithmetic the primary idea of a fraction is "broken number."

Thus,,,, are fractions in this sense.

In the course of arithmetic work, combinations appear such as †, f, 4, etc., which look like fractions, and behave like fractions, but which are not in the original sense "broken numbers." They are not properly fractions, and are accordingly called "improper fractions."

The Principle of No Exception is then applied, and such combinations as 1, 4, 1, 8, etc., are all, without exception, spoken of as fractions, without specifying whether they are proper or improper fractions, so-called.

3. It will now be shown that the application of this idea of No Exception leads us to an extension of our previous notions concerning number, and to the invention of a new kind of number, a kind of number which does not appear, as did the primary numbers, as a result of counting, but which nevertheless may be used in our calculations in such a way as always to give sense.

POSITIVE AND NEGATIVE QUANTITIES

4. Certain words, such as

forward-backward,

upward - downward,
north-south,

profit loss,

earning - spending,
increasing-diminishing,

suggest to us a condition of two things such that each tends to destroy the effect produced by the other. One tends to increase whatever the other tends to decrease. The terms are merely relative and imply that, from some point of view, one thing tends to oppose another.

E. g. If travelling east takes us away from some particular place, then from the same point of view, travelling west will take us toward that same place.

In trade, the effect produced by profits offsets the effect produced by losses.

5. Without multiplying illustrations we will remark simply that the terms positive and negative are used in mathematics in such a way as to imply that there is some opposition such that if, in a calculation, the things denoted as positive should be added, then

those called negative should be subtracted. This may be due either to the nature of the things considered, or to the point of view from which we regard them.

6. The opposition between two sets of things is often such that it is of no consequence which is considered as positive. The selection. being once made, so long as the things of one set in a calculation are considered as positive quantities, those of the other set must in opposition remain as negative quantities.

7. By the absolute value of a quantity expressed in terms of some unit of the same kind, is meant the number of times the unit is contained in the given quantity. This is without regard to the quality of either the quantity or the unit, that is, as to whether both are positive or both negative.

8. If two quantities are such that, when combined or considered as parts of one whole, any given amount of one destroys the effect produced by an amount of the other equal in absolute value to that of the first, these two quantities are called opposites.

In mathematical calculations one of two opposite quantities is called positive and the remaining one negative.

9. If, in any calculation, we choose to regard some quantity as being positive, then all other quantities which tend to increase it must be considered as positive also, and all those which tend to diminish it must be taken as negative.

It is merely a matter of choice which one of two opposite quantities is regarded as positive. On one occasion we may regard motion in one direction, say toward the right, as being positive, and on another we may equally as well choose to regard motion toward the left as being positive. In either case motion in a direction directly opposite to that chosen as positive would be considered as negative motion.

Also, if we choose to call the capital invested in a business positive, then all profits will be positive, since they may be added to and used to increase the capital; all losses and expenses will be negative, for they tend to diminish the capital, since they must be subtracted from it.

10. It is not essential to positive quantities that they be numerically greater than those which are negative. Thus, losses in business, regarded as negative quantities, might greatly exceed gains, which would then be positive quantities.

11. From the nature of things, we may treat positive and negative quantities according to the following Principles:

Principle I. If a positive and a negative quantity of the same kind are equal in absolute value, either will destroy the effect of the other when both are taken together or combined by addition.

E. g. Items of income and expense may be regarded as being opposite quantities, and we may call one positive and the other negative; for any item of expense reduces by just an equal amount the effective income.

Principle II. Positive quantities alone may be added in any order; also negative quantities alone may be added in any order.

E. g. Since negative quantities are those which are considered as tending to diminish the effect of certain others called positive quantities, the combined effect of several negative quantities will be a negative quantity which is equal to their sum.

There is no contradiction in speaking of adding negative quantities, for the idea suggested by the terms positive and negative is one of nature or quality, not number or amount.

If incomes be regarded as positive, expenses must be treated as negative quantities, and we may add all of our expenses and then subtract the sum total from our income to determine our financial condition.

A single negative quantity may "oppose" a positive quantity to produce a decreased "value" indicated by subtraction, while taken with another negative quantity there will be produced an "increased negative effect" which would have to be indicated by addition.

Thus, as before, the total expense results from adding several expenses. Principle III. The resultant effect of several combined positive and negative quantities is equal to the numerical difference between the total positive and total negative effects, and has the quality or nature of the greater total.

E. g. The result of combining expenses of $5 and $10 with items of income of $3, $2, $3, $6, $4, $2, and $1, may be obtained by finding the difference between the total expense. $15, and the total income, $21. This difference would be a balance of $6 in favor of the income. This balance may be taken as a positive quantity.

Principle IV. The removal or subtraction of a positive quantity has the same effect on an expression in which it occurs as the addition of a negative quantity equal in absolute value to the positive quantity.